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Ft(\))31 b(\(b)s(ounded)h(mean)113 4984 y(oscillation,)g(see)i (references)i([Ga,)c(GCRF])i(for)f(its)g(de\014nition)h(and)g(more)e (information\).)313 5106 y(By)e(F)-8 b(e\013erman's)30 b(dualit)m(y)h(theorem)f(BMO)i(is)f(the)g(dual)g(of)g(the)h(Hardy)e (space)i(H)3395 5063 y Fs(1)3471 5106 y Ft(th)m(us)113 5226 y(one)h(can)g(add)g(an)f Fp(L)875 5189 y Fm(1)992 5226 y Ft(function)h(to)f Fp(B)37 b Ft(so)32 b(that)g(the)h(harmonic)e (conjugate)j(of)e(the)h(sum)f(will)113 5345 y(also)37 b(b)s(e)g Fp(L)524 5309 y Fm(1)608 5345 y Ft(.)55 b(This)37 b(suggests)g(to)g(lo)s(ok)e(for)i(an)g(holomorphic)f(function)i Fo(B)i Ft(de\014ned)e(on)f(the)113 5465 y(upp)s(er)31 b(half)f(plane)g(whic)m(h)h(is)e Fq(Z)p Ft({p)s(erio)s(dic)e(and)k (whose)f(trace)g(on)g Fq(R)40 b Ft(has)30 b(for)g(imaginary)e(part)113 5584 y(the)f(Brjuno)h(function)f Fp(B)5 b Ft(.)42 b(The)27 b(function)h Fo(B)i Ft(will)c(b)s(e)h(called)g(the)g Fl(c)-5 b(omplex)30 b(Brjuno)g(function)p Ft(.)p eop %%Page: 3 3 3 2 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2320 b(3)313 383 y(Another)54 b(motiv)-6 b(ation)53 b(for)i(the)g(in)m(tro)s (duction)f(of)h(the)g(complex)f(Brjuno)h(function)113 503 y(comes)60 b(from)g(results)h(concerning)h(the)e(linearization)g (of)g(the)h(quadratic)f(p)s(olynomial)113 622 y Fp(P)177 637 y Fn(\025)230 622 y Ft(\()p Fp(z)t Ft(\))27 b(=)i Fp(\025)p Ft(\()p Fp(z)d Fo(\000)d Fp(z)810 586 y Fs(2)855 622 y Ft(\))33 b(\([Y)-8 b(o1],)31 b(Chapter)j(I)s(I\).)e(One)j(has)e (the)h(follo)m(wing)f(results)g(:)136 742 y(\(1\))49 b(there)36 b(exists)g(a)f(b)s(ounded)j(holomorphic)d(function)h Fp(U)60 b Ft(:)31 b Fq(D)57 b Fo(!)32 b Fq(C)61 b Ft(suc)m(h)38 b(that)d Fo(j)p Fp(U)11 b Ft(\()p Fp(\025)p Ft(\))p Fo(j)35 b Ft(is)313 861 y(equal)e(to)f(the)i(radius)g(of)f(con)m(v)m(ergence)i (of)f(the)f(normalized)g(linearization)f(of)h Fp(P)3375 876 y Fn(\025)3448 861 y Ft(;)136 981 y(\(2\))49 b(for)31 b(all)f Fp(\025)657 996 y Fs(0)729 981 y Fo(2)e Fq(S)884 945 y Fs(1)923 981 y Ft(,)j Fo(j)p Fp(U)11 b Ft(\()p Fp(\025)p Ft(\))p Fo(j)30 b Ft(has)h(a)f(non{tangen)m(tial)h(limit)e (in)h Fp(\025)2621 996 y Fs(0)2697 981 y Ft(\(whic)m(h)h(is)g(still)f (equal)g(to)313 1100 y(the)j(radius)h(of)f(con)m(v)m(ergence)i(of)f (the)f(normalized)g(linearization)f(of)i Fp(P)2991 1115 y Fn(\025)3038 1125 y Fi(0)3082 1100 y Ft(\))19 b(;)136 1220 y(\(3\))49 b(if)30 b Fp(\025)e Ft(=)g Fp(e)638 1184 y Fs(2)p Fn(\031)s(i\013)812 1220 y Ft(,)j Fp(\013)c Fo(2)i Fq(R)21 b Fo(n)16 b Fq(Q)h Ft(,)37 b Fp(P)1411 1235 y Fn(\025)1493 1220 y Ft(is)31 b(linearizable)f(if)g(and)h(only)f (if)g Fp(\013)g Ft(is)h(a)f(Brjuno)g(n)m(um)m(b)s(er.)313 1340 y(Moreo)m(v)m(er)43 b(there)h(exists)e(a)h(univ)m(ersal)g(constan) m(t)g Fp(C)2315 1355 y Fs(1)2404 1340 y Fp(>)h Ft(0)f(and)g(for)g(all)f Fp(")i(>)g Ft(0)f(there)313 1459 y(exists)33 b Fp(C)659 1474 y Fn(")729 1459 y Fp(>)28 b Ft(0)33 b(suc)m(h)i(that)e(for)g(all)g (Brjuno)h(n)m(um)m(b)s(ers)g Fp(\013)f Ft(one)h(has)963 1659 y(\(1)22 b Fo(\000)g Fp(")p Ft(\))p Fp(B)5 b Ft(\()p Fp(\013)p Ft(\))22 b Fo(\000)g Fp(C)1673 1674 y Fn(")1744 1659 y Fo(\024)28 b(\000)17 b Ft(log)g Fo(j)p Fp(U)11 b Ft(\()p Fp(\025)p Ft(\))p Fo(j)27 b(\024)h Fp(B)5 b Ft(\()p Fp(\013)p Ft(\))21 b(+)i Fp(C)2906 1674 y Fs(1)2979 1659 y Fp(:)113 1859 y Ft(In)47 b([MMY])e(the)i(authors)f(prop)s(osed)h (the)g(follo)m(wing)e(conjecture)j(\(see)f(also)e([Ma]\))h(:)69 b(the)113 1978 y(function)50 b(de\014ned)h(on)f(the)f(set)g(of)h (Brjuno)f(n)m(um)m(b)s(ers)h(b)m(y)g Fp(\013)k Fo(7!)f Fp(B)5 b Ft(\()p Fp(\013)p Ft(\))32 b(+)i(log)17 b Fo(j)p Fp(U)11 b Ft(\()p Fp(e)3417 1942 y Fs(2)p Fn(\031)s(i\013)3590 1978 y Ft(\))p Fo(j)113 2098 y Ft(extends)41 b(to)e(a)g(1)p Fp(=)p Ft(2{H\177)-50 b(older)39 b(con)m(tin)m(uous)i(function)g(as)e Fp(\013)h Ft(v)-6 b(aries)40 b(in)f Fq(R)5 b Ft(.)69 b(If)40 b(this)g(w)m(ere)g(true)113 2217 y(then)32 b(the)g(function)g Fo(\000)p Fp(i)p Fo(B)s Ft(\()p Fp(z)t Ft(\))20 b(+)e(log)f Fp(U)11 b Ft(\()p Fp(e)1627 2181 y Fs(2)p Fn(\031)s(iz)1790 2217 y Ft(\))31 b(w)m(ould)h(also)f(extend)g(to)g(a)g(H\177)-50 b(older)31 b(con)m(tin)m(uous)113 2337 y(function)k(on)p 642 2256 78 4 v 33 w Fq(H)26 b Ft(.)113 2559 y Fr(1.2)39 b(The)e(complex)h(Brjuno)h(function)313 2704 y Ft(A)32 b(natural)h(question)f(no)m(w)h(is)g(ho)m(w)g(to)f(extend)i(the)f(op)s (erator)f Fp(T)47 b Ft(to)32 b(complex)g(analytic)113 2823 y(functions.)94 b(This)49 b(is)g(ac)m(hiev)m(ed)i(as)e(follo)m(ws) g(:)76 b(the)50 b(op)s(erator)f Fp(T)63 b Ft(extends)50 b(to)f(the)h(space)113 2943 y Fp(A)188 2907 y Fm(0)216 2943 y Ft(\([0)p Fp(;)17 b Ft(1]\))28 b(of)i(h)m(yp)s(erfunctions)i Fp(u)e Ft(with)g(supp)s(ort)h(con)m(tained)g(in)f([0)p Fp(;)17 b Ft(1])29 b(\(see)h(section)h(1.4)e(for)i(a)113 3062 y(pro)s(of)g(of)f(this)g(fact)h(and)f(App)s(endix)h(2)f(for)g(a)g (v)m(ery)g(brief)h(in)m(tro)s(duction)f(to)f(h)m(yp)s(erfunctions\).) 113 3182 y(This)43 b(space)h(is)f(canonically)f(isomorphic)g(to)h(the)g (complex)f(v)m(ector)g(space)i Fo(O)3156 3146 y Fs(1)3201 3182 y Ft(\()p 3240 3101 72 4 v Fq(C)54 b Fo(n)29 b Ft([0)p Fp(;)17 b Ft(1]\))113 3302 y(of)44 b(holomorphic)e(functions)j(on)p 1406 3221 V 44 w Fq(C)55 b Fo(n)28 b Ft([0)p Fp(;)17 b Ft(1])42 b(whic)m(h)i(v)-6 b(anish)44 b(at)f(in\014nit)m(y)-8 b(.)74 b(The)44 b(connection)113 3421 y(b)s(et)m(w)m(een)f Fp(u)e Ft(and)g(the)g(asso)s(ciated)g(holomorphic)f(function)i Fp(')f Ft(is)g(commonly)d(written)j(as)g(:)113 3541 y Fp(u)p Ft(\()p Fp(x)p Ft(\))32 b(=)472 3501 y Fs(1)p 458 3518 68 4 v 458 3575 a(2)p Fn(i)538 3541 y Ft(\()p Fp(')p Ft(\()p Fp(x)24 b Ft(+)g Fp(i)p Ft(0\))g Fo(\000)g Fp(')p Ft(\()p Fp(x)g Fo(\000)g Fp(i)p Ft(0\)\),)36 b(whic)m(h)g(is)g (also)f(equal)h(to)f Fo(=)p Fp(m)17 b(')p Ft(\()p Fp(x)24 b Ft(+)g Fp(i)p Ft(0\))36 b(when)h Fp(')f Ft(is)113 3680 y(real)d(\(i.e.)44 b Fp(')p Ft(\()p 622 3625 51 4 v Fp(z)t Ft(\))28 b(=)p 844 3593 194 4 v 28 w Fp(')p Ft(\()p Fp(z)t Ft(\)\).)44 b(On)34 b Fo(O)1396 3643 y Fs(1)1441 3680 y Ft(\()p 1480 3599 72 4 v Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))31 b(the)i(form)m(ula)g(for)g Fp(T)48 b Ft(reads)607 3959 y(\()p Fp(T)14 b(')p Ft(\)\()p Fp(z)t Ft(\))27 b(=)i Fo(\000)p Fp(z)1273 3835 y Fm(1)1241 3865 y Fk(X)1227 4076 y Fn(m)p Fs(=1)1416 3819 y Fk(\024)1468 3959 y Fp(')1550 3819 y Fk(\022)1636 3892 y Ft(1)p 1635 3936 51 4 v 1635 4028 a Fp(z)1720 3959 y Fo(\000)23 b Fp(m)1907 3819 y Fk(\023)2003 3959 y Fo(\000)f Fp(')p Ft(\()p Fo(\000)p Fp(m)p Ft(\))2409 3819 y Fk(\025)2485 3959 y Ft(+)2630 3835 y Fm(1)2598 3865 y Fk(X)2584 4076 y Fn(m)p Fs(=1)2772 3959 y Fp(')2837 3918 y Fm(0)2865 3959 y Ft(\()p Fo(\000)p Fp(m)p Ft(\))29 b Fp(:)288 b Ft(\(1)p Fp(:)p Ft(3\))113 4245 y(F)-8 b(ormally)31 b(w)m(e)j(ha)m(v)m (e)848 4459 y(\(1)21 b Fo(\000)i Fp(T)14 b Ft(\))1169 4418 y Fm(\000)p Fs(1)1276 4459 y Fp(')p Ft(\()p Fp(z)t Ft(\))28 b(=)1602 4365 y Fk(X)1603 4579 y Fn(r)r Fm(\025)p Fs(0)1746 4459 y Ft(\()p Fp(T)1857 4418 y Fn(r)1901 4459 y Fp(')p Ft(\)\()p Fp(z)t Ft(\))g(=)2288 4365 y Fk(X)2266 4579 y Fn(g)s Fm(2M)2454 4459 y Ft(\()p Fp(L)2561 4474 y Fn(g)2607 4459 y Fp(')p Ft(\)\()p Fp(z)t Ft(\))g Fp(;)557 b Ft(\(1)p Fp(:)p Ft(4\))113 4762 y(where)34 b(the)g(monoid)113 4996 y Fo(M)28 b Ft(=)366 4856 y Fk(\032)441 4996 y Fp(g)i Ft(=)625 4856 y Fk(\022)714 4937 y Fp(a)104 b(b)719 5056 y(c)h(d)935 4856 y Fk(\023)1036 4996 y Fo(2)28 b Ft(GL)17 b(\(2)p Fp(;)g Fq(Z)p Ft(\))g Fp(;)40 b(d)27 b Fo(\025)i Fp(b)e Fo(\025)h Fp(a)g Fo(\025)g Ft(0)17 b Fp(;)43 b Ft(and)28 b Fp(d)g Fo(\025)g Fp(c)g Fo(\025)g Fp(a)2927 4856 y Fk(\033)3002 4996 y Fo([)3068 4856 y Fk(\032\022)3233 4937 y Ft(1)99 b(0)3233 5056 y(0)g(1)3449 4856 y Fk(\023\033)3630 4996 y Fp(;)113 5270 y Ft(acts)34 b(on)f Fo(O)539 5234 y Fs(1)584 5270 y Ft(\()p 623 5189 72 4 v Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))31 b(according)i(to)268 5529 y(\()p Fp(L)375 5544 y Fn(g)421 5529 y Fp(')p Ft(\)\()p Fp(z)t Ft(\))27 b(=)i(\()p Fp(a)21 b Fo(\000)i Fp(cz)t Ft(\))1148 5389 y Fk(\024)1201 5529 y Fp(')1283 5389 y Fk(\022)1368 5462 y Fp(dz)k Fo(\000)22 b Fp(b)p 1368 5507 269 4 v 1368 5598 a(a)f Fo(\000)i Fp(cz)1648 5389 y Fk(\023)1743 5529 y Fo(\000)g Fp(')1925 5389 y Fk(\022)1998 5529 y Fo(\000)2087 5462 y Fp(d)p 2087 5507 52 4 v 2091 5598 a(c)2151 5389 y Fk(\023)q(\025)2299 5529 y Fo(\000)g Ft(det\()p Fp(g)t Ft(\))p Fp(c)2710 5488 y Fm(\000)p Fs(1)2816 5529 y Fp(')2881 5488 y Fm(0)2926 5389 y Fk(\022)2999 5529 y Fo(\000)3088 5462 y Fp(d)p 3088 5507 V 3092 5598 a(c)3152 5389 y Fk(\023)3270 5529 y Fp(:)154 b Ft(\(1)p Fp(:)p Ft(5\))p eop %%Page: 4 4 4 3 bop 113 144 a Ft(4)2455 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y(The)26 b(series)h(\(1.4\))d(actually)h(con)m(v)m (erges)h(in)g Fo(O)1801 347 y Fs(1)1846 383 y Ft(\()p 1885 303 72 4 v Fq(C)32 b Fo(n)7 b Ft([0)p Fp(;)17 b Ft(1]\))23 b(to)i(a)h(function)2855 308 y Fk(P)2960 413 y Fm(M)3075 383 y Fp(')p Ft(.)42 b(T)-8 b(o)25 b(reco)m(v)m(er)113 503 y(a)33 b(holomorphic)g(p)s(erio)s(dic)g(function)h(on)g Fq(H)59 b Ft(one)34 b(sums)f(o)m(v)m(er)g(in)m(teger)h(translates)f(:) 1228 845 y Fo(B)1293 860 y Fn(')1350 845 y Ft(\()p Fp(z)t Ft(\))28 b(=)1614 751 y Fk(X)1611 965 y Fn(n)p Fm(2)p Fg(Z)1777 675 y Fk( )1856 751 y(X)1881 965 y Fm(M)2017 845 y Fp(')2082 675 y Fk(!)2177 845 y Ft(\()p Fp(z)f Fo(\000)c Fp(n)p Ft(\))k Fp(:)909 b Ft(\(1)p Fp(:)p Ft(6\))113 1214 y(T)-8 b(o)42 b(construct)h(the)f(complex)f(Brjuno)i(function)g (one)g(has)f(to)g(tak)m(e)g Fp(')2836 1229 y Fs(0)2880 1214 y Ft(\()p Fp(z)t Ft(\))h(=)g Fo(\000)3265 1175 y Fs(1)p 3261 1191 50 4 v 3261 1248 a Fn(\031)3322 1214 y Ft(Li)3412 1229 y Fs(2)3473 1133 y Fk(\000)3531 1175 y Fs(1)p 3530 1191 41 4 v 3530 1248 a Fn(z)3583 1133 y Fk(\001)3629 1214 y Ft(,)113 1333 y(where)34 b(Li)491 1348 y Fs(2)569 1333 y Ft(is)f(the)h(dilogarithm)d(\(App)s(endix)j(3,)e ([O]\).)h(Then)h(the)f(ab)s(o)m(v)m(e)h(form)m(ulas)f(giv)m(e)290 1652 y Fo(B)s Ft(\()p Fp(z)t Ft(\))28 b(=)g Fo(\000)714 1585 y Ft(1)p 708 1629 61 4 v 708 1720 a Fp(\031)840 1557 y Fk(X)798 1776 y Fn(p=q)s Fm(2)p Fg(Q)1043 1511 y Fk(\032)1118 1652 y Ft(\()p Fp(p)1207 1611 y Fm(0)1257 1652 y Fo(\000)22 b Fp(q)1404 1611 y Fm(0)1432 1652 y Fp(z)t Ft(\))1538 1511 y Fk(\024)1591 1652 y Ft(Li)1681 1667 y Fs(2)1742 1511 y Fk(\022)1828 1585 y Fp(p)1878 1548 y Fm(0)1928 1585 y Fo(\000)g Fp(q)2075 1548 y Fm(0)2103 1585 y Fp(z)p 1828 1629 327 4 v 1856 1720 a(q)t(z)k Fo(\000)d Fp(p)2166 1511 y Fk(\023)2262 1652 y Fo(\000)f Ft(Li)2451 1667 y Fs(2)2512 1511 y Fk(\022)2586 1652 y Fo(\000)2675 1585 y Fp(q)2723 1548 y Fm(0)p 2675 1629 76 4 v 2689 1720 a Fp(q)2763 1511 y Fk(\023)q(\025)515 1987 y Ft(+\()p Fp(p)681 1946 y Fm(00)755 1987 y Fo(\000)g Fp(q)902 1946 y Fm(00)953 1987 y Fp(z)t Ft(\))1059 1847 y Fk(\024)1112 1987 y Ft(Li)1202 2002 y Fs(2)1263 1847 y Fk(\022)1348 1920 y Fp(p)1398 1884 y Fm(00)1471 1920 y Fo(\000)h Fp(q)1619 1884 y Fm(00)1670 1920 y Fp(z)p 1348 1964 373 4 v 1399 2056 a(q)t(z)k Fo(\000)c Fp(p)1733 1847 y Fk(\023)1828 1987 y Fo(\000)g Ft(Li)2018 2002 y Fs(2)2079 1847 y Fk(\022)2152 1987 y Fo(\000)2241 1920 y Fp(q)2289 1884 y Fm(00)p 2242 1964 99 4 v 2267 2056 a Fp(q)2353 1847 y Fk(\023\025)2501 1987 y Ft(+)2612 1920 y(1)p 2612 1964 50 4 v 2613 2056 a Fp(q)2691 1987 y Ft(log)2848 1920 y Fp(q)j Ft(+)d Fp(q)3066 1884 y Fm(00)p 2848 1964 269 4 v 2859 2056 a Fp(q)k Ft(+)22 b Fp(q)3077 2027 y Fm(0)3129 1847 y Fk(\033)3248 1987 y Fp(;)3452 1820 y Ft(\(1)p Fp(:)p Ft(7\))113 2346 y(where)412 2235 y Fk(h)471 2301 y Fn(p)512 2271 y Ff(0)p 471 2323 69 4 v 472 2380 a Fn(q)512 2360 y Ff(0)551 2346 y Fp(;)607 2301 y Fn(p)648 2271 y Ff(00)p 607 2323 90 4 v 608 2380 a Fn(q)648 2360 y Ff(00)709 2235 y Fk(i)800 2346 y Ft(is)44 b(the)g(F)-8 b(arey)44 b(in)m(terv)-6 b(al)44 b(suc)m(h)h(that)2225 2301 y Fn(p)p 2225 2323 42 4 v 2226 2380 a(q)2323 2346 y Ft(=)2458 2301 y Fn(p)2499 2271 y Ff(0)2526 2301 y Fs(+)p Fn(p)2628 2271 y Ff(00)p 2458 2323 220 4 v 2460 2380 a Fn(q)2500 2360 y Ff(0)2526 2380 y Fs(+)p Fn(q)2627 2360 y Ff(00)2733 2346 y Ft(\(with)e(the)i(con)m(v)m(en)m(tion)113 2495 y Fp(p)163 2459 y Fm(0)219 2495 y Ft(=)28 b Fp(p)23 b Fo(\000)f Ft(1,)33 b Fp(q)655 2459 y Fm(0)711 2495 y Ft(=)28 b(1,)k Fp(p)976 2459 y Fm(00)1055 2495 y Ft(=)c(1,)33 b Fp(q)1319 2459 y Fm(00)1398 2495 y Ft(=)28 b(0)33 b(if)g Fp(q)f Ft(=)c(1\).)113 2767 y Fr(1.3)39 b(Main)e(results)g(:)51 b(Prop)s(erties)38 b(of)g(the)f(complex)h(Brjuno)h(function)313 2936 y Ft(Our)g(main)e(result)h(\(Corollary)f(5.8\))g(is)h(that)g Fl(the)i(r)-5 b(e)g(al)41 b(p)-5 b(art)40 b(of)g Fo(B)j Fl(is)d(b)-5 b(ounde)g(d)38 b Ft(on)g(the)113 3056 y(upp)s(er)44 b(half)g(plane)f(\(note)g(that)f(this)h(statemen)m(t)g(is)f(stronger)i (than)f(the)g(result)g(obtained)113 3175 y(b)m(y)f(the)f(ab)s(o)m(v)m (e)h(men)m(tioned)f(general)h(prop)s(erties)g(of)f(the)h(harmonic)e (conjugates)j(of)e(BMO)113 3295 y(functions\))21 b(;)35 b(moreo)m(v)m(er)e Fl(the)k(tr)-5 b(ac)g(e)36 b(of)g Fo(<)p Fp(e)16 b Fo(B)40 b Fl(on)d Fq(R)47 b Fl(is)36 b(c)-5 b(ontinuous)37 b(at)f(al)5 b(l)38 b(irr)-5 b(ational)36 b(p)-5 b(oints)113 3414 y(and)36 b(has)f(a)h(jump)g(of)f Fp(\031)t(=q)k Fl(at)c(e)-5 b(ach)35 b(r)-5 b(ational)36 b(p)-5 b(oint)35 b Fp(p=q)d Fo(2)c Fq(Q)50 b Ft(\(see)34 b(Section)g(5.2.9\).)313 3540 y(A)42 b(n)m(umerical)g(study)h(of)g(the) g(function)h(arg)18 b Fp(U)11 b Ft(\()p Fp(e)2205 3504 y Fs(2)p Fn(\031)s(i\013)2379 3540 y Ft(\))42 b(seems)h(to)f(indicate)h (a)g(similar)113 3660 y(b)s(eha)m(viour)34 b(\(see)g(Figure)f(1\).)313 3786 y(Concerning)g(the)h(b)s(oundary)f(b)s(eha)m(viour)g(of)h(the)f (imaginary)e(part)i(of)g Fo(B)j Ft(w)m(e)e(pro)m(v)m(e)f(the)113 3906 y(follo)m(wing)g(\(Theorem)g(5.19\))f(:)158 4032 y(\(i\))49 b(if)38 b Fp(\013)h Ft(is)f(a)g(Brjuno)h(n)m(um)m(b)s(er)h (then)f Fo(=)p Fp(m)17 b Fo(B)s Ft(\()p Fp(\013)26 b Ft(+)g Fp(w)s Ft(\))38 b(con)m(v)m(erges)i(to)e Fp(B)5 b Ft(\()p Fp(\013)p Ft(\))37 b(as)i Fp(w)g Fo(!)d Ft(0)j(in)313 4151 y(an)m(y)33 b(domain)f(with)h(a)g(\014nite)h(order)f(of)h (tangency)f(to)g(the)h(real)f(axis)19 b(;)130 4277 y(\(ii\))49 b(if)30 b Fp(\013)h Ft(is)f(diophan)m(tine)i(one)f(can)g(allo)m(w)f (domains)f(with)i(in\014nite)g(order)g(of)f(tangency)h(\(see)313 4397 y(\(5.58\)\))313 4523 y(The)i(precise)i(b)s(eha)m(viour)e(of)h(at) e(rational)h(p)s(oin)m(ts)g(is)g(describ)s(ed)i(b)m(y)e(Theorem)g (5.10.)p eop %%Page: 5 5 5 4 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2320 b(5)113 383 y Fr(1.4)39 b(Hyp)s(erfunctions)e(and)g(op)s(erator)g Fp(T)313 526 y Ft(Let)c Fp(u;)17 b( )31 b Fo(2)d Fp(L)851 490 y Fs(2)895 526 y Ft(\([0)p Fp(;)17 b Ft(1]\),)31 b Fp(m)d Fo(2)g Fq(N)9 b Ft(,)39 b Fp(m)28 b Fo(\025)g Ft(1.)44 b(W)-8 b(e)34 b(consider)882 778 y Fp(T)940 793 y Fn(m)1016 778 y Fp(u)p Ft(\()p Fp(x)p Ft(\))27 b(=)1341 637 y Fk(\032)1432 738 y Fp(xu)p Ft(\(1)p Fp(=x)21 b Fo(\000)i Fp(m)p Ft(\))127 b(if)34 b Fp(x)27 b Fo(2)2386 628 y Fk(h)2511 699 y Fs(1)p 2445 715 172 4 v 2445 773 a Fn(m)p Fs(+1)2629 738 y Fp(;)2701 699 y Fs(1)p 2685 715 71 4 v 2685 773 a Fn(m)2768 628 y Fk(i)2848 738 y Ft(,)1699 882 y(0)528 b(otherwise.)3452 778 y(\(1)p Fp(:)p Ft(8\))113 1029 y(Note)37 b(that)f Fp(T)635 1044 y Fn(m)711 1029 y Fp(u)e Ft(=)g(\()p Fp(T)14 b(u)p Ft(\))33 b Fo(j)1181 1070 y Ft([)1276 1035 y Fi(1)p 1220 1046 147 4 v 1220 1086 a Fj(m)p Fi(+1)1378 1062 y Fn(;)1427 1036 y Fi(1)p 1414 1047 62 4 v 1414 1086 a Fj(m)1487 1069 y Ft(])1520 1029 y(,)k(th)m(us)h Fp(T)47 b Ft(=)2024 955 y Fk(P)2129 1059 y Fn(m)p Fm(\025)p Fs(1)2324 1029 y Fp(T)2382 1044 y Fn(m)2458 1029 y Ft(.)54 b(W)-8 b(e)37 b(de\014ne)i(the)e(adjoin)m(t) g Fp(T)3595 993 y Fm(\003)3581 1054 y Fn(m)113 1174 y Ft(b)m(y)993 1224 y Fk(Z)1093 1248 y Fs(1)1048 1450 y(0)1154 1359 y Fp(T)1212 1374 y Fn(m)1288 1359 y Fp(u)p Ft(\()p Fp(x)p Ft(\))p Fp( )t Ft(\()p Fp(x)p Ft(\))p Fp(dx)26 b Ft(=)1924 1224 y Fk(Z)2023 1248 y Fs(1)1979 1450 y(0)2085 1359 y Fp(u)p Ft(\()p Fp(x)p Ft(\))p Fp(T)2349 1318 y Fm(\003)2335 1384 y Fn(m)2410 1359 y Fp( )t Ft(\()p Fp(x)p Ft(\))p Fp(dx)g(;)113 1573 y Ft(whic)m(h)34 b(giv)m(es)1147 1747 y Fp(T)1219 1706 y Fm(\003)1205 1772 y Fn(m)1281 1747 y Fp( )t Ft(\()p Fp(x)p Ft(\))27 b(=)1798 1680 y(1)p 1629 1724 389 4 v 1629 1816 a(\()p Fp(m)22 b Ft(+)h Fp(x)p Ft(\))1973 1787 y Fs(3)2029 1747 y Fp( )2114 1607 y Fk(\022)2308 1680 y Ft(1)p 2199 1724 267 4 v 2199 1816 a Fp(m)g Ft(+)g Fp(x)2478 1607 y Fk(\023)2595 1747 y Fp(:)113 1971 y Ft(The)33 b(previous)g(form)m(ula)f(with)g Fp( )k Ft(analytic)31 b(in)h(a)g(neigh)m(b)s(orho)s(o)s(d)h(of)g([0)p Fp(;)17 b Ft(1])30 b(allo)m(ws)i(to)g(extend)113 2091 y(the)27 b(domain)e(of)h(de\014nition)h(of)f Fp(T)1331 2106 y Fn(m)1433 2091 y Ft(to)f(the)i(h)m(yp)s(erfunctions)h Fp(u)f Fo(2)h Fp(A)2634 2054 y Fm(0)2662 2091 y Ft(\([0)p Fp(;)17 b Ft(1]\))24 b(\(see)i(App)s(endix)h(5)113 2240 y(for)h(a)e(v)m(ery)h(short)g(summary)e(of)i(h)m(yp)s(erfunctions\))i (and)e(to)g(obtain)g Fp(T)2708 2255 y Fn(m)2783 2240 y Fp(u)h Fo(2)g Fp(A)3037 2204 y Fm(0)3082 2129 y Fk(\020h)3266 2201 y Fs(1)p 3200 2217 172 4 v 3200 2274 a Fn(m)p Fs(+1)3384 2240 y Fp(;)3455 2201 y Fs(1)p 3440 2217 71 4 v 3440 2274 a Fn(m)3522 2129 y Fk(i)q(\021)3629 2240 y Ft(.)113 2429 y(More)41 b(generally)-8 b(,)41 b(if)f Fp(u)f Fo(2)h Fp(A)1205 2393 y Fm(0)1232 2429 y Ft(\([)p Fp(\015)1351 2444 y Fs(0)1395 2429 y Fp(;)17 b(\015)1492 2444 y Fs(1)1536 2429 y Ft(]\),)41 b Fp(\015)1724 2444 y Fs(0)1807 2429 y Fp(>)f Fo(\000)p Ft(1,)i(then)f Fp(T)2413 2444 y Fn(m)2488 2429 y Fp(u)f Fo(2)f Fp(A)2765 2393 y Fm(0)2810 2319 y Fk(\020h)3015 2390 y Fs(1)p 2928 2406 213 4 v 2928 2464 a Fn(m)p Fs(+)p Fn(\015)3102 2474 y Fi(1)3153 2429 y Fp(;)3295 2390 y Fs(1)p 3209 2406 V 3209 2464 a Fn(m)p Fs(+)p Fn(\015)3383 2474 y Fi(0)3433 2319 y Fk(i\021)3579 2429 y Fo(\032)113 2619 y Fp(A)188 2582 y Fm(0)233 2508 y Fk(\020h)339 2619 y Ft(0)p Fp(;)516 2579 y Fs(1)p 445 2596 182 4 v 445 2653 a(1+)p Fn(\015)588 2663 y Fi(0)639 2508 y Fk(i\021)745 2619 y Ft(.)44 b(One)35 b(has)929 2776 y Fk(Z)1029 2801 y Fs(1)985 3003 y(0)1090 2912 y Fp(T)14 b(u)p Ft(\()p Fp(x)p Ft(\))p Fp( )t Ft(\()p Fp(x)p Ft(\))p Fp(dx)26 b Ft(=)1813 2817 y Fk(X)1798 3031 y Fn(m)p Fm(\025)p Fs(1)1988 2776 y Fk(Z)2087 2801 y Fs(1)2043 3003 y(0)2148 2912 y Fp(u)p Ft(\()p Fp(x)p Ft(\))p Fp(T)2412 2871 y Fm(\003)2398 2936 y Fn(m)2474 2912 y Fp( )t Ft(\()p Fp(x)p Ft(\))p Fp(dx)g(:)611 b Ft(\(1)p Fp(:)p Ft(9\))113 3201 y(If)38 b Fp( )j Ft(is)c(holomorphic)g(in)g(a)h(neigh)m(b)s(orho)s (o)s(d)g Fp(V)59 b Ft(of)38 b([0)p Fp(;)17 b Ft(1],)36 b(then)i(also)f Fp(T)2847 3165 y Fm(\003)2833 3226 y Fn(m)2909 3201 y Fp( )k Ft(is)d(holomorphic)113 3321 y(in)c Fp(V)55 b Ft(and)34 b(one)f(has)1291 3462 y(sup)1334 3546 y Fn(V)1458 3462 y Fo(j)p Fp(T)1558 3421 y Fm(\003)1544 3487 y Fn(m)1619 3462 y Fp( )t Fo(j)27 b(\024)1860 3395 y Ft(1)p 1860 3439 50 4 v 1860 3531 a(2)1922 3462 y Fp(m)2009 3421 y Fm(\000)p Fs(3)2133 3462 y Ft(sup)2176 3546 y Fn(V)2300 3462 y Fo(j)p Fp( )t Fo(j)f Fp(;)923 b Ft(\(1)p Fp(:)p Ft(10\))113 3685 y(\(this)37 b(follo)m(ws)g(immediately)d(from)j (the)g(estimates)f(of)i(section)f(3.2)g(c)m(ho)s(osing)g Fp(V)59 b Ft(to)37 b(b)s(e)g(the)113 3805 y(complemen)m(t)46 b(of)i(a)f(neigh)m(b)s(orho)s(o)s(d)g(of)g Fp(D)1756 3820 y Fm(1)1888 3805 y Ft(with)g(resp)s(ect)h(to)f(the)g(P)m(oincar)m (\023)-47 b(e)47 b(metric)g(on)p 113 3844 72 4 v 113 3924 a Fq(C)29 b Fo(n)s Ft([0)p Fp(;)17 b Ft(1]\).)38 b(Therefore)25 b(the)f(series)1408 3849 y Fk(P)1513 3954 y Fn(m)p Fm(\025)p Fs(1)1708 3924 y Fp(T)1766 3939 y Fn(m)1841 3924 y Fp(u)g Ft(con)m(v)m(erges)h(in)e Fp(A)2540 3888 y Fm(0)2568 3924 y Ft(\([0)p Fp(;)17 b Ft(1]\))k(to)i(a)g(h)m(yp)s (erfunction)113 4044 y(whic)m(h)34 b(will)f(b)s(e)g(denoted)i Fp(T)14 b(u)p Ft(.)313 4163 y(Let)37 b Fp(u)c Fo(2)h Fp(A)760 4127 y Fm(0)787 4163 y Ft(\([)p Fp(\015)906 4178 y Fs(0)950 4163 y Fp(;)17 b(\015)1047 4178 y Fs(1)1091 4163 y Ft(]\),)36 b Fp(\015)1274 4178 y Fs(0)1352 4163 y Fp(>)e Fo(\000)p Ft(1,)j(and)g(let)g Fp(')c Fo(2)h(O)2280 4127 y Fs(1)2325 4163 y Ft(\()p 2364 4083 V Fq(C)50 b Fo(n)24 b Ft([)p Fp(\015)2614 4178 y Fs(0)2658 4163 y Fp(;)17 b(\015)2755 4178 y Fs(1)2799 4163 y Ft(]\))36 b(b)s(e)h(the)g(asso)s(ciated)113 4283 y(holomorphic)24 b(function,)j(i.e.)40 b Fp(')25 b Ft(is)f(holomorphic)f(outside)i([)p Fp(\015)2411 4298 y Fs(0)2455 4283 y Fp(;)17 b(\015)2552 4298 y Fs(1)2595 4283 y Ft(])24 b(and)h(v)-6 b(anishes)26 b(at)d(in\014nit)m(y)-8 b(.)113 4402 y(F)g(or)44 b(all)e Fp(m)k Fo(\025)f Ft(1)f(the)g(holomorphic)e(function)j(asso)s(ciated)f (to)f Fp(T)2638 4417 y Fn(m)2714 4402 y Fp(u)h Ft(is)f Fp(L)2993 4420 y Fn(g)s Fs(\()p Fn(m)p Fs(\))3172 4402 y Fp(')h Ft(giv)m(en)g(b)m(y)113 4522 y(equation)33 b(\(1.5\))f(with)h Fp(a)27 b Ft(=)i(0,)j Fp(b)c Ft(=)g Fp(c)g Ft(=)g(1,)33 b Fp(d)27 b Ft(=)h Fp(m)p Ft(.)45 b(Indeed)34 b(if)g Fp(z)43 b(=)-61 b Fo(2)28 b Ft([)p Fp(\015)2755 4537 y Fs(0)2799 4522 y Fp(;)17 b(\015)2896 4537 y Fs(1)2940 4522 y Ft(])1409 4704 y(\()p Fp(T)1506 4719 y Fn(m)1581 4704 y Fp(u)p Ft(\)\()p Fp(c)1759 4719 y Fn(z)1805 4704 y Ft(\))27 b(=)h Fp(u)p Ft(\()p Fp(T)2144 4663 y Fm(\003)2130 4729 y Fn(m)2206 4704 y Fp(c)2249 4719 y Fn(z)2295 4704 y Ft(\))p Fp(;)1040 b Ft(\(1)p Fp(:)p Ft(11\))113 4905 y(where)34 b Fp(c)444 4920 y Fn(z)490 4905 y Ft(\()p Fp(x)p Ft(\))28 b(=)775 4866 y(1)p 770 4882 61 4 v 770 4939 a Fp(\031)944 4866 y Ft(1)p 854 4882 230 4 v 854 4944 a Fp(x)22 b Fo(\000)h Fp(z)1095 4905 y Ft(,)33 b(and)320 5170 y Fp(\031)t(T)453 5128 y Fm(\003)439 5194 y Fn(m)515 5170 y Fp(c)558 5185 y Fn(z)604 5170 y Ft(\()p Fp(x)p Ft(\))27 b(=)1052 5102 y(1)p 883 5147 389 4 v 883 5238 a(\()p Fp(m)22 b Ft(+)h Fp(x)p Ft(\))1227 5209 y Fs(3)1457 5102 y Ft(1)p 1295 5147 374 4 v 1376 5203 a Fs(1)p 1307 5220 178 4 v 1307 5277 a Fn(m)p Fs(+)p Fn(x)1518 5243 y Fo(\000)g Fp(z)1708 5170 y Ft(=)1994 5102 y(1)p 1825 5147 389 4 v 1825 5238 a(\()p Fp(m)g Ft(+)f Fp(x)p Ft(\))2169 5209 y Fs(2)2248 5170 y Fo(\000)g Fp(z)2415 5029 y Fk(\022)2701 5102 y Ft(1)p 2500 5147 453 4 v 2500 5243 a Fp(m)g Ft(+)h Fp(x)f Fo(\000)2900 5203 y Fs(1)p 2900 5220 41 4 v 2900 5277 a Fn(z)2987 5170 y Fo(\000)3207 5102 y Ft(1)p 3098 5147 267 4 v 3098 5238 a Fp(m)h Ft(+)g Fp(x)3376 5029 y Fk(\023)766 5500 y Ft(=)28 b Fp(\031)948 5329 y Fk(")1055 5432 y Fp(@)p 1030 5477 110 4 v 1030 5568 a(@)6 b(z)1151 5500 y(c)1194 5515 y Fn(z)1240 5500 y Ft(\()p Fp(x)p Ft(\))1375 5355 y Fk(\014)1375 5415 y(\014)1375 5475 y(\014)1375 5534 y(\014)1408 5599 y Fn(z)s Fs(=)p Fm(\000)p Fn(m)1670 5500 y Fo(\000)22 b Fp(z)1837 5389 y Fk(\020)1896 5500 y Fp(c)1952 5498 y Fi(1)p 1951 5509 37 4 v 1951 5548 a Fj(z)1999 5524 y Fm(\000)p Fn(m)2137 5500 y Ft(\()p Fp(x)p Ft(\))g Fo(\000)g Fp(c)2436 5515 y Fm(\000)p Fn(m)2575 5500 y Ft(\()p Fp(x)p Ft(\))2710 5389 y Fk(\021)2768 5329 y(#)2871 5500 y Fp(:)p eop %%Page: 6 6 6 5 bop 113 144 a Ft(6)2455 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y(Th)m(us)49 b(w)m(e)g(are)f(led)g(to)f(de\014ne)j Fp(T)14 b(')p Ft(,)51 b(according)d(to)g(\(1.3\),)i(as)e(an)g(elemen)m (t)f(of)h(the)h(space)113 503 y Fo(O)195 466 y Fs(1)240 503 y Ft(\()p 279 422 72 4 v Fq(C)f Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1)p Fp(=)p Ft(\(1)j(+)j Fp(\015)929 518 y Fs(0)973 503 y Ft(\)]\).)313 623 y(T)-8 b(o)36 b(construct)h(the)g (complex)f(analytic)g(extension)h(of)f(the)i(functions)f Fp(B)3100 638 y Fn(f)3188 623 y Ft(\(de\014ned)i(in)113 743 y(Section)34 b(1.1\))e(our)i(strategy)e(is)h(the)h(follo)m(wing)f (:)174 863 y(1\))50 b(tak)m(e)32 b(the)i(restriction)f(of)h(the)f(p)s (erio)s(dic)g(function)i Fp(f)43 b Ft(to)33 b(the)h(in)m(terv)-6 b(al)33 b([0)p Fp(;)17 b Ft(1])h(;)174 983 y(2\))50 b(consider)41 b(its)e(asso)s(ciated)h(h)m(yp)s(erfunction)i Fp(u)2035 998 y Fn(f)2126 983 y Ft(and)f(its)e(holomorphic)h(represen)m(tativ)m (e)313 1103 y Fp(')27 b Fo(2)i(O)582 1067 y Fs(1)626 1103 y Ft(\()p 665 1022 V Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\).)313 1223 y(Then)51 b(the)f(series)h(\(1.6\))e(con)m (v)m(erges)i(\(thanks)f(to)g(Corollary)e(3.6\))i(to)f(the)i(complex)113 1343 y(extension)40 b Fo(B)625 1358 y Fn(f)716 1343 y Ft(of)f(the)h(function)g Fp(B)1485 1358 y Fn(f)1536 1343 y Ft(.)61 b(The)40 b(main)e(di\016cult)m(y)h(\(unless)h Fp(f)49 b Ft(b)s(elongs)40 b(to)e(some)113 1463 y Fp(L)181 1426 y Fn(p)264 1463 y Ft(space,)h(see)f(Section)f(4.3\))f(w)m(ould)i (b)s(e)f(to)f(reco)m(v)m(er)i Fp(B)2263 1478 y Fn(f)2351 1463 y Ft(as)g(non{tangen)m(tial)e(limit)f(of)i(the)113 1582 y(imaginary)31 b(part)i(of)h Fo(B)976 1597 y Fn(f)1061 1582 y Ft(as)g Fo(=)p Fp(m)17 b(z)32 b Fo(!)27 b Ft(0.)113 1826 y Fr(1.5)39 b(Summary)g(of)f(the)f(con)m(ten)m(ts)313 1978 y Ft(Let)c(us)h(no)m(w)g(brie\015y)f(describ)s(e)i(the)e(con)m (ten)m(ts)i(of)e(this)g(article.)313 2098 y(In)49 b(Section)g(2)f(w)m (e)h(discuss)h(the)f(relation)f(b)s(et)m(w)m(een)i(the)f(monoid)f Fo(M)g Ft(and)i(the)f(full)113 2218 y(mo)s(dular)32 b(group)i(GL)17 b(\(2)p Fp(;)g Fq(Z)p Ft(\).)40 b(W)-8 b(e)34 b(then)f(describ)s(e)i(v) -6 b(arious)33 b(automorphic)g(actions)g(of)g Fo(M)p Ft(.)313 2338 y(In)28 b(Section)h(3,)g(the)g(in)m(tro)s(duction)f(of)h (a)f(complex)f(analogue)h(of)h(the)g(con)m(tin)m(ued)h(fraction)113 2458 y(expansion)36 b(of)g(a)g(real)f(n)m(um)m(b)s(er)h(allo)m(ws)f(us) h(to)f(pro)m(v)m(e)i(the)e(con)m(v)m(ergence)j(of)e(the)g(series)g (\(1.4\))113 2577 y(and)h(\(1.6\))e(\(Corollary)f(3.6\).)52 b(The)37 b(main)e(feature)i(of)f(the)h(complex)e(con)m(tin)m(ued)i (fraction)g(is)113 2697 y(that)30 b(it)g(reduces)j(to)d(the)h(real)f (con)m(tin)m(ued)i(fraction)e(when)i(the)f(n)m(um)m(b)s(er)g(is)f(real) g(and)h(it)f(stops)113 2816 y(after)i(a)f(\014nite)g(n)m(um)m(b)s(er)g (of)h(iterations)e(when)i(the)f(n)m(um)m(b)s(er)h(is)f(rational)f(or)h (complex.)42 b(In)31 b(the)113 2936 y(latter)k(case)h(the)f(absolute)h (v)-6 b(alue)35 b(of)g(the)h(imaginary)d(part)i(of)g(the)h(iterates)f (gro)m(ws)g(at)g(least)113 3055 y(exp)s(onen)m(tially)c(with)h(the)h(n) m(um)m(b)s(er)f(of)g(iterations)g(and)g(when)h(it)e(reac)m(hes)j(1)p Fp(=)p Ft(2)d(the)i(iteration)113 3175 y(stops.)313 3295 y(In)e(Section)h(4)f(w)m(e)h(use)g(the)f(complex)g(con)m(tin)m(ued)i (fraction)e(to)g(study)g(the)h(b)s(eha)m(viour)f(of)113 3415 y(the)36 b(series)f(\(1.4\))f(when)i Fp(z)j Ft(is)c(close)g(to)g ([0)p Fp(;)17 b Ft(1].)47 b(This)35 b(is)g(in)m(teresting)g(in)g (itself)g(and)g(it)f(will)g(b)s(e)113 3534 y(v)m(ery)28 b(imp)s(ortan)m(t)e(when)i(applied)g(to)f(the)h(complex)e(Brjuno)i (function)h(in)e(order)h(to)g(pro)m(v)m(e)g(our)113 3654 y(main)k(results.)45 b(Our)33 b(study)h(allo)m(ws)e(us)i(to)e(pro)m(v)m (e)i(that)e(the)h(restriction)g(of)g Fp(T)47 b Ft(to)33 b(the)g(Hardy)113 3774 y(spaces)27 b Fp(H)502 3737 y Fn(p)548 3774 y Ft(\()p 587 3693 V Fq(C)k Fo(n)6 b Ft([0)p Fp(;)17 b Ft(1]\))6 b Fo(\\)g(O)1120 3737 y Fs(1)1162 3774 y Ft(\()p 1201 3693 V Fq(C)31 b Fo(n)6 b Ft([0)p Fp(;)17 b Ft(1]\),)24 b(1)j Fo(\024)i Fp(p)e Fo(\024)i Ft(+)p Fo(1)p Ft(,)d(is)f(con)m(tin)m(uous)h(with)f(sp)s(ectral)g (radius)113 3913 y(b)s(ounded)42 b(ab)s(o)m(v)m(e)d(b)m(y)972 3815 y Fm(p)p 1037 3815 40 4 v 58 x Fs(5)p Fm(\000)p Fs(1)p 972 3890 208 4 v 1056 3947 a(2)1191 3913 y Ft(.)63 b(The)40 b(same)f(results)h(holds)g(also)f(on)h(the)g(space)h(of)e (functions)113 4045 y Fp(')28 b Fo(2)g(O)382 4009 y Fs(1)427 4045 y Ft(\()p 466 3965 72 4 v Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))31 b(with)i(b)s(ounded)i(real)e(part.)313 4166 y(The)28 b(complex)e(Brjuno)i(function)h(is)e(\014nally)g(in)m (tro)s(duced)i(in)e(Section)h(5)g(where)g(w)m(e)g(state)113 4285 y(and)34 b(pro)m(v)m(e)g(our)f(main)f(results.)313 4406 y(In)27 b(the)g(App)s(endices)h(w)m(e)f(recall)g(the)g(results)g (w)m(e)g(need)h(on)f(the)g(real)g(con)m(tin)m(ued)h(fraction,)113 4525 y(on)39 b(the)g(h)m(yp)s(erfunctions)h(and)f(on)g(the)g (dilogarithm.)58 b(Then)39 b(w)m(e)g(sho)m(w)g(ho)m(w)g(to)f(relate)g (the)113 4645 y(complex)i(Brjuno)i(function)g(with)f(the)g(ev)m(en)h (real)f(Brjuno)g(function)h(treated)f(in)g([MMY].)113 4765 y(Finally)28 b(w)m(e)g(describ)s(e)i(ho)m(w)f(the)g(real)f(Brjuno) h(function)g(can)g(b)s(e)g(view)m(ed)f(as)h(a)f(co)s(cycle)g(under)113 4884 y(the)34 b(action)f(of)g(the)h(mo)s(dular)e(group.)313 5066 y Fl(A)-5 b(cknow)5 b(le)-5 b(dgements.)84 b Ft(This)48 b(w)m(ork)e(b)s(egun)i(during)g(a)e(visit)g(of)h(the)h(\014rst)f (author)g(at)113 5186 y(the)41 b(S.Ph.T.{CEA)d(Sacla)m(y)i(and)g(at)f (the)h(Dept.)65 b(of)40 b(Mathematics)f(of)h(Orsa)m(y)g(during)g(the) 113 5305 y(academic)32 b(y)m(ear)g(1993{1994.)41 b(This)33 b(researc)m(h)g(has)g(b)s(een)f(supp)s(orted)i(b)m(y)e(the)g(CNR,)g (CNRS,)113 5425 y(INFN,)h(MURST)h(and)f(a)g(EEC)g(gran)m(t.)p eop %%Page: 7 7 7 6 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2320 b(7)113 383 y Fr(2.)51 b(Mo)s(dular)38 b(group,)g(the)f(monoid)h Fo(M)g Fr(and)f(its)h(action)113 528 y Ft(According)52 b(to)g(\(1.4\))e(the)i(in)m(v)m(ersion)h(of)e(\(1)35 b Fo(\000)f Fp(T)14 b Ft(\))52 b(leads)g(to)f(consider)i(the)f(monoid)f Fo(M)113 648 y Ft(of)d(matrices)f(of)i(GL\(2)p Fp(;)17 b Fq(Z)p Ft(\).)84 b(In)48 b(this)g(section)g(w)m(e)g(study)h(its)e (algebraic)h(prop)s(erties)g(and)113 767 y(w)m(e)55 b(describ)s(e)h(v) -6 b(arious)55 b(actions)f(of)h(the)g(mo)s(dular)e(group)i(and)g(of)g (the)g(monoid)e Fo(M)i Ft(on)113 887 y(meromorphic)32 b(or)h(holomorphic)g(functions.)113 1057 y Fr(2.1)45 b(Algebraic)d(prop)s(erties,)i(notations,)g(structure)f(of)h(the)f (monoid)h Fo(M)23 b Fr(;)46 b(rela-)113 1176 y(tions)38 b(of)h Fo(M)e Fr(with)g(the)g(mo)s(dular)h(group)h(and)e(with)g(F)-10 b(arey)38 b(in)m(terv)-6 b(als)313 1321 y(2.1.1)33 b Ft(Notations)g(:)184 1476 y Fp(G)51 b Ft(=)28 b(GL\(2)p Fp(;)17 b Fq(Z)p Ft(\))24 b(=)k Fo(f)979 1335 y Fk(\022)1069 1416 y Fp(a)104 b(b)1074 1535 y(c)g(d)1290 1335 y Fk(\023)1396 1476 y Fp(;)44 b(a;)17 b(b;)g(c;)g(d)25 b Fo(2)j Fq(Z)17 b Fp(;)41 b(")2114 1491 y Fn(g)2189 1476 y Ft(:=)27 b Fp(ad)22 b Fo(\000)h Fp(bc)k Ft(=)h Fo(\006)p Ft(1)p Fo(g)20 b Ft(;)172 1725 y Fp(H)58 b Ft(is)45 b(the)h(subgroup)h(of)e (order)h(8)f(of)h(matrices)f(of)g(the)h(form)2642 1584 y Fk(\022)2733 1665 y Fp(")115 b Ft(0)2732 1784 y(0)99 b Fp(")2927 1748 y Fm(0)2972 1584 y Fk(\023)3091 1725 y Ft(or)3225 1584 y Fk(\022)3328 1665 y Ft(0)113 b Fp(")3315 1784 y(")3361 1748 y Fm(0)3489 1784 y Ft(0)3556 1584 y Fk(\023)3629 1725 y Ft(,)313 1904 y(where)34 b Fp(";)17 b(")738 1868 y Fm(0)794 1904 y Fo(2)28 b(f\000)p Ft(1)p Fp(;)17 b Ft(+1)p Fo(g)i Ft(;)143 2084 y Fo(M)50 b Ft(is)39 b(the)g(monoid)f(with)h(unit)1405 1943 y Fk(\022)1495 2024 y Ft(1)99 b(0)1495 2143 y(0)g(1)1711 1943 y Fk(\023)1823 2084 y Ft(made)39 b(of)g(matrices)f Fp(g)j Ft(=)2823 1943 y Fk(\022)2913 2024 y Fp(a)103 b(b)2917 2143 y(c)i(d)3133 1943 y Fk(\023)3244 2084 y Fo(2)38 b Fp(G)h Ft(suc)m(h)313 2257 y(that,)32 b(if)h Fp(g)e Fo(6)p Ft(=)45 b(id)17 b(,)33 b(w)m(e)g(ha)m(v)m(e)h Fp(d)27 b Fo(\025)i Fp(b)e Fo(\025)h Fp(a)g Fo(\025)g Ft(0)33 b(and)g Fp(d)28 b Fo(\025)g Fp(c)g Fo(\025)g Fp(a)p Ft(.)188 2431 y Fp(Z)57 b Ft(is)33 b(the)g(subgroup)i(of)e(matrices)g(of)g(the)h(form)2050 2291 y Fk(\022)2140 2371 y Ft(1)100 b Fp(n)2140 2491 y Ft(0)105 b(1)2366 2291 y Fk(\023)2439 2431 y Ft(,)33 b Fp(n)28 b Fo(2)g Fq(Z)p Ft(.)313 2706 y Fr(2.1.2)i Ft(Let)g Fp(g)t Ft(\()p Fp(m)p Ft(\))d(=)1102 2565 y Fk(\022)1192 2646 y Ft(0)118 b(1)1192 2765 y(1)100 b Fp(m)1446 2565 y Fk(\023)1519 2706 y Ft(,)30 b(where)h Fp(m)d Fo(\025)g Ft(1.)43 b(Clearly)29 b Fp(g)t Ft(\()p Fp(m)p Ft(\))e Fo(2)h(M)p Ft(.)42 b(Moreo)m(v)m(er,)31 b Fo(M)113 2885 y Ft(is)i(the)g Fl(fr)-5 b(e)g(e)35 b(monoid)d Ft(generated)i(b)m(y)f(the)g(elemen)m(ts)g Fp(g)t Ft(\()p Fp(m)p Ft(\),)f Fp(m)c Fo(\025)g Ft(1)33 b(:)44 b(eac)m(h)33 b(elemen)m(t)g Fp(g)j Ft(of)d Fo(M)113 3004 y Ft(can)i(b)s(e)h(written) e(as)h Fp(g)e Ft(=)d Fp(g)t Ft(\()p Fp(m)1275 3019 y Fs(1)1319 3004 y Ft(\))17 b Fo(\001)g(\001)g(\001)e Fp(g)t Ft(\()p Fp(m)1686 3019 y Fn(r)1729 3004 y Ft(\))30 b Fp(;)76 b(r)33 b Fo(\025)e Ft(0)f Fp(;)46 b(m)2329 3019 y Fn(i)2393 3004 y Fo(\025)30 b Ft(1)g Fp(;)35 b Ft(and)g(this)f (decomp)s(osition)113 3124 y(is)g(unique)f(\(see)h(Prop)s(osition)e (A1.2\).)313 3269 y Fr(2.1.3)h Ft(One)i(has)1537 3388 y Fp(G)28 b Ft(=)g Fp(Z)h Fo(\001)22 b(M)g(\001)g Fp(H)35 b(;)113 3559 y Ft(i.e.)44 b(the)34 b(application)e Fp(Z)e Fo(\002)22 b(M)g(\002)h Fp(H)35 b Fo(!)27 b Fp(G)h(;)72 b Ft(\()p Fp(z)t(;)17 b(m;)g(h)p Ft(\))27 b Fo(7!)h Fp(g)i Ft(=)f Fp(z)d Fo(\001)c Fp(m)h Fo(\001)f Fp(h)33 b Ft(is)h(a)f Fl(bije)-5 b(ction)p Ft(.)313 3764 y Fr(2.1.4)43 b Ft(The)f(subset)i Fp(Z)36 b Fo(\001)28 b(M)42 b Ft(of)g Fp(G)h Ft(is)f(made)g(of)h (matrices)e Fp(g)46 b Ft(=)2814 3624 y Fk(\022)2904 3704 y Fp(a)104 b(b)2908 3824 y(c)h(d)3124 3624 y Fk(\023)3240 3764 y Ft(suc)m(h)44 b(that)113 3938 y Fp(d)28 b Fo(\025)g Fp(c)g Fo(\025)g Ft(0)i(with)f(the)i(follo)m(wing)e(additional)g (restrictions)h(:)43 b Fp(a)27 b Ft(=)h(1)i(if)g Fp(c)e Ft(=)g(0,)i(and,)h Fp(b)c Ft(=)i Fp(a)16 b Ft(+)g(1)113 4057 y(if)34 b Fp(d)27 b Ft(=)h Fp(c)g Ft(=)g(1.)313 4177 y(W)-8 b(e)43 b(will)f(also)h(often)h(use)g(the)g(follo)m(wing)f (remark,)h(whic)m(h)g(is)f(an)g(immediate)e(conse-)113 4297 y(quence)35 b(of)e(the)h(structure)g(of)g Fo(M)e Ft(and)i(of)g(the)f(relation)1277 4549 y Fp(g)t Ft(\()p Fp(m)p Ft(\))1511 4408 y Fk(\022)1600 4489 y Ft(1)99 b(1)1600 4609 y(0)g(1)1816 4408 y Fk(\023)1917 4549 y Ft(=)28 b Fp(g)t Ft(\()p Fp(m)21 b Ft(+)i(1\))55 b(:)113 4801 y(one)34 b(has)g(the)f Fl(p)-5 b(artition)860 5053 y Fp(Z)29 b Fo(\001)22 b(M)27 b Ft(=)h Fp(Z)h Fo(\001)22 b(M)g(\001)1598 4913 y Fk(\022)1688 4993 y Ft(1)99 b(1)1688 5113 y(0)g(1)1904 4913 y Fk(\023)1993 4959 y(G)2121 5053 y Fp(Z)29 b Fo(\001)22 b(M)g(\001)2459 4913 y Fk(\022)2549 4993 y Ft(0)100 b(1)2549 5113 y(1)g(1)2765 4913 y Fk(\023)2883 5053 y Fp(:)313 5350 y Fr(2.1.5)26 b Ft(Let)f(us)h(consider)h(the)f (usual)g(action)f(of)h Fp(G)f Ft(on)p 2293 5270 72 4 v 26 w Fq(C)54 b Ft(=)28 b Fq(C)f Fo([)7 b(f1g)30 b Ft(b)m(y)c (homographies)f(:)113 5389 y Fk(\022)203 5470 y Fp(a)104 b(b)208 5589 y(c)h(d)424 5389 y Fk(\023)520 5529 y Fo(\001)21 b Fp(z)33 b Ft(=)765 5490 y Fp(az)26 b Ft(+)d Fp(b)p 765 5507 268 4 v 765 5580 a(cz)k Ft(+)c Fp(d)1045 5529 y Ft(.)44 b(The)33 b(follo)m(wing)g(facts)h(are)f(easy)g(to)g(c)m(hec)m (k)i(:)p eop %%Page: 8 8 8 7 bop 113 144 a Ft(8)2455 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)185 383 y(1.)50 b Fp(g)20 b Fo(\001)d Ft([0)p Fp(;)g Ft(1])26 b(=)i([0)p Fp(;)17 b Ft(1])29 b(if)i(and)h(only)e(if)h Fp(g)j Ft(b)s(elongs)d(to)f(the)h(subgroup)h(of)g(order)f(4)f(of)i (matrices)313 562 y(of)h(the)h(form)e Fo(\006)927 422 y Fk(\022)1017 503 y Ft(1)100 b(0)1017 622 y(0)g(1)1233 422 y Fk(\023)1306 562 y Ft(,)33 b Fo(\006)1461 422 y Fk(\022)1551 503 y Fo(\000)p Ft(1)100 b(1)1590 622 y(0)138 b(1)1845 422 y Fk(\023)1918 562 y Ft(.)185 747 y(2.)50 b(The)33 b(monoid)g(of)g(the)h(elemen)m(ts)f Fp(g)j Ft(suc)m(h)f(that)e Fp(g)25 b Fo(\001)c Ft([0)p Fp(;)c Ft(1])26 b Fo(\032)j Ft([0)p Fp(;)17 b Ft(1])31 b(admits)h(the)i(partition)745 1063 y Fo(M)882 968 y Fk(G)1009 1063 y Fo(M)1146 922 y Fk(\022)1235 1003 y Fo(\000)p Ft(1)138 b(0)1273 1122 y(0)h Fo(\000)p Ft(1)1606 922 y Fk(\023)1695 968 y(G)1823 1063 y Fo(M)1960 922 y Fk(\022)2049 1003 y Fo(\000)p Ft(1)100 b(1)2088 1122 y(0)138 b(1)2342 922 y Fk(\023)2432 968 y(G)2560 1063 y Fo(M)2697 922 y Fk(\022)2786 1003 y Ft(1)99 b Fo(\000)p Ft(1)2786 1122 y(0)g Fo(\000)p Ft(1)3079 922 y Fk(\023)3197 1063 y Fp(:)313 1448 y Ft(Note)32 b(that)769 1307 y Fk(\022)859 1388 y Fo(\000)p Ft(1)100 b(1)898 1508 y(0)138 b(1)1153 1307 y Fk(\023)1254 1448 y Ft(=)1359 1307 y Fk(\022)1449 1388 y Ft(1)99 b(1)1449 1508 y(0)g(1)1665 1307 y Fk(\023)16 b(\022)1844 1388 y Fo(\000)p Ft(1)100 b(0)1883 1508 y(0)138 b(1)2138 1307 y Fk(\023)2211 1448 y Ft(.)185 1666 y(3.)50 b(The)38 b(application)f Fp(g)g Fo(7!)e Fp(g)28 b Fo(\001)c Ft(1)35 b(=)1598 1627 y Fp(a)22 b Ft(+)g Fp(b)p 1598 1643 218 4 v 1598 1717 a(c)g Ft(+)h Fp(d)1865 1666 y Ft(is)37 b(a)g Fl(bije)-5 b(ction)36 b Ft(of)h Fp(Z)7 b Fo(M)37 b Ft(o)m(v)m(er)h Fq(Q)55 b Ft(whic)m(h)38 b(maps)313 1810 y Fo(M)32 b Ft(on)m(to)i Fq(Q)39 b Fo(\\)23 b Ft(\(0)p Fp(;)17 b Ft(1].)185 1935 y(4.)50 b(The)33 b(application)f Fp(g)f Fo(7!)c Fp(g)d Fo(\001)d Ft(0)28 b(=)g Fp(b=d)k Ft(maps)g Fp(Z)7 b Fo(M)33 b Ft(on)m(to)f Fq(Q)51 b Ft(and)33 b(eac)m(h)g(rational)f(n)m(um)m(b)s(er)313 2054 y(has)46 b(exactly)e Fl(two)j(inverse)f(images)p Ft(.)81 b(The)46 b(t)m(w)m(o)f(elemen)m(ts)h(whic)m(h)g(map)f(0)g(on)h(1)g(are)313 2088 y Fk(\022)403 2168 y Ft(1)99 b(1)403 2288 y(0)g(1)618 2088 y Fk(\023)725 2228 y Ft(and)919 2088 y Fk(\022)1009 2168 y Ft(0)g(1)1009 2288 y(1)g(1)1224 2088 y Fk(\023)1298 2228 y Ft(.)44 b(This)33 b(mak)m(es)g(the)g(partition)g(of)g(2.1.4)f (less)i(m)m(ysterious.)185 2413 y(5.)50 b(The)45 b(application)f Fp(g)50 b Fo(7!)d Fp(g)33 b Fo(\001)c Ft([0)p Fp(;)17 b Ft(+)p Fo(1)p Ft(])44 b(is)g(a)h Fl(bije)-5 b(ction)43 b Ft(of)i Fp(Z)7 b Fo(M)44 b Ft(on)h(the)g(set)g(of)g(F)-8 b(arey)313 2532 y(in)m(terv)i(als)55 b(\(the)g(con)m(v)m(en)m(tion)h(w) m(e)g(adopt)f(here)h(implies)f(that)f([)p Fp(n;)17 b Ft(+)p Fo(1)p Ft(])54 b(is)h(a)g(F)-8 b(arey)313 2652 y(in)m(terv)i(al,)38 b(but)g([)p Fo(\0001)p Fp(;)17 b(n)p Ft(])37 b(is)h(not\).)57 b(F)-8 b(or)38 b(the)g(de\014nition)g(and)h (prop)s(erties)f(of)g(the)g(F)-8 b(arey)313 2772 y(partition)32 b(of)h([0)p Fp(;)17 b Ft(1])32 b(w)m(e)i(refer)g(the)f(reader)h(to)f ([HW].)185 2897 y(6.)50 b(The)35 b(application)f Fp(g)f Fo(7!)d Fp(g)c Fo(\001)c(1)30 b Ft(=)h Fp(a=c)e Fo(2)p 1879 2816 78 4 v 30 w Fq(Q)48 b Ft(=)30 b Fq(Q)41 b Fo([)23 b(f1g)34 b Ft(of)h Fp(Z)7 b Fo(M)34 b Ft(on)p 3002 2816 V 35 w Fq(Q)52 b Ft(is)35 b(surjectiv)m(e.)313 3016 y(Moreo)m(v)m(er) 412 3141 y Fo(\017)50 b Fp(g)25 b Fo(\001)d(1)27 b Ft(=)h Fo(1)33 b Ft(if)h(and)f(only)g(if)g Fp(g)e Fo(2)d Fp(Z)f Ft(;)412 3266 y Fo(\017)50 b Fp(g)25 b Fo(\001)d(1)27 b Ft(=)h Fp(n)g Fo(2)g Fq(Z)j Ft(if)i(and)h(only)e(if)755 3576 y Fp(g)e Ft(=)939 3436 y Fk(\022)1028 3516 y Fp(n)100 b Ft(1)22 b(+)g Fp(k)s(n)1033 3636 y Ft(1)221 b Fp(k)1491 3436 y Fk(\023)1609 3576 y Fp(;)71 b(k)31 b Fo(\025)d Ft(1)f(or)h Fp(g)j Ft(=)2274 3436 y Fk(\022)2364 3516 y Fp(n)99 b Fo(\000)p Ft(1)23 b(+)f Fp(k)s(n)2369 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b(in)m(terv)-6 b(al)33 b(whic)m(h)h(con)m(tains)g Fp(a=c)p Ft(.)p eop %%Page: 9 9 9 8 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2320 b(9)113 383 y Fr(2.2)39 b(Actions)e(of)i Fo(M)f Fr(on)g(some)g(spaces)g (of)g(holomorphic)g(functions)313 558 y(2.2.1)h Ft(Let)f Fp(U)49 b Ft(b)s(e)39 b(an)g(op)s(en)g(subset)g(of)p 1847 478 72 4 v 39 w Fq(C)26 b Ft(.)59 b(W)-8 b(e)39 b(will)e(denote)j(b)m(y)e Fo(O)s Ft(\()p Fp(U)11 b Ft(\))38 b(the)h(complex)113 678 y(v)m(ector)31 b(space)h(of)f(holomorphic)f (functions)i(on)f Fp(U)11 b Ft(.)43 b(Let)31 b Fp(I)k Fo(\032)28 b Fq(R)42 b Ft(b)s(e)31 b(a)g(compact)f(in)m(terv)-6 b(al)30 b(and)113 798 y(let)e Fp(k)j Fo(2)d Fq(Z)p Ft(.)40 b(If)28 b Fp(k)j Fo(\025)d Ft(0)g(\()p Fp(k)j(<)d Ft(0)g(resp)s(ectiv)m (ely\),)g(w)m(e)h(will)e(denote)i(with)f Fo(O)2781 761 y Fn(k)2830 798 y Ft(\()p 2869 717 V Fq(C)38 b Fo(n)12 b Fp(I)c Ft(\))27 b(the)i(complex)113 917 y(v)m(ector)k(space)h(of)f (functions)h(holomorphic)d(in)i Fq(C)47 b Fo(n)21 b Fp(I)8 b Ft(,)32 b(meromorphic)f(in)p 2900 837 V 33 w Fq(C)47 b Fo(n)21 b Fp(I)8 b Ft(,)32 b(whic)m(h)h(ha)m(v)m(e)113 1037 y(a)g(zero)h(at)f(in\014nit)m(y)g(of)g(order)h(at)f(least)g Fp(k)j Ft(\(resp.)45 b(a)33 b(p)s(ole)g(of)g(order)h(at)f(most)f Fo(j)p Fp(k)s Fo(j)p Ft(\).)313 1216 y(Let)h Fp(g)e Ft(=)675 1076 y Fk(\022)765 1156 y Fp(a)104 b(b)770 1276 y(c)g(d)986 1076 y Fk(\023)1087 1216 y Fo(2)28 b Fp(G)33 b Ft(and)h(assume)f(that)g Fp(')h Ft(is)f(meromorphic)f(in)h Fp(U)11 b Ft(.)44 b(W)-8 b(e)33 b(de\014ne)1108 1567 y Fp(L)1176 1526 y Fs(\()p Fn(k)r Fs(\))1176 1591 y Fn(g)1287 1567 y Fp(')p Ft(\()p Fp(z)t Ft(\))28 b(=)g(\()p Fp(a)21 b Fo(\000)i Fp(cz)t Ft(\))1958 1526 y Fm(\000)p Fn(k)2070 1567 y Fp(')2152 1426 y Fk(\022)2238 1499 y Fp(dz)j Fo(\000)d Fp(b)p 2237 1544 269 4 v 2237 1635 a(a)f Fo(\000)g Fp(cz)2517 1426 y Fk(\023)2635 1567 y Fp(:)789 b Ft(\(2)p Fp(:)p Ft(1\))113 1877 y(The)34 b(function)g Fp(L)776 1825 y Fs(\()p Fn(k)r Fs(\))776 1889 y Fn(g)887 1877 y Fp(')g 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Ft(th)m(us,)34 b(for)g Fp(k)c Ft(=)f Fo(\000)p Ft(1)1430 2617 y(\()p Fp(L)1537 2576 y Fs(\()p Fm(\000)p Fs(1\))1537 2642 y Fn(g)1706 2617 y Fp(')p Ft(\))1810 2576 y Fm(00)1888 2617 y Ft(=)g Fp(L)2062 2576 y Fs(\(3\))2062 2642 y Fn(g)2168 2617 y Fp(')2233 2576 y Fm(00)2312 2617 y Fp(:)1112 b Ft(\(2)p Fp(:)p Ft(2\))113 2831 y(Note)33 b(also)g(that)g(if)g Fp(g)e Fo(2)d Fp(Z)40 b Ft(then)34 b Fp(L)1437 2780 y Fs(\()p Fn(k)r Fs(\))1437 2843 y Fn(g)1582 2831 y Ft(do)s(es)f(not)g (dep)s(end)i(on)f Fp(k)s Ft(.)313 2951 y(The)27 b(form)m(ula)e(\(2.1\)) h(ab)s(o)m(v)m(e)g(clearly)g(de\014nes)i(an)f(action)f(of)h Fp(G)f Ft(:)41 b(if)27 b Fp(g)t(;)17 b(g)2946 2915 y Fm(0)2999 2951 y Fo(2)28 b Fp(G)p Ft(,)g Fp(k)i Fo(2)e Fq(Z)c Ft(and)113 3090 y Fp(')40 b Ft(is)f(meromorphic)f(in)h Fp(U)50 b Ft(then)40 b(the)g(functions)h Fp(L)2089 3038 y Fs(\()p Fn(k)r Fs(\))2089 3102 y Fn(g)2200 3090 y Ft(\()p Fp(L)2307 3038 y Fs(\()p Fn(k)r Fs(\))2307 3119 y Fn(g)2349 3099 y Ff(0)2417 3090 y Fp(')p Ft(\))f(and)f Fp(L)2828 3038 y Fs(\()p Fn(k)r Fs(\))2828 3119 y Fn(g)s(g)2912 3099 y Ff(0)2943 3090 y Fp(')g Ft(\(meromorphic)113 3209 y(on)34 b Fp(g)t(g)356 3173 y Fm(0)404 3209 y Fo(\001)22 b Fp(U)11 b Ft(\))33 b(coincide.)313 3357 y Fr(2.2.2)d Ft(Let)g Fp(J)40 b Ft(denote)31 b(a)f(compact)f(in)m(terv)-6 b(al)30 b(of)g Fq(R)41 b Ft(and)31 b(let)e Fp(')f Fo(2)g(O)2810 3321 y Fs(2)2855 3357 y Ft(\()p 2894 3276 72 4 v Fq(C)41 b Fo(n)16 b Fp(J)10 b Ft(\).)42 b(The)31 b(series)1076 3503 y Fk(X)1074 3718 y Fn(n)p Fm(2)p Fg(Z)1240 3598 y Fp(')p Ft(\()p Fp(z)26 b Fo(\000)d Fp(n)p Ft(\))k(=)1753 3503 y Fk(X)1747 3717 y Fn(g)s Fm(2)p Fn(Z)1918 3598 y Fp(L)1986 3557 y Fs(\()p Fn(k)r Fs(\))1986 3623 y Fn(g)2097 3598 y Fp(')56 b Ft(\(for)33 b(all)f Fp(k)s Ft(\))c Fp(;)755 b Ft(\(2)p Fp(:)p Ft(3\))113 3927 y(con)m(v)m(erges)60 b(uniformly)e(on)g(compact)g(subsets)i(of)f Fq(C)65 b Fo(n)39 b Fq(R)70 b Ft(and)59 b(also)f(on)g(the)h(domains)113 4047 y Fo(f)p Fp(z)53 b Fo(2)48 b Fq(C)74 b Fp(;)64 b Fo(j)17 b(=)p Fp(m)f(z)t Fo(j)48 b(\025)h Fp(\016)i(>)e Ft(0)e Fp(;)112 b Fo(j)17 b(<)p Fp(e)g(z)t Fo(j)48 b(\024)g Fp(A)p Fo(g)p Ft(.)80 b(The)46 b(sum)f(will)f(b)s(e)i(denoted)3358 3972 y Fk(P)3463 4077 y Fn(Z)3544 4047 y Fp(')20 b Ft(;)113 4166 y(it)33 b(is)g(a)g(function)i(holomorphic)d(in)h Fq(C)48 b Fo(n)22 b Fq(R)5 b Ft(,)39 b(p)s(erio)s(dic)34 b(of)f(p)s(erio)s(d)g(1)g(and)h(v)-6 b(anishing)34 b(at)f Fo(\006)p Fp(i)p Fo(1)p Ft(.)113 4286 y(Th)m(us)47 b(taking)c(the)j (quotien)m(t)f(b)m(y)g Fq(Z)d Ft(it)i(can)i(b)s(e)f(represen)m(ted)j(b) m(y)d(means)f(of)i(the)f(v)-6 b(ariable)113 4405 y Fp(q)32 b Ft(=)c Fp(e)340 4369 y Fm(\006)p Fs(2)p Fn(\031)s(iz)566 4405 y Ft(.)313 4525 y(If)33 b Fp(')28 b Fo(2)g(O)681 4489 y Fs(1)726 4525 y Ft(\()p 765 4444 V Fq(C)48 b Fo(n)22 b Fp(J)10 b Ft(\))33 b(with)f([0)p Fp(;)17 b Ft(1])26 b Fo(\032)j Fp(J)43 b Ft(then)34 b(one)f(can)h(decomp)s(ose)f(in)h(a)f (unique)g(w)m(a)m(y)1262 4772 y Fp(')p Ft(\()p Fp(z)t Ft(\))28 b(=)g Fp(a)1641 4787 y Fs(0)1702 4772 y Ft(log)1945 4704 y Fp(z)p 1859 4749 223 4 v 1859 4840 a(z)f Fo(\000)c Ft(1)2115 4772 y(+)g Fp(')2280 4787 y Fs(0)2325 4772 y Ft(\()p Fp(z)t Ft(\))28 b Fp(;)943 b Ft(\(2)p Fp(:)p Ft(4\))113 5049 y(where)35 b Fp(a)455 5064 y Fs(0)529 5049 y Fo(2)30 b Fq(C)20 b Ft(,)40 b Fp(')824 5064 y Fs(0)898 5049 y Fo(2)30 b(O)1076 5012 y Fs(2)1120 5049 y Ft(\()p 1159 4968 72 4 v Fq(C)49 b Fo(n)22 b Fp(J)10 b Ft(\))34 b(and)h(w)m(e)f(consider)i(the)e(main)f(branc)m(h)j(of)e (the)h(logarithm)113 5168 y(in)f Fq(C)48 b Fo(n)22 b Fq(R)462 5132 y Fm(\000)535 5168 y Ft(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)1186 5362 y Fn(N)1150 5392 y Fk(X)1100 5606 y Fn(n)p Fs(=)p Fm(\000)p Fn(N)1360 5487 y Ft(log)1603 5419 y Fp(z)27 b Fo(\000)22 b Fp(n)p 1517 5464 404 4 v 1517 5555 a(z)27 b Fo(\000)c Fp(n)f Fo(\000)g Ft(1)1961 5487 y(=)28 b(log)2309 5419 y Fp(z)f Ft(+)22 b Fp(N)p 2223 5464 435 4 v 2223 5555 a(z)27 b Fo(\000)c Fp(N)33 b Fo(\000)22 b Ft(1)p eop %%Page: 10 10 10 9 bop 113 144 a Ft(10)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y(and)34 b(this)f(leads)h(to)f(the)g(de\014nition) 845 522 y Fk(X)887 736 y Fn(Z)1006 617 y Fp(')p Ft(\()p Fp(z)t Ft(\))28 b(:=)1360 522 y Fk(X)1402 736 y Fn(Z)1520 617 y Fp(')1585 632 y Fs(0)1630 617 y Ft(\()p Fp(z)t Ft(\))22 b(+)1880 476 y Fk(\032)1971 559 y Fo(\000)p Fp(a)2101 574 y Fs(0)2146 559 y Fp(\031)t(i)99 b Ft(if)34 b Fo(=)p Fp(m)17 b(z)32 b(>)c Ft(0)1971 679 y(+)p Fp(a)2101 694 y Fs(0)2146 679 y Fp(\031)t(i)99 b Ft(if)34 b Fo(=)p Fp(m)17 b(z)32 b(<)c Ft(0)2897 617 y Fp(:)527 b Ft(\(2)p Fp(:)p Ft(5\))113 939 y(Note)38 b(that)g(in)g(order)h(to)e(insure)j (the)e(con)m(v)m(ergence)i(of)f(the)f(series)2692 865 y Fk(P)2797 969 y Fn(g)s Fm(2)p Fn(Z)2973 939 y Fp(L)3041 888 y Fs(\()p Fn(k)r Fs(\))3041 951 y Fn(g)3152 939 y Fp(')g Ft(one)h(m)m(ust)113 1137 y(regroup)24 b(together)f(the)g(terms) 1281 997 y Fk(\022)1371 1077 y Ft(1)99 b Fp(n)1371 1197 y Ft(0)104 b(1)1596 997 y Fk(\023)1693 1137 y Ft(and)1876 997 y Fk(\022)1966 1077 y Ft(1)c Fo(\000)p Fp(n)1966 1197 y Ft(0)143 b(1)2270 997 y Fk(\023)2343 1137 y Ft(,)25 b Fp(n)i Fo(\025)h Ft(1)23 b(\(symmetric)e(summation)113 1317 y(or)33 b(Eisenstein's)h(summation\).)313 1462 y Fr(2.2.3)44 b Ft(Let)h Fp(k)j Fo(2)f Fq(Z)41 b Ft(and)k Fp(g)k Fo(2)d(M)p Ft(.)76 b(Since)45 b Fp(g)33 b Fo(\001)c Ft([0)p Fp(;)17 b Ft(1])44 b Fo(\032)i Ft([0)p Fp(;)17 b Ft(1])43 b(if)h Fp(')g Ft(is)g(meromorphic)113 1600 y(in)p 252 1520 72 4 v 56 w Fq(C)63 b Fo(n)37 b Ft([0)p Fp(;)17 b Ft(1])54 b(then)j Fp(L)1021 1549 y Fs(\()p Fn(k)r Fs(\))1021 1612 y Fn(g)1132 1600 y Fp(')f Ft(will)e(still)h(b)s (e)h(meromorphic)f(in)p 2603 1520 V 55 w Fq(C)64 b Fo(n)36 b Ft([0)p Fp(;)17 b Ft(1].)110 b(Moreo)m(v)m(er)57 b(if)113 1740 y Fp(')43 b Fo(2)g(O)412 1704 y Fn(k)461 1740 y Ft(\()p 500 1659 V Fq(C)54 b Fo(n)28 b Ft([0)p Fp(;)17 b Ft(1]\))40 b(then)j(also)e Fp(L)1470 1688 y Fs(\()p Fn(k)r Fs(\))1470 1751 y Fn(g)1581 1740 y Fp(')i Fo(2)g(O)1880 1704 y Fn(k)1929 1740 y Ft(\()p 1968 1659 V Fq(C)54 b Fo(n)28 b Ft([0)p Fp(;)17 b Ft(1]\).)68 b(Th)m(us)43 b(one)g(has)f(an)h(action)e(of)113 1859 y(the)34 b(monoid)e Fo(M)h Ft(on)g Fo(O)1012 1823 y Fn(k)1062 1859 y Ft(\()p 1101 1779 V Fq(C)47 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\).)313 2004 y Fr(2.2.4)29 b Ft(W)-8 b(e)30 b(will)e(no)m(w)i(de\014ne)h(a)e (new)h(action)f Fp(L)2060 2019 y Fn(g)2135 2004 y Ft(of)h Fo(M)f Ft(on)h Fo(O)2611 1968 y Fs(1)2655 2004 y Ft(\()p 2694 1924 V Fq(C)40 b Fo(n)14 b Ft([0)p Fp(;)j Ft(1]\))28 b(whic)m(h)i(di\013ers)113 2147 y(from)j(the)g(action)g Fp(L)887 2095 y Fs(\()p Fm(\000)p Fs(1\))887 2158 y Fn(g)1090 2147 y Ft(on)g Fo(O)1310 2111 y Fm(\000)p Fs(1)1417 2147 y Ft(\()p 1456 2066 V Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))31 b(b)m(y)i(an)h(a\016ne)f(correction.)313 2266 y(All)f(functions)i Fp(')28 b Fo(2)g(O)1174 2230 y Fm(\000)p Fs(1)1281 2266 y Ft(\()p 1320 2186 V Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))31 b(can)j(b)s(e)g(uniquely)f (written)g(as)803 2465 y Fp(')p Ft(\()p Fp(z)t Ft(\))28 b(=)h Fp(Az)d Ft(+)d Fp(B)j Ft(+)d Fp(p)p Ft(\()p Fp(')p Ft(\)\()p Fp(z)t Ft(\))28 b Fp(;)71 b(p)p Ft(\()p Fp(')p Ft(\))28 b Fo(2)g(O)2424 2424 y Fs(1)2469 2465 y Ft(\()p 2508 2385 V Fq(C)47 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))26 b Fp(:)485 b Ft(\(2)p Fp(:)p Ft(6\))113 2724 y(Note)33 b(that)g(if)g Fp(g)e Ft(=)845 2583 y Fk(\022)935 2664 y Fp(a)104 b(b)940 2784 y(c)g(d)1156 2583 y Fk(\023)1257 2724 y Fo(2)28 b Fp(G)34 b Ft(and)f Fp( )t Ft(\()p Fp(z)t Ft(\))28 b(=)g Fp(Az)e Ft(+)d Fp(b)33 b Ft(then)1118 3001 y Fp(L)1186 2960 y Fs(\()p Fm(\000)p Fs(1\))1186 3026 y Fn(g)1355 3001 y Fp( )t Ft(\()p Fp(z)t Ft(\))27 b(=)i Fp(A)p Ft(\()p Fp(dz)d Fo(\000)c Fp(b)p Ft(\))g(+)h Fp(B)5 b Ft(\()p Fp(a)21 b Fo(\000)h Fp(cz)t Ft(\))113 3200 y(is)34 b(still)e(an)h(a\016ne)h(function.)45 b(Th)m(us)34 b(the)g(form)m(ula)1473 3399 y Fp(L)1541 3414 y Fn(g)1587 3399 y Fp(')28 b Ft(:=)f Fp(p)p Ft(\()p Fp(L)1969 3358 y Fs(\()p Fm(\000)p Fs(1\))1969 3424 y Fn(g)2138 3399 y Fp(')p Ft(\))h Fp(;)1154 b Ft(\(2)p Fp(:)p Ft(7\))113 3615 y(where)37 b Fp(g)f Fo(2)d(M)p Ft(,)k Fp(')32 b Fo(2)h(O)1050 3579 y Fs(1)1095 3615 y Ft(\()p 1134 3535 V Fq(C)50 b Fo(n)24 b Ft([0)p Fp(;)17 b Ft(1]\),)35 b(de\014nes)j(an)e Fl(action)f Ft(of)i Fo(M)f Ft(on)g Fo(O)2877 3579 y Fs(1)2922 3615 y Ft(\()p 2961 3535 V Fq(C)50 b Fo(n)24 b Ft([0)p Fp(;)17 b Ft(1]\))34 b(whic)m(h)113 3735 y(mak)m(es)f(the)g(follo)m (wing)g(diagram)f(comm)m(ute)984 4131 y Fo(O)1066 4095 y Fm(\000)p Fs(1)1173 4131 y Ft(\()p 1212 4051 V Fq(C)47 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))1784 4054 y Fn(L)1839 4024 y Fi(\()p Ff(\000)p Fi(1\))1839 4074 y Fj(g)1715 4131 y Fo(\000)-40 b(\000)-21 b(\000)f(\000)g(\000)-40 b(!)100 b(O)2237 4095 y Fm(\000)p Fs(1)2344 4131 y Ft(\()p 2383 4051 V Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))1203 4430 y Fn(p)1260 4256 y Fk(?)1260 4316 y(?)1260 4376 y(?)1260 4435 y(?)1260 4495 y(y)2431 4256 y(?)2431 4316 y(?)2431 4376 y(?)2431 4435 y(?)2431 4495 y(y)2526 4430 y Fn(p)1015 4743 y Fo(O)1097 4707 y Fs(1)1141 4743 y Ft(\()p 1180 4662 V Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))129 b Fo(\000)-40 b(\000)-21 b(\000)f(\000)g(\000)-40 b(!)1837 4807 y Fn(L)1892 4817 y Fj(g)2186 4743 y Fo(O)2268 4707 y Fs(1)2313 4743 y Ft(\()p 2352 4662 V Fq(C)47 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))113 5201 y(More)34 b(explicitly)-8 b(,)31 b(if)i Fp(g)e Ft(=)1097 5061 y Fk(\022)1187 5141 y Fp(a)104 b(b)1192 5261 y(c)h(d)1408 5061 y Fk(\023)1509 5201 y Fo(2)28 b(M)394 5529 y Fp(L)462 5544 y Fn(g)508 5529 y Fp(')p Ft(\()p Fp(z)t Ft(\))g(=)g(\()p Fp(a)21 b Fo(\000)i Fp(cz)t Ft(\))1196 5389 y Fk(\024)1249 5529 y Fp(')1331 5389 y Fk(\022)1416 5462 y Fp(dz)k Fo(\000)c Fp(b)p 1416 5507 269 4 v 1416 5598 a(a)f Fo(\000)g Fp(cz)1696 5389 y Fk(\023)1792 5529 y Fo(\000)g Fp(')1973 5389 y Fk(\022)2046 5529 y Fo(\000)2135 5462 y Fp(d)p 2135 5507 52 4 v 2139 5598 a(c)2200 5389 y Fk(\023\025)2348 5529 y Fo(\000)g Fp(")2493 5544 y Fn(g)2540 5529 y Fp(c)2583 5488 y Fm(\000)p Fs(1)2690 5529 y Fp(')2755 5488 y Fm(0)2800 5389 y Fk(\022)2873 5529 y Fo(\000)2962 5462 y Fp(d)p 2962 5507 V 2966 5598 a(c)3026 5389 y Fk(\023)3144 5529 y Fp(:)280 b Ft(\(2)p Fp(:)p Ft(8\))p eop %%Page: 11 11 11 10 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(11)113 388 y(Since)45 b(this)f(de\014nition)g(di\013ers)h(from)e (that)h(of)g Fp(L)2015 336 y Fs(\()p Fm(\000)p Fs(1\))2015 399 y Fn(g)2228 388 y Ft(only)f(b)m(y)h(an)g(a\016ne)g(correction)g (one)113 507 y(clearly)33 b(has)1492 647 y(\()p Fp(L)1599 662 y Fn(g)1645 647 y Fp(')p Ft(\))1749 606 y Fm(00)1827 647 y Ft(=)28 b Fp(L)2000 606 y Fs(\(3\))2000 671 y Fn(g)2107 647 y Fp(')2172 606 y Fm(00)2251 647 y Fp(;)1173 b Ft(\(2)p Fp(:)p Ft(9\))113 873 y(where)28 b Fp(')460 837 y Fm(00)539 873 y Fo(2)g(O)715 837 y Fs(3)760 873 y Ft(\()p 799 793 72 4 v Fq(C)35 b Fo(n)10 b Ft([0)p Fp(;)17 b Ft(1]\).)40 b(Equiv)-6 b(alen)m(tly)25 b(one)j(can)f(sa)m(y)g(that)g Fp(L)2615 888 y Fn(g)2661 873 y Fp(')g Ft(is)g(obtained)h(b)m(y)f (taking)113 1016 y(the)34 b(double)g(primitiv)m(e)d(of)j Fp(L)1214 964 y Fs(\(3\))1214 1028 y Fn(g)1321 1016 y Fp(')1386 980 y Fm(00)1470 1016 y Ft(whic)m(h)g(v)-6 b(anishes)34 b(at)f(in\014nit)m(y)-8 b(.)313 1142 y(Tw)m(o)23 b(other)h(form)m(ulas)f(will)f(b)s(e)i(used)h(throughout)f(what)f (follo)m(ws)h(:)39 b(if)23 b Fp(')28 b Fo(2)g(O)3180 1106 y Fs(1)3225 1142 y Ft(\()p 3264 1062 V Fq(C)g Fo(n)s Ft([0)p Fp(;)17 b Ft(1]\),)113 1322 y Fp(g)31 b Ft(=)297 1181 y Fk(\022)387 1262 y Fp(a)104 b(b)392 1381 y(c)h(d)608 1181 y Fk(\023)709 1322 y Fo(2)28 b(M)33 b Ft(and)h Fp(z)43 b(=)-61 b Fo(2)1322 1241 y Fk(\002)1379 1282 y Fn(b)p 1376 1299 42 4 v 1376 1356 a(d)1429 1322 y Fp(;)1485 1282 y Fn(a)p 1485 1299 44 4 v 1489 1356 a(c)1540 1241 y Fk(\003)1610 1322 y Ft(=)28 b Fp(g)d Fo(\001)d Ft([0)p Fp(;)17 b Ft(+)p Fo(1)p Ft(])32 b(one)h(has)395 1722 y Fp(L)463 1737 y Fn(g)509 1722 y Fp(')p Ft(\()p Fp(z)t Ft(\))28 b(=)g Fp(")881 1737 y Fn(g)928 1722 y Fp(c)971 1681 y Fm(\000)p Fs(1)1095 1581 y Fk(\024)1147 1586 y(Z)1247 1611 y Fs(1)1203 1813 y(0)1308 1722 y Fp(')1373 1681 y Fm(0)1418 1581 y Fk(\022)1491 1722 y Fo(\000)1580 1654 y Fp(d)p 1580 1699 52 4 v 1584 1790 a(c)1667 1722 y Ft(+)22 b Fp(")1812 1737 y Fn(g)2048 1654 y Fp(t)p 1871 1699 389 4 v 1871 1790 a(c)p Ft(\()p Fp(a)g Fo(\000)g Fp(cz)t Ft(\))2272 1581 y Fk(\023)2362 1722 y Fp(dt)g Fo(\000)h Fp(')2637 1681 y Fm(0)2681 1581 y Fk(\022)2755 1722 y Fo(\000)2844 1654 y Fp(d)p 2844 1699 52 4 v 2848 1790 a(c)2908 1581 y Fk(\023\025)730 2012 y Ft(=)28 b Fp(c)878 1971 y Fm(\000)p Fs(2)985 2012 y Ft(\()p Fp(a)22 b Fo(\000)h Fp(cz)t Ft(\))1331 1971 y Fm(\000)p Fs(1)1455 1877 y Fk(Z)1554 1901 y Fs(1)1510 2103 y(0)1616 2012 y Fp(')1681 1971 y Fm(00)1748 1872 y Fk(\022)1822 2012 y Fo(\000)1911 1945 y Fp(d)p 1911 1989 V 1915 2081 a(c)1997 2012 y Ft(+)g Fp(")2143 2027 y Fn(g)2378 1945 y Fp(t)p 2201 1989 389 4 v 2201 2081 a(c)p Ft(\()p Fp(a)f Fo(\000)h Fp(cz)t Ft(\))2602 1872 y Fk(\023)2692 2012 y Ft(\(1)f Fo(\000)h Fp(t)p Ft(\))p Fp(dt)k(:)3402 1861 y Ft(\(2)p Fp(:)p Ft(10\))113 2326 y(Note)33 b(that)g(the)h(assumption)f(ab)s(out)g Fp(z)38 b Ft(means)33 b(that)g(the)h(segmen)m(t)g(whose)g(extremities)e (are)113 2445 y Fo(\000)p Fp(d=c)g Ft(and)g(\()p Fp(dz)23 b Fo(\000)c Fp(b)p Ft(\))p Fp(=)p Ft(\()p Fp(a)e Fo(\000)i Fp(cz)t Ft(\))32 b(do)s(es)f(not)h(in)m(tersect)g(the)g(in)m(terv)-6 b(al)31 b([0,1].)42 b(The)32 b(t)m(w)m(o)f(form)m(ulas)113 2565 y(are)e(th)m(us)g(nothing)f(else)h(than)f(T)-8 b(a)m(ylor's)27 b(form)m(ulas)h(of)h(\014rst)f(and)h(second)g(order)g(with)f(in)m (tegral)113 2684 y(remainder.)313 2847 y Fr(2.2.5)40 b Ft(If)h Fp(')e Fo(2)h(O)s Ft(\()p 1026 2767 72 4 v Fq(C)52 b Fo(n)27 b Fp(I)8 b Ft(\))39 b(w)m(e)i(denote)g(b)m(y)f Fp(\033)30 b Fo(\001)c Fp(')p Ft(\()p Fp(z)t Ft(\))40 b(=)p 2449 2761 194 4 v 40 w Fp(')p Ft(\()6 b(\026)-56 b Fp(z)t Ft(\))39 b Fp(:)h(\033)k Ft(is)c(an)g(in)m(v)m(olution)f(of) 113 2967 y Fo(O)s Ft(\()p 234 2886 72 4 v Fq(C)48 b Fo(n)22 b Fp(I)8 b Ft(\))33 b(whic)m(h)h(preserv)m(es)h(all)e(the)g(subspaces)j Fo(O)2085 2931 y Fn(k)2134 2967 y Ft(\()p 2173 2886 V Fq(C)48 b Fo(n)22 b Fp(I)8 b Ft(\),)32 b(comm)m(utes)h(with)g Fp(p)g Ft(and)h(with)113 3086 y(the)f(actions)g(of)g Fo(M)f Ft(on)h Fo(O)1107 3050 y Fn(k)1156 3086 y Ft(.)44 b(If)33 b Fp(\033)25 b Fo(\001)c Fp(')28 b Ft(=)g Fp(')p Ft(,)k(then)i Fp(')p Ft(\()p Fp(x)p Ft(\))27 b Fo(2)h Fq(R)44 b Ft(for)33 b(all)f Fp(x)27 b Fo(2)h Fq(R)k Fo(n)21 b Fp(I)8 b Ft(,)32 b(and)i(w)m(e)f(sa)m(y)113 3206 y(that)g Fp(')h Ft(is)f(real)g(holomorphic.)113 3479 y Fr(3.)51 b(Complex)39 b(con)m(tin)m(ued)c(fractions)313 3648 y Ft(Exactly)c(as)h(in)h(the)g(real)f(case)i(treated)f(in)f([MMY],)g (where)i(the)f(use)g(of)g(the)g(con)m(tin)m(ued)113 3768 y(fraction)40 b(expansion)h(w)m(as)f(imp)s(ortan)m(t)f(for)h(the)g (study)g(of)h(the)f(real)g(Brjuno)g(function,)i(the)113 3887 y(in)m(tro)s(duction)52 b(of)f(a)g(complex)f(analogue)h(of)g (Gauss')h(algorithm)d(of)i(con)m(tin)m(ued)i(fraction)113 4007 y(expansion)23 b(of)f(a)g(real)g(n)m(um)m(b)s(er)h(will)e(b)s(e)i (essen)m(tial)g(for)f(the)h(study)f(of)g(the)h(b)s(oundary)f(b)s(eha)m (viour)113 4127 y(of)238 4052 y Fk(P)343 4156 y Fm(M)442 4127 y Ft(\()p Fp(L)549 4142 y Fn(g)595 4127 y Fp(')p Ft(\)\()p Fp(z)t Ft(\))45 b(and)g(the)g(construction)g(of)g(the)g (complex)e(Brjuno)j(function.)79 b(In)45 b(this)113 4246 y(section)36 b(w)m(e)f(\014rst)g(de\014ne)i(our)e(complex)f(v)m(ersion) h(of)g(the)g(con)m(tin)m(ued)i(fraction)e(\(section)g(3.1\))113 4366 y(then)h(w)m(e)e(use)i(it)e(to)g(estimate)f(the)i(sp)s(ectral)f (radius)h(of)g Fp(T)48 b Ft(\(section)35 b(3.2\))e(and)i(to)f(pro)m(v)m (e)h(the)113 4485 y(con)m(v)m(ergence)g(of)f(the)f(series)h(\(1.4\))e (and)i(\(1.6\))e(\(Corollary)f(3.6\).)p eop %%Page: 12 12 12 11 bop 113 144 a Ft(12)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y Fr(3.1)39 b(De\014nition)d(of)i(the)f(complex)i (con)m(tin)m(ued)c(fractions)313 527 y(3.1.1)e Ft(W)-8 b(e)34 b(consider)g(the)g(follo)m(wing)e(domains)h(:)184 798 y Fp(D)266 813 y Fs(0)339 798 y Ft(=)444 628 y Fk(\()524 798 y Fp(z)g Fo(2)28 b Fq(C)37 b Fp(;)17 b Fo(j)p Fp(z)31 b Ft(+)23 b(1)p Fo(j)k(\024)h Ft(1)17 b Fp(;)32 b Fo(<)p Fp(e)17 b(z)33 b Fo(\025)1698 649 y(p)p 1781 649 50 4 v 82 x Ft(3)p 1698 775 133 4 v 1739 867 a(2)1865 798 y Fo(\000)22 b Ft(1)2014 628 y Fk(\))2139 798 y Fp(;)184 1107 y(D)266 1122 y Fs(1)339 1107 y Ft(=)444 967 y Fk(\032)519 1107 y Fp(z)32 b Fo(2)c Fq(C)37 b Fp(;)17 b Fo(j)p Fp(z)t Fo(j)33 b(\025)28 b Ft(1)17 b Fp(;)1190 963 y Fk(\014)1190 1023 y(\014)1190 1082 y(\014)1190 1142 y(\014)1223 1107 y Fp(z)27 b Fo(\000)1449 1040 y Ft(1)p 1408 1084 V 1408 1104 a Fo(p)p 1491 1104 50 4 v 83 x Ft(3)1553 963 y Fk(\014)1553 1023 y(\014)1553 1082 y(\014)1553 1142 y(\014)1614 1107 y Fo(\024)1772 1040 y Ft(1)p 1731 1084 133 4 v 1731 1104 a Fo(p)p 1814 1104 50 4 v 83 x Ft(3)1875 967 y Fk(\033)1994 1107 y Fp(;)226 1326 y(D)k Ft(=)d Fo(f)p Fp(z)k Fo(2)c Fq(C)37 b Fp(;)17 b Fo(j)p Fp(z)t Fo(j)33 b(\024)28 b Ft(1)17 b Fp(;)32 b Fo(j)p Fp(z)27 b Fo(\000)c Fp(i)p Fo(j)k(\025)h Ft(1)17 b Fp(;)33 b Fo(j)p Fp(z)26 b Ft(+)d Fp(i)p Fo(j)k(\025)i Ft(1)17 b Fp(;)32 b Fo(<)p Fp(e)17 b(z)32 b(>)c Ft(0)p Fo(g)g Fp(;)184 1476 y(H)267 1491 y Fs(0)339 1476 y Ft(=)g Fo(f)p Fp(z)k Fo(2)c Fq(C)37 b Fp(;)17 b Fo(j)p Fp(z)32 b Fo(\000)22 b Fp(i)p Fo(j)28 b(\024)g Ft(1)17 b Fp(;)32 b Fo(j)p Fp(z)27 b Ft(+)c(1)p Fo(j)k(\025)h Ft(1)17 b Fp(;)32 b Fo(=)p Fp(m)17 b(z)33 b Fo(\024)28 b Ft(1)p Fp(=)p Ft(2)p Fo(g)f Fp(;)184 1625 y(H)275 1584 y Fm(0)267 1650 y Fs(0)339 1625 y Ft(=)h Fo(f)p Fp(z)k Fo(2)c Fq(C)37 b Fp(;)p 816 1570 51 4 v 39 w(z)32 b Fo(2)c Fp(H)1071 1640 y Fs(0)1116 1625 y Fo(g)228 1775 y Ft(\001)g(=)g Fp(H)527 1790 y Fs(0)594 1775 y Fo([)22 b Fp(H)773 1734 y Fm(0)765 1799 y Fs(0)832 1775 y Fo([)g Fp(D)31 b Ft(=)e Fo(f)p Fp(z)j Fo(2)c Fq(C)37 b Fp(;)17 b Fo(j)p Fp(z)t Fo(j)32 b(\024)d Ft(1)17 b Fp(;)32 b Fo(j)p Fp(z)26 b Ft(+)d(1)p Fo(j)k(\025)h Ft(1)17 b Fp(;)33 b Fo(j)17 b(=)p Fp(m)f(z)t Fo(j)28 b(\024)g Ft(1)p Fp(=)p Ft(2)p Fo(g)f Fp(;)144 1938 y(D)226 1953 y Fm(1)339 1938 y Ft(=)p 444 1857 72 4 v 28 w Fq(C)48 b Fo(n)22 b Ft(\()p Fp(D)731 1953 y Fs(0)798 1938 y Fo([)h Ft(\001)f Fo([)g Fp(D)1162 1953 y Fs(1)1208 1938 y Ft(\))339 2170 y(=)28 b Fo(fj)17 b(=)p Fp(m)f(z)t Fo(j)28 b Fp(>)g Ft(1)p Fp(=)p Ft(2)p Fo(g)22 b([)g(f<)p Fp(e)17 b(z)32 b(<)1616 2021 y Fo(p)p 1699 2021 50 4 v 82 x Ft(3)p 1616 2148 133 4 v 1657 2239 a(2)1783 2170 y Fo(\000)22 b Ft(1)p Fo(g)g([)g(f<)p Fp(e)17 b(z)33 b(>)2473 2021 y Fo(p)p 2556 2021 50 4 v 82 x Ft(3)p 2473 2148 133 4 v 2514 2239 a(2)2634 2170 y Fp(;)g Fo(j)p Fp(z)27 b Fo(\000)22 b Ft(1)p Fp(=)2995 2083 y Fo(p)p 3078 2083 50 4 v 87 x Ft(3)p Fo(j)27 b Fp(>)h Ft(1)p Fp(=)3388 2083 y Fo(p)p 3471 2083 V 87 x Ft(3)p Fo(g)f Fp(:)113 2391 y Ft(Figures)37 b(2)f(and)g(3)g(sho)m(w)h(these)g(domains)e(and)i(their)f(image)e (under)k(the)e(in)m(v)m(ersion)h Fp(S)6 b Ft(\()p Fp(z)t Ft(\))32 b(=)113 2511 y(1)p Fp(=z)t Ft(.)45 b(A)33 b(fundamen)m(tal)g (prop)s(ert)m(y)h(is)f(the)g(follo)m(wing)213 2630 y Fo(\017)50 b Ft(if)33 b Fp(z)43 b(=)-61 b Fo(2)28 b Fp(D)e Fo([)c Fp(D)854 2645 y Fs(1)933 2630 y Ft(\(in)33 b(particular)g(if)g Fp(z)f Fo(2)c Fp(D)1893 2645 y Fm(1)1978 2630 y Ft(\))33 b(then)h(1)p Fp(=z)27 b Fo(\000)22 b Fp(m)29 b Fo(2)f Fp(D)2841 2645 y Fm(1)2959 2630 y Ft(for)33 b(all)g Fp(m)28 b Fo(\025)g Ft(1)20 b(;)213 2750 y Fo(\017)50 b Ft(if)33 b Fp(z)f Fo(2)c Fp(D)658 2765 y Fs(1)704 2750 y Ft(,)k(then)i(1)p Fp(=z)27 b Fo(\000)c Ft(1)k Fo(2)h Fp(D)1517 2765 y Fs(0)1595 2750 y Ft(and)34 b(1)p Fp(=z)27 b Fo(\000)22 b Fp(m)28 b Fo(2)h Fp(D)2353 2765 y Fm(1)2471 2750 y Ft(for)k(all)g Fp(m)28 b Fo(\025)g Ft(2.)313 2870 y(Observ)m(e)k(that)f Fp(S)6 b(D)31 b Ft(=)d Fo([)1258 2885 y Fn(m)p Fm(\025)p Fs(1)1436 2870 y Ft(\(\001)17 b(+)i Fp(m)p Ft(\))28 b Fp(;)i Ft(where)i(the)g(domains)e(ha)m(v)m(e)i(disjoin)m(t)f(in)m (terior.)113 2989 y(Th)m(us,)j(for)g Fp(z)e Fo(2)c Fp(D)s Ft(,)34 b(w)m(e)f(de\014ne)1220 3218 y Fp(A)p Ft(\()p Fp(z)t Ft(\))27 b(=)1568 3151 y(1)p 1567 3195 51 4 v 1567 3286 a Fp(z)1652 3218 y Fo(\000)c Fp(m)28 b Ft(=)g(\()p Fp(g)t Ft(\()p Fp(m)p Ft(\)\))2267 3177 y Fm(\000)p Fs(1)2395 3218 y Fo(\001)21 b Fp(z)33 b(;)901 b Ft(\(3)p Fp(:)p Ft(1\))113 3509 y(\(w)m(e)34 b(recall)f(that)f Fp(g)t Ft(\()p Fp(m)p Ft(\))27 b(=)1129 3368 y Fk(\022)1219 3449 y Ft(0)119 b(1)1219 3568 y(1)100 b Fp(m)1473 3368 y Fk(\023)1546 3509 y Ft(,)33 b Fp(m)28 b Fo(\025)g Ft(1\))33 b(where)h Fp(m)28 b Fo(\025)g Ft(1)33 b(is)g(the)h(unique)g(in)m(teger) f(suc)m(h)113 3688 y(that)g Fp(A)p Ft(\()p Fp(z)t Ft(\))28 b Fo(2)g Ft(\001)g Fp(;)99 b Fo(j)p Fp(A)p Ft(\()p Fp(z)t Ft(\))p Fo(j)27 b Fp(<)h Ft(1)g Fp(:)k Ft(Iterating)h(from)g Fp(z)2116 3703 y Fs(0)2188 3688 y Fo(2)28 b Fp(D)s Ft(,)33 b(w)m(e)h(de\014ne)1344 3875 y Fp(z)1390 3890 y Fn(i)p Fs(+1)1552 3875 y Ft(=)28 b Fp(A)p Ft(\()p Fp(z)1817 3890 y Fn(i)1850 3875 y Ft(\))f(=)h Fp(A)2096 3834 y Fn(i)p Fs(+1)2230 3875 y Ft(\()p Fp(z)2315 3890 y Fs(0)2360 3875 y Ft(\))1053 b(\(3)p Fp(:)p Ft(2\))113 4063 y(as)46 b(long)g(as)g Fp(z)658 4078 y Fn(i)740 4063 y Ft(=)j Fp(A)941 4027 y Fn(i)974 4063 y Ft(\()p Fp(z)t Ft(\))g Fo(2)g Fp(D)s Ft(.)82 b(The)46 b(iteration)f(pro)s(cess)i(stops)f(when) h(one)f(of)h(the)f(t)m(w)m(o)113 4182 y(follo)m(wing)33 b(conditions)h(is)f(v)m(eri\014ed)h(:)213 4302 y Fo(\017)50 b Fp(z)359 4317 y Fn(l)417 4302 y Ft(=)28 b(0)33 b(for)h(some)f Fp(l)c Fo(\025)f Ft(0)20 b(;)33 b(this)g(happ)s(ens)i(if)e(and)h(only)e (if)i Fp(z)2529 4317 y Fs(0)2601 4302 y Fo(2)28 b Fq(Q)12 b Ft(,)213 4421 y Fo(\017)50 b Fp(z)359 4436 y Fn(l)428 4421 y Fp(=)-61 b Fo(2)28 b Ft(\()p Fp(D)d Fo([)e(f)p Ft(0)p Fo(g)p Ft(\),)32 b(for)i(some)e Fp(l)e Fo(\025)e Ft(0)19 b(;)33 b(this)h(happ)s(ens)g(if)g(and)f(only)g(if)g Fp(z)2918 4436 y Fs(0)3002 4421 y Fp(=)-61 b Fo(2)28 b Fq(R)5 b Ft(.)313 4541 y(F)-8 b(or)32 b(all)h(0)27 b Fo(\024)h Fp(i)g(<)h(l)r Ft(,)j(w)m(e)i(will)e(denote)i Fp(m)1815 4556 y Fn(i)p Fs(+1)1983 4541 y Ft(the)f(in)m(teger)h(suc)m (h)h(that)1231 4774 y Fp(z)1277 4789 y Fn(i)p Fs(+1)1439 4774 y Ft(=)1571 4706 y(1)p 1556 4751 80 4 v 1556 4842 a Fp(z)1602 4857 y Fn(i)1670 4774 y Fo(\000)22 b Fp(m)1856 4789 y Fn(i)p Fs(+1)2008 4774 y Fp(;)44 b(m)2167 4789 y Fn(i)p Fs(+1)2329 4774 y Fo(\025)28 b Ft(1)f Fp(:)913 b Ft(\(3)p Fp(:)p Ft(3\))313 5103 y Fr(3.1.2)40 b Ft(Let)773 4963 y Fk(\022)863 5044 y Fp(p)913 5059 y Fn(i)p Fm(\000)p Fs(1)1148 5044 y Fp(p)1198 5059 y Fn(i)866 5163 y Fp(q)910 5178 y Fn(i)p Fm(\000)p Fs(1)1151 5163 y Fp(q)1195 5178 y Fn(i)1248 4963 y Fk(\023)1359 5103 y Ft(=)1475 4963 y Fk(\022)1565 5044 y Ft(0)141 b(1)1565 5163 y(1)100 b Fp(m)1802 5178 y Fs(1)1864 4963 y Fk(\023)1954 5103 y Fp(:)17 b(:)g(:)2086 4963 y Fk(\022)2176 5044 y Ft(0)135 b(1)2176 5163 y(1)100 b Fp(m)2413 5178 y Fn(i)2463 4963 y Fk(\023)2575 5103 y Fo(2)39 b(M)17 b Fp(;)54 b Ft(0)38 b Fo(\024)h Fp(i)g Fo(\024)g Fp(l)h(:)g Ft(Then)113 5277 y(one)34 b(has)g(the)f(same)g(recurrence)i(relations)e(as)g(for)h(the)g (real)f(con)m(tin)m(ued)h(fraction)1393 5440 y Fp(p)1443 5455 y Fn(i)p Fs(+1)1605 5440 y Ft(=)28 b Fp(m)1797 5455 y Fn(i)p Fs(+1)1931 5440 y Fp(p)1981 5455 y Fn(i)2037 5440 y Ft(+)23 b Fp(p)2187 5455 y Fn(i)p Fm(\000)p Fs(1)2349 5440 y Fp(;)1399 5589 y(q)1443 5604 y Fn(i)p Fs(+1)1605 5589 y Ft(=)28 b Fp(m)1797 5604 y Fn(i)p Fs(+1)1931 5589 y Fp(q)1975 5604 y Fn(i)2031 5589 y Ft(+)23 b Fp(q)2175 5604 y Fn(i)p Fm(\000)p Fs(1)2338 5589 y Fp(;)3452 5515 y Ft(\(3)p Fp(:)p Ft(4\))p eop %%Page: 13 13 13 12 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(13)113 383 y(with)33 b(initial)f(data)h Fp(p)905 398 y Fm(\000)p Fs(1)1040 383 y Ft(=)28 b Fp(q)1189 398 y Fs(0)1262 383 y Ft(=)g(1)33 b(and)h Fp(p)1694 398 y Fs(0)1766 383 y Ft(=)28 b Fp(q)1915 398 y Fm(\000)p Fs(1)2050 383 y Ft(=)h(0.)43 b(Moreo)m(v)m(er)1025 690 y Fp(z)1071 705 y Fs(0)1144 690 y Ft(=)1261 623 y Fp(p)1311 638 y Fn(i)p Fm(\000)p Fs(1)1446 623 y Fp(z)1492 638 y Fn(i)1548 623 y Ft(+)23 b Fp(p)1698 638 y Fn(i)p 1261 667 470 4 v 1267 759 a Fp(q)1311 774 y Fn(i)p Fm(\000)p Fs(1)1446 759 y Fp(z)1492 774 y Fn(i)1548 759 y Ft(+)g Fp(q)1692 774 y Fn(i)1759 690 y Fp(;)100 b(z)1933 705 y Fn(i)1994 690 y Ft(=)2213 623 y Fp(p)2263 638 y Fn(i)2318 623 y Fo(\000)23 b Fp(q)2462 638 y Fn(i)2496 623 y Fp(z)2542 638 y Fs(0)p 2111 667 578 4 v 2111 759 a Fp(q)2155 774 y Fn(i)p Fm(\000)p Fs(1)2291 759 y Fp(z)2337 774 y Fs(0)2404 759 y Fo(\000)f Fp(p)2553 774 y Fn(i)p Fm(\000)p Fs(1)2717 690 y Fp(;)707 b Ft(\(3)p Fp(:)p Ft(5\))113 1010 y(and)34 b(if)f(one)h(p)s(oses)1130 1249 y Fp(\014)1186 1264 y Fn(i)1219 1249 y Ft(\()p Fp(z)1304 1264 y Fs(0)1349 1249 y Ft(\))27 b(=)1575 1124 y Fn(i)1526 1154 y Fk(Y)1520 1366 y Fn(j)t Fs(=0)1675 1249 y Fp(z)1721 1264 y Fn(j)1791 1249 y Ft(=)h(\()p Fo(\000)p Ft(1\))2101 1208 y Fn(i)2134 1249 y Ft(\()p Fp(q)2217 1264 y Fn(i)2250 1249 y Fp(z)2296 1264 y Fs(0)2364 1249 y Fo(\000)22 b Fp(p)2513 1264 y Fn(i)2546 1249 y Ft(\))28 b Fp(;)811 b Ft(\(3)p Fp(:)p Ft(6\))113 1568 y(then)1063 1738 y Fp(\014)1119 1753 y Fn(i)1153 1738 y Ft(\()p Fp(z)1238 1753 y Fs(0)1282 1738 y Ft(\))28 b(=)1656 1671 y Fp(z)1702 1686 y Fn(i)p 1466 1715 459 4 v 1466 1807 a Fp(q)1510 1822 y Fn(i)1566 1807 y Ft(+)22 b Fp(q)1709 1822 y Fn(i)p Fm(\000)p Fs(1)1845 1807 y Fp(z)1891 1822 y Fn(i)1964 1738 y Ft(=)2336 1671 y(1)p 2081 1715 559 4 v 2081 1807 a Fp(q)2125 1822 y Fn(i)p Fs(+1)2282 1807 y Ft(+)h Fp(q)2426 1822 y Fn(i)2459 1807 y Fp(z)2505 1822 y Fn(i)p Fs(+1)2679 1738 y Fp(:)745 b Ft(\(3)p Fp(:)p Ft(7\))113 2008 y(Finally)32 b(one)i(has)651 2273 y(\()p Fo(\000)p Ft(1\))856 2232 y Fn(i)906 2273 y Fo(=)p Fp(m)17 b(z)1128 2288 y Fs(0)1200 2273 y Ft(=)29 b Fo(j)p Fp(\014)1390 2288 y Fn(i)p Fm(\000)p Fs(1)1525 2273 y Ft(\()p Fp(z)1610 2288 y Fs(0)1655 2273 y Ft(\))p Fo(j)1722 2232 y Fs(2)1782 2273 y Fo(=)p Fp(m)17 b(z)2004 2288 y Fn(i)2065 2273 y Ft(=)29 b Fo(j)p Fp(q)2243 2288 y Fn(i)2298 2273 y Ft(+)23 b Fp(q)2442 2288 y Fn(i)p Fm(\000)p Fs(1)2577 2273 y Fp(z)2623 2288 y Fn(i)2657 2273 y Fo(j)2685 2232 y Fm(\000)p Fs(2)2808 2273 y Fo(=)p Fp(m)17 b(z)3030 2288 y Fn(i)3091 2273 y Fp(;)1024 2417 y(dz)1122 2432 y Fn(i)p 1018 2461 143 4 v 1018 2553 a Fp(dz)1116 2568 y Fs(0)1200 2484 y Ft(=)29 b(\()p Fo(\000)p Ft(1\))1511 2443 y Fn(i)1544 2484 y Ft(\()p Fp(\014)1639 2499 y Fn(i)p Fm(\000)p Fs(1)1774 2484 y Ft(\()p Fp(z)1859 2499 y Fs(0)1904 2484 y Ft(\)\))1982 2443 y Fm(\000)p Fs(2)2116 2484 y Ft(=)f(\()p Fo(\000)p Ft(1\))2426 2443 y Fn(i)2459 2484 y Ft(\()p Fp(q)2542 2499 y Fn(i)2597 2484 y Ft(+)23 b Fp(q)2741 2499 y Fn(i)p Fm(\000)p Fs(1)2877 2484 y Fp(z)2923 2499 y Fn(i)2956 2484 y Ft(\))2995 2443 y Fs(2)3067 2484 y Fp(:)113 2830 y Ft(Observ)m(e)33 b(that,)e(as)h Fo(j)p Fp(z)931 2845 y Fn(i)p Fs(+1)1084 2830 y Ft(+)19 b(1)p Fo(j)27 b(\025)h Ft(1)j(and)h Fo(<)p Fp(e)17 b(z)1844 2845 y Fn(i)p Fs(+1)2006 2830 y Fo(\025)2123 2733 y Fm(p)p 2189 2733 40 4 v 2189 2791 a Fs(3)p 2123 2807 106 4 v 2156 2865 a(2)2259 2830 y Fo(\000)i Ft(1)31 b(for)h Fp(i)c(<)g(l)r Ft(,)j(w)m(e)h(ha)m(v)m(e)g(from)f(\(3.7\))992 3181 y Fo(j)p Fp(\014)1076 3196 y Fn(i)1109 3181 y Ft(\()p Fp(z)1194 3196 y Fs(0)1239 3181 y Ft(\))p Fo(j)c(\024)h Fp(q)1486 3138 y Fm(\000)p Fs(1)1482 3209 y Fn(i)p Fs(+1)1617 3181 y Ft([cos)16 b Fp(\031)t(=)p Ft(12])2033 3140 y Fm(\000)p Fs(1)2166 3181 y Ft(=)2344 3114 y(2)2394 3031 y Fo(p)p 2477 3031 50 4 v 83 x Ft(2)p 2283 3158 305 4 v 2283 3260 a(1)22 b(+)2455 3178 y Fo(p)p 2538 3178 50 4 v 82 x Ft(3)2600 3181 y Fp(q)2648 3138 y Fm(\000)p Fs(1)2644 3209 y Fn(i)p Fs(+1)3399 3181 y Ft(\(3)p Fp(:)p Ft(8)p Fp(a)p Ft(\))113 3507 y(and,)34 b(as)f Fp(q)501 3522 y Fn(i)562 3507 y Fo(\024)c Fp(q)712 3522 y Fn(i)p Fs(+1)846 3507 y Ft(,)k Fo(j)p Fp(z)981 3522 y Fn(i)p Fs(+1)1115 3507 y Fo(j)27 b(\024)i Ft(1,)1508 3822 y Fo(j)p Fp(\014)1592 3837 y Fn(i)1625 3822 y Ft(\()p Fp(z)1710 3837 y Fs(0)1755 3822 y Ft(\))p Fo(j)e(\025)1966 3755 y Ft(1)p 1966 3799 V 1966 3890 a(2)2028 3822 y Fp(q)2076 3779 y Fm(\000)p Fs(1)2072 3850 y Fn(i)p Fs(+1)2234 3822 y Fp(:)1147 b Ft(\(3)p Fp(:)p Ft(8)p Fp(b)p Ft(\))p eop %%Page: 14 14 14 13 bop 113 144 a Ft(14)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y Fr(3.2)39 b(The)e(op)s(erator)h Fp(T)14 b Fr(,)38 b(its)g(sp)s(ectral)e(radius)i(and)f(the)g(sum)h(o)m (v)m(er)h(the)e(monoid)313 532 y(3.2.1)c Ft(Let)h Fp(\015)812 547 y Fs(1)884 532 y Fp(>)28 b(\015)1041 547 y Fs(0)1113 532 y Fp(>)g Fo(\000)p Ft(1,)33 b Fp(I)j Ft(=)28 b([)p Fp(\015)1671 547 y Fs(0)1715 532 y Fp(;)17 b(\015)1812 547 y Fs(1)1855 532 y Ft(],)33 b Fp(')28 b Fo(2)g(O)2213 496 y Fs(1)2257 532 y Ft(\()p 2296 452 72 4 v Fq(C)48 b Fo(n)22 b Fp(I)8 b Ft(\).)313 652 y(F)-8 b(or)32 b(all)h Fp(m)28 b Fo(\025)g Ft(1)33 b(one)h(has)f(\(compare)g(with)g (\(2.10\)\))753 948 y Fp(L)821 966 y Fn(g)s Fs(\()p Fn(m)p Fs(\))1000 948 y Fp(')p Ft(\()p Fp(z)t Ft(\))28 b(=)g Fo(\000)p Fp(z)1471 808 y Fk(\022)1545 948 y Fp(')1627 808 y Fk(\022)1712 881 y Ft(1)p 1712 925 51 4 v 1712 1016 a Fp(z)1796 948 y Fo(\000)23 b Fp(m)1983 808 y Fk(\023)2079 948 y Fo(\000)g Fp(')p Ft(\()p Fo(\000)p Fp(m)p Ft(\))2486 808 y Fk(\023)2582 948 y Ft(+)f Fp(')2746 907 y Fm(0)2775 948 y Ft(\()p Fo(\000)p Fp(m)p Ft(\))1221 1238 y(=)28 b Fo(\000)p Fp(z)1453 1197 y Fm(\000)p Fs(1)1578 1103 y Fk(Z)1678 1127 y Fs(1)1634 1329 y(0)1739 1238 y Fp(')1804 1197 y Fm(00)1872 1098 y Fk(\022)1945 1238 y Fo(\000)p Fp(m)23 b Ft(+)2251 1171 y Fp(t)p 2244 1216 V 2244 1307 a(z)2307 1098 y Fk(\023)2396 1238 y Ft(\(1)f Fo(\000)h Fp(t)p Ft(\))p Fp(dt)k(;)3452 1093 y Ft(\(3)p Fp(:)p Ft(9\))113 1535 y(pro)m(vided)38 b(that)f Fp(z)51 b(=)-61 b Fo(2)35 b Ft([0)p Fp(;)17 b Ft(1)p Fp(=)p Ft(\()p Fp(\015)1248 1550 y Fs(0)1315 1535 y Ft(+)26 b Fp(m)p Ft(\)],)38 b(and)g(an)f(ev)m (en)h(simpler)f(form)m(ula)g(for)g(the)h(action)f(at)113 1654 y(the)d(lev)m(el)f(of)g(second)i(deriv)-6 b(ativ)m(es)1185 1926 y Fp(L)1253 1874 y Fs(\(3\))1253 1959 y Fn(g)s Fs(\()p Fn(m)p Fs(\))1432 1926 y Fp( )t Ft(\()p Fp(z)t Ft(\))28 b(=)g Fo(\000)p Fp(z)1889 1884 y Fm(\000)p Fs(3)1997 1926 y Fp( )2082 1785 y Fk(\022)2168 1858 y Ft(1)p 2168 1903 V 2168 1994 a Fp(z)2252 1926 y Fo(\000)23 b Fp(m)2439 1785 y Fk(\023)2557 1926 y Fp(:)817 b Ft(\(3)p Fp(:)p Ft(10\))313 2245 y Fr(3.2.2)33 b Ft(Let)h Fp(")28 b(>)g Ft(0,)454 2481 y Fp(U)522 2496 y Fn(")592 2481 y Ft(=)g Fo(f)p Fp(z)k Fo(2)c Fq(C)37 b Fp(;)17 b Fo(<)p Fp(e)22 b(z)32 b Fo(\024)d Fp(\015)1423 2496 y Fs(0)1489 2481 y Fo(\000)23 b Fp(")17 b Ft(or)33 b Fo(<)p Fp(e)17 b(z)32 b Fo(\025)c Fp(\015)2144 2496 y Fs(1)2211 2481 y Ft(+)22 b Fp(")17 b Ft(or)g Fo(j)g(=)p Fp(m)f(z)t Fo(j)28 b(\025)g Fp(")p Fo(g)g Fp(:)340 b Ft(\(3)p Fp(:)p Ft(11\))113 2718 y(W)-8 b(e)34 b(ha)m(v)m(e)g(the)f(follo)m(wing)113 2896 y Fr(Prop)s(osition)39 b(3.1)57 b Fe(Let)34 b Fp(\015)1167 2911 y Fs(0)1211 2896 y Fp(;)17 b(\015)1308 2911 y Fs(1)1385 2896 y Fe(as)33 b(ab)s(o)m(v)m(e,)g Fp(I)j Ft(=)28 b([)p Fp(\015)2082 2911 y Fs(0)2126 2896 y Fp(;)17 b(\015)2223 2911 y Fs(1)2266 2896 y Ft(])p Fe(,)33 b Fp(J)k Ft(=)28 b([0)p Fp(;)17 b Ft(1)p Fp(=)p Ft(\(1)k(+)h Fp(\015)3036 2911 y Fs(0)3080 2896 y Ft(\)])p Fe(.)185 3016 y(1.)50 b(F)-8 b(or)22 b(all)g Fp(')28 b Fo(2)g(O)878 2980 y Fs(1)923 3016 y Ft(\()p 962 2935 72 4 v Fq(C)g Fo(n)r Fp(I)8 b Ft(\))p Fe(,)23 b(the)h(series)1648 2941 y Fk(P)1754 3046 y Fn(m)p Fm(\025)p Fs(1)1948 3016 y Fp(L)2016 3034 y Fn(g)s Fs(\()p Fn(m)p Fs(\))2195 3016 y Fp(')f Fe(con)m(v)m(erges)h (uniformly)e(on)h(compact)313 3157 y(subsets)36 b Fp(K)41 b Fe(of)p 900 3077 V 35 w Fq(C)49 b Fo(n)23 b Fp(J)44 b Fe(to)34 b(a)g(function)i Fp(T)14 b(')30 b Fo(2)g(O)2111 3121 y Fs(1)2156 3157 y Ft(\()p 2195 3077 V Fq(C)49 b Fo(n)23 b Fp(J)10 b Ft(\))34 b Fe(and)h(there)g(exists)f Fp(")d(>)f Ft(0)k Fe(and)313 3277 y Fp(C)384 3292 y Fn(K)488 3277 y Fp(>)28 b Ft(0)34 b Fe(suc)m(h)g(that)f(:)44 b Ft(sup)1339 3301 y Fn(K)1432 3277 y Fo(j)p Fp(T)14 b(')p Fo(j)28 b(\024)g Fp(C)1829 3292 y Fn(K)1922 3277 y Ft(sup)2072 3301 y Fn(U)2126 3311 y Fj(")2187 3277 y Fo(j)p Fp(')p Fo(j)f Fp(:)185 3425 y Fe(2.)50 b(F)-8 b(or)24 b(all)g Fp( )31 b Fo(2)d(O)885 3389 y Fs(3)930 3425 y Ft(\()p 969 3345 V Fq(C)j Fo(n)5 b Fp(I)j Ft(\))24 b Fe(the)h(series)1638 3350 y Fk(P)1743 3455 y Fn(m)p Fm(\025)p Fs(1)1937 3425 y Fp(L)2005 3373 y Fs(\(3\))2005 3459 y Fn(g)s Fs(\()p Fn(m)p Fs(\))2185 3425 y Fp( )j Fe(con)m(v)m(erges)e(uniformly)d(on)i (compact)313 3575 y(subsets)40 b Fp(K)45 b Fe(of)p 912 3494 V 39 w Fq(C)52 b Fo(n)26 b Fp(J)48 b Fe(to)38 b(a)h(function)h Fp(T)1873 3539 y Fs(\(3\))1980 3575 y Fp( )g Fo(2)d(O)2270 3539 y Fs(1)2315 3575 y Ft(\()p 2354 3494 V Fq(C)51 b Fo(n)26 b Fp(J)10 b Ft(\))38 b Fe(and)h(there)h(exists)e Fp(")g(>)f Ft(0)313 3695 y Fe(and)c Fp(C)577 3710 y Fn(K)682 3695 y Fp(>)28 b Ft(0)33 b Fe(suc)m(h)i(that)e(:)44 b Ft(sup)1533 3719 y Fn(K)1626 3695 y Fo(j)p Fp(T)1726 3658 y Fs(\(3\))1833 3695 y Fp( )t Fo(j)27 b(\024)h Fp(C)2133 3710 y Fn(K)2226 3695 y Ft(sup)2376 3719 y Fn(U)2430 3729 y Fj(")2491 3695 y Fo(j)p Fp( )t Fo(j)e Fp(:)185 3827 y Fe(3.)50 b(F)-8 b(or)32 b(all)h Fp(')28 b Fo(2)g(O)899 3791 y Fs(1)943 3827 y Ft(\()p 982 3747 V Fq(C)48 b Fo(n)22 b Fp(I)8 b Ft(\))33 b Fe(one)g(has)67 b Fp(T)1737 3791 y Fs(\(3\))1844 3827 y Fp(')1909 3791 y Fm(00)1988 3827 y Ft(=)28 b(\()p Fp(T)14 b(')p Ft(\))2308 3791 y Fm(00)2387 3827 y Fp(:)113 4036 y Fl(Pr)-5 b(o)g(of.)34 b Ft(Let)g Fp(")28 b(>)g Ft(0.)44 b(There)34 b(exist)f Fp(c)1458 4051 y Fn(")1528 4036 y Fp(>)28 b Ft(0)33 b(suc)m(h)i(that)e(for)g Fp( )f Fo(2)c(O)2582 4000 y Fs(3)2626 4036 y Ft(\()p 2665 3955 V Fq(C)48 b Fo(n)22 b Fp(I)8 b Ft(\),)32 b Fp(z)h Fo(2)28 b Fp(U)3223 4051 y Fn(")3299 4036 y Ft(one)33 b(has)1362 4272 y Fo(j)p Fp( )t Ft(\()p Fp(z)t Ft(\))p Fo(j)27 b(\024)h Fp(c)1790 4287 y Fn(")1833 4272 y Fo(j)p Fp(z)t Fo(j)1939 4231 y Fm(\000)p Fs(3)2062 4272 y Ft(sup)2091 4356 y Fn(U)2145 4366 y Fj(")2229 4272 y Fo(j)p Fp( )t Fo(j)e Fp(:)113 4591 y Ft(If)41 b Fp(K)47 b Ft(is)40 b(a)g(compact)g(subset)h(of)p 1387 4511 V 40 w Fq(C)53 b Fo(n)27 b Fp(J)50 b Ft(there)41 b(exists)f Fp(")f Ft(=)h Fp(")2460 4606 y Fn(K)2578 4591 y Ft(suc)m(h)h(that)f(1)p Fp(=z)31 b Fo(\000)d Fp(m)39 b Fo(2)h Fp(U)3614 4606 y Fn(")113 4711 y Ft(for)f(all)f Fp(m)g Fo(\025)f Ft(1)i(and)g Fp(z)j Fo(2)37 b Fp(K)7 b Ft(.)61 b(Moreo)m(v)m(er)39 b(there)g(exists)g Fp(c)2341 4726 y Fn(K)2455 4711 y Fp(>)e Ft(0)i(and)g Fp(M)48 b Ft(=)37 b Fp(M)11 b Ft(\()p Fp(\015)3315 4726 y Fs(0)3359 4711 y Ft(\),)39 b(suc)m(h)113 4830 y(that)d Fo(j)p Ft(1)p Fp(=z)27 b Fo(\000)e Fp(m)p Fo(j)750 4794 y Fm(\000)p Fs(1)888 4830 y Fo(\024)32 b Fp(c)1040 4845 y Fn(K)1117 4830 y Fp(m)1204 4794 y Fm(\000)p Fs(1)1347 4830 y Ft(for)k(all)f Fp(z)h Fo(2)c Fp(K)7 b Ft(,)36 b Fp(m)c Fo(\025)g Fp(M)11 b Ft(.)51 b(Consequen)m(tly)-8 b(,)36 b(for)f Fp(z)i Fo(2)32 b Fp(K)42 b Ft(and)113 4950 y Fp( )32 b Fo(2)c(O)386 4914 y Fs(3)430 4950 y Ft(\()p 469 4869 V Fq(C)48 b Fo(n)22 b Fp(I)8 b Ft(\))33 b(w)m(e)g(ha)m(v)m(e)369 5151 y Fk(X)355 5365 y Fn(m)p Fm(\025)p Fs(1)544 5102 y Fk(\014)544 5161 y(\014)544 5221 y(\014)544 5281 y(\014)577 5246 y Fp( )662 5106 y Fk(\022)748 5179 y Ft(1)p 748 5223 51 4 v 748 5314 a Fp(z)832 5246 y Fo(\000)23 b Fp(m)1019 5106 y Fk(\023)1093 5102 y(\014)1093 5161 y(\014)1093 5221 y(\014)1093 5281 y(\014)1154 5246 y Fo(\024)28 b Fp(c)1302 5205 y Fm(0)1302 5271 y Fn(K)1395 5246 y Ft(sup)1424 5330 y Fn(U)1478 5340 y Fj(")1562 5246 y Fo(j)p Fp( )t Fo(j)98 b Fp(;)116 b Ft(with)100 b Fp(c)2266 5205 y Fm(0)2266 5271 y Fn(K)2370 5246 y Ft(=)28 b Fp(M)33 b Ft(+)23 b Fp(c)2748 5261 y Fn(")2790 5246 y Fp(c)2833 5205 y Fs(3)2833 5271 y Fn(K)2963 5151 y Fk(X)2927 5365 y Fn(m)p Fm(\025)p Fn(M)3160 5246 y Fp(m)3247 5205 y Fm(\000)p Fs(3)3388 5246 y Fp(;)113 5584 y Ft(whic)m(h)34 b(pro)m(v)m(es)g(the)g(second)h (part)e(of)g(the)h(prop)s(osition.)p eop %%Page: 15 15 15 14 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(15)313 383 y(By)37 b(in)m(tegrating)i(t)m(wice)f(from)g Fo(1)h Ft(one)g(deduces)i(that)d(for)h Fp(')e Fo(2)g(O)2860 347 y Fs(1)2905 383 y Ft(\()p 2944 303 72 4 v Fq(C)51 b Fo(n)26 b Fp(I)8 b Ft(\))38 b(the)h(series)113 428 y Fk(P)219 533 y Fn(m)p Fm(\025)p Fs(1)413 503 y Fp(L)481 521 y Fn(g)s Fs(\()p Fn(m)p Fs(\))660 503 y Fp(')49 b Ft(con)m(v)m(erges)h(uniformly)e(on)h(compact)f(subsets)i(of)p 2763 422 V 49 w Fq(C)59 b Fo(n)32 b Fp(J)59 b Ft(to)48 b(a)h(function)113 644 y Fp(T)14 b(')39 b Fo(2)h(O)477 608 y Fs(1)522 644 y Ft(\()p 561 563 V Fq(C)52 b Fo(n)26 b Fp(J)10 b Ft(\).)64 b(Moreo)m(v)m(er)40 b(for)h(an)m(y)f(compact)f Fp(K)45 b Fo(\032)p 2382 563 V 40 w Fq(C)52 b Fo(n)27 b Fp(J)49 b Ft(there)41 b(exist)f Fp(c)3209 659 y Fn(")3290 644 y Fp(>)f Ft(0)h(and)113 764 y Fp(c)156 779 y Fn(K)276 764 y Fp(>)i Ft(0)g(suc)m(h)i(that)d(for)i Fp(z)k Fo(2)42 b Fp(K)7 b Ft(,)44 b Fo(j)p Ft(\()p Fp(T)14 b(')p Ft(\))1715 727 y Fm(00)1765 764 y Ft(\()p Fp(z)t Ft(\))p Fo(j)42 b Ft(=)h Fo(j)p Ft(\()p Fp(T)2222 727 y Fs(\(3\))2329 764 y Fp(')2394 727 y Fm(00)2445 764 y Ft(\)\()p Fp(z)t Ft(\))p Fo(j)f(\024)g Fp(c)2844 779 y Fn(K)2921 764 y Fo(j)p Fp(z)t Fo(j)3027 727 y Fm(\000)p Fs(3)3151 764 y Ft(sup)3301 788 y Fn(U)3355 798 y Fj(")3415 764 y Fo(j)p Fp(')3508 727 y Fm(00)3559 764 y Fo(j)g Fp(:)113 883 y Ft(On)d(the)f(other)g(hand)g(there)h(exists)e Fp(c)1560 847 y Fm(00)1560 908 y Fn(")1647 883 y Fp(>)e Ft(0)j(suc)m(h)h(that,)f (sup)2475 907 y Fn(U)2529 917 y Fj(")2590 883 y Fo(j)p Fp(')2683 847 y Fm(00)2733 883 y Fo(j)d(\024)h Fp(c)2952 847 y Fm(00)2952 908 y Fn(")3020 883 y Ft(sup)3170 907 y Fn(U)3224 920 y Fj("=)p Fi(2)3355 883 y Fo(j)p Fp(')p Fo(j)h Ft(\(b)m(y)113 1024 y(Cauc)m(h)m(y's)j(form)m(ula\),)g(hence)h (w)m(e)e(get)g(for)h Fp(z)i Fo(2)d Fp(K)7 b Ft(,)40 b Fo(j)p Fp(T)14 b(')p Ft(\()p Fp(z)t Ft(\))p Fo(j)37 b(\024)k Ft(~)-53 b Fp(c)2618 1039 y Fn(K)2695 1024 y Fo(j)p Fp(z)t Fo(j)2801 988 y Fm(\000)p Fs(1)2925 1024 y Ft(sup)3075 1049 y Fn(U)3129 1062 y Fj("=)p Fi(2)3260 1024 y Fo(j)p Fp(')p Fo(j)37 b Fp(:)i Ft(The)113 1144 y(third)34 b(part)f(of)g(the)h (Prop)s(osition)e(is)h(immediate.)1595 b Fd(\003)313 1344 y Fr(3.2.3)53 b Ft(The)g(op)s(en)g(set)p 1260 1264 V 53 w Fq(C)61 b Fo(n)35 b Ft([0)p Fp(;)17 b Ft(1])51 b(is)i(an)g(h)m(yp)s(erb)s(olic)g(Riemann)f(surface)i(whic)m(h)g(is)113 1464 y(naturally)43 b(equipp)s(ed)i(with)e(a)h(P)m(oincar)m(\023)-47 b(e)44 b(metric.)74 b(The)45 b(follo)m(wing)e(w)m(ell-kno)m(wn)h(fact)g (will)113 1583 y(b)s(e)d(crucial)g(for)f(the)h(pro)s(of)g(of)f(Lemma)f (3.2)h(:)58 b(giv)m(en)41 b(t)m(w)m(o)f(h)m(yp)s(erb)s(olic)h(Riemann)f (surfaces)113 1703 y Fp(M)5 b(;)17 b(N)53 b Ft(and)43 b(an)f(analytic)g(map)f Fp(f)70 b Ft(:)60 b Fp(M)53 b Fo(!)43 b Fp(N)53 b Ft(either)43 b(its)f(di\013eren)m(tial)g Fp(d)-17 b(f)53 b Ft(con)m(tracts)43 b(the)113 1823 y(h)m(yp)s(erb)s (olic)49 b(metric)e(or)h Fp(f)59 b Ft(is)48 b(a)g(surjectiv)m(e)i(lo)s (cal)d(isometry)-8 b(.)87 b(In)49 b(what)f(follo)m(ws)g(w)m(e)h(will) 113 1942 y(denote)34 b Fp(d)486 1957 y Fn(hy)s(per)732 1942 y Ft(the)g(P)m(oincar)m(\023)-47 b(e)33 b(metric)f(on)i(the)f (Riemann)g(surface)i(under)f(consideration.)313 2062 y(Giv)m(en)f Fp(\032)27 b(>)h Ft(0)34 b(w)m(e)f(denote)865 2275 y Fp(V)923 2290 y Fn(\032)969 2275 y Ft(\()p Fp(D)1090 2290 y Fm(1)1175 2275 y Ft(\))28 b(=)g Fo(f)p Fp(z)k Fo(2)p 1569 2194 V 28 w Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1])g Fp(;)31 b(d)2064 2290 y Fn(hy)s(per)2276 2275 y Ft(\()p Fp(z)t(;)17 b(D)2492 2290 y Fm(1)2577 2275 y Ft(\))27 b Fp(<)h(\032)p Fo(g)f Fp(:)497 b Ft(\(3)p Fp(:)p Ft(12\))113 2488 y(the)34 b Fp(\032)p Ft({neigh)m(b)s(orho)s(o)s (d)f(of)g Fp(D)1207 2503 y Fm(1)1326 2488 y Ft(in)p 1442 2407 V 33 w Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1].)113 2661 y Fr(Lemma)37 b(3.2)57 b Fe(Let)32 b Fp(\032)27 b Fo(\025)h Ft(0)p Fe(.)43 b(F)-8 b(or)31 b(all)g Fp(m)d Fo(\025)g Ft(1)j Fe(and)h Fp(z)h Fo(2)28 b Fp(V)2241 2676 y Fn(\032)2287 2661 y Ft(\()p Fp(D)2408 2676 y Fm(1)2493 2661 y Ft(\))j Fe(one)h(has)2932 2622 y Fs(1)p 2931 2638 41 4 v 2931 2696 a Fn(z)3003 2661 y Fo(\000)19 b Fp(m)28 b Fo(2)g Fp(V)3366 2676 y Fn(\032)3412 2661 y Ft(\()p Fp(D)3533 2676 y Fm(1)3618 2661 y Ft(\))113 2875 y Fl(Pr)-5 b(o)g(of.)36 b Ft(The)g(M\177)-50 b(obius)36 b(transformation)e Fp(z)i Fo(7!)1881 2836 y Fs(1)p 1880 2852 V 1880 2909 a Fn(z)1957 2875 y Fo(\000)24 b Fp(m)35 b Ft(maps)g Fp(D)2525 2890 y Fm(1)2645 2875 y Ft(in)m(to)g(itself)h(and)p 3289 2795 72 4 v 36 w Fq(C)49 b Fo(n)23 b Ft([0)p Fp(;)17 b Ft(1])113 3008 y(on)m(to)p 328 2927 V 24 w Fq(C)29 b Fo(n)s Ft([1)s Fo(\000)s Fp(m;)17 b Ft(+)p Fo(1)p Ft(])22 b(whic)m(h)i(is)f(con)m(tained)i(in)p 1889 2927 V 23 w Fq(C)k Fo(n)s Ft([0)p Fp(;)17 b Ft(1].)39 b(Th)m(us)24 b(it)f(decreases)j(the)e(h)m(yp)s(erb)s(olic)113 3128 y(distance)30 b(and)f(the)g Fp(\032)p Ft({neigh)m(b)s(orho)s(o)s(d)f (of)h(the)g(image)e(of)i Fp(D)2325 3143 y Fm(1)2439 3128 y Ft(w.r.t.)e(the)i(P)m(oincar)m(\023)-47 b(e)29 b(metric)e(of)p 113 3167 V 113 3247 a Fq(C)50 b Fo(n)24 b Ft([1)f Fo(\000)h Fp(m;)17 b Ft(+)p Fo(1)p Ft(])35 b(is)h(con)m(tained)g(in)m(to)g(the)f Fp(\032)p Ft({neigh)m(b)s(orho)s(o)s(d)h(of)g Fp(D)2719 3262 y Fm(1)2839 3247 y Ft(w.r.t.)e(the)i(P)m(oincar)m(\023)-47 b(e)113 3367 y(metric)33 b(of)p 537 3286 V 33 w Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1].)2647 b Fd(\003)313 3567 y Fr(3.2.4)54 b Ft(Once)h(the)g(existence)f(of)h(the)f(op)s (erator)f Fp(T)68 b Ft(on)54 b(holomorphic)g(functions)h(is)113 3687 y(established)38 b(\(Prop)s(osition)d(3.1\))g(one)i(can)g(ask)f (for)g(more)g(information)f(on)h(Hardy)g(spaces.)113 3806 y(The)f(results)g(w)m(e)g(will)e(pro)m(v)m(e)i(are)f(completely)f (analogous)h(to)g(those)h(obtained)g(for)f(the)h(real)113 3926 y(Brjuno)f(op)s(erator)f(in)g([MMY].)113 4099 y Fr(Prop)s(osition)49 b(3.3)58 b Fe(Let)43 b Fp(\032)f Fo(\025)h Ft(0)p Fe(.)72 b(There)43 b(exists)f Fp(c)2123 4063 y Fm(0)2123 4124 y Fn(\032)2212 4099 y Fp(>)i Ft(0)e Fe(suc)m(h)i(that)e(for)g(all)g Fp(r)k Fo(\025)d Ft(0)f Fe(and)113 4236 y Fp( )32 b Fo(2)c(O)386 4200 y Fs(3)430 4236 y Ft(\()p 469 4155 V Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))31 b Fe(one)j(has)788 4544 y Ft(sup)718 4632 y Fn(V)765 4642 y Fj(\032)807 4632 y Fs(\()p Fn(D)903 4642 y Ff(1)976 4632 y Fs(\))1023 4544 y Fo(j)p Ft(\(\()p Fp(T)1201 4503 y Fs(\(3\))1308 4544 y Ft(\))1347 4503 y Fn(r)1391 4544 y Fp( )t Ft(\)\()p Fp(z)t Ft(\))p Fo(j)26 b(\024)i Fp(c)1829 4503 y Fm(0)1829 4569 y Fn(\032)1892 4374 y Fk( )1983 4394 y Fo(p)p 2066 4394 50 4 v 83 x Ft(5)22 b Fo(\000)h Ft(1)p 1983 4521 305 4 v 2111 4613 a(2)2300 4374 y Fk(!)2379 4394 y Fn(r)2508 4544 y Ft(sup)2439 4632 y Fn(V)2486 4642 y Fj(\032)2527 4632 y Fs(\()p Fn(D)2623 4642 y Ff(1)2696 4632 y Fs(\))2744 4544 y Fo(j)p Fp( )t Ft(\()p Fp(z)t Ft(\))p Fo(j)k Fp(:)113 4880 y Fl(Pr)-5 b(o)g(of.)33 b Ft(First)f(of)g(all)f(note)i(that)e(if)h Fp(z)h Fo(2)28 b Fp(D)1681 4895 y Fm(1)1798 4880 y Ft(and)k Fp(g)f Fo(2)d(M)k Ft(then)g Fp(g)2592 4844 y Fm(\000)p Fs(1)2718 4880 y Fo(\001)19 b Fp(z)33 b Fo(2)28 b Fp(D)3020 4895 y Fm(1)3105 4880 y Ft(,)k(th)m(us)h(taking)113 5000 y(in)m(to)g(accoun)m(t)h(Lemma)e(3.2,)g(if)i Fp(z)e Fo(2)c Fp(V)1549 5015 y Fn(\032)1595 5000 y Ft(\()p Fp(D)1716 5015 y Fm(1)1801 5000 y Ft(\))33 b(then)h Fp(g)2152 4964 y Fm(\000)p Fs(1)2258 5000 y Fp(z)e Fo(2)d Fp(V)2489 5015 y Fn(\032)2535 5000 y Ft(\()p Fp(D)2656 5015 y Fm(1)2741 5000 y Ft(\).)313 5119 y(Giv)m(en)41 b(an)h(in)m(teger)f Fp(r)j Fo(\025)d Ft(0)g(w)m(e)h(denote)g Fo(M)1995 5083 y Fs(\()p Fn(r)r Fs(\))2142 5119 y Ft(the)g(set)f(of)h(elemen)m(ts)f Fp(g)k Ft(of)c Fo(M)g Ft(of)g(the)113 5239 y(form)33 b Fp(g)t Ft(\()p Fp(m)527 5254 y Fs(1)571 5239 y Ft(\))17 b Fp(:)g(:)g(:)d(g)t Ft(\()p Fp(m)937 5254 y Fn(r)981 5239 y Ft(\),)32 b Fp(m)1167 5254 y Fn(i)1228 5239 y Fo(\025)d Ft(1)k(for)g(all)f(1)c Fo(\024)g Fp(i)g Fo(\024)g Fp(r)s Ft(.)44 b(Then)34 b(one)g(has)812 5472 y(\()p Fp(T)923 5431 y Fs(\(3\))1029 5472 y Ft(\))1068 5431 y Fn(r)1112 5472 y Fp( )d Ft(=)1383 5377 y Fk(X)1314 5601 y Fn(g)s Fm(2M)1504 5581 y Fi(\()p Fj(r)r Fi(\))1613 5472 y Fp(L)1681 5431 y Fs(\(3\))1681 5496 y Fn(g)1788 5472 y Fp( )g Ft(=)2116 5377 y Fk(X)1989 5601 y Fn(g)2031 5581 y Ff(0)2057 5601 y Fm(2M)2205 5581 y Fi(\()p Fj(r)r Ff(\000)p Fi(1\))2403 5472 y Fp(L)2471 5420 y Fs(\(3\))2471 5501 y Fn(g)2513 5481 y Ff(0)2578 5472 y Ft(\()p Fp(T)2689 5431 y Fs(\(3\))2796 5472 y Fp( )t Ft(\))c Fp(:)p eop %%Page: 16 16 16 15 bop 113 144 a Ft(16)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 428 y(Let)31 b Fp(g)341 392 y Fm(0)396 428 y Ft(=)501 287 y Fk(\022)591 368 y Fp(a)644 332 y Fm(0)776 368 y Fp(b)819 332 y Fm(0)596 488 y Fp(c)639 452 y Fm(0)771 488 y Fp(d)823 452 y Fm(0)867 287 y Fk(\023)969 428 y Fo(2)d(M)1183 392 y Fs(\()p Fn(r)r Fm(\000)p Fs(1\))1421 428 y Ft(and)j Fp(z)h Fo(2)c Fp(V)1842 443 y Fn(\032)1889 428 y Ft(\()p Fp(D)2010 443 y Fm(1)2094 428 y Ft(\))20 b(;)31 b(let)f Fp(z)2403 392 y Fm(0)2459 428 y Ft(=)2578 389 y Fn(d)2619 359 y Ff(0)2646 389 y Fn(z)s Fm(\000)p Fn(b)2784 359 y Ff(0)p 2576 405 236 4 v 2576 462 a Fn(a)2619 442 y Ff(0)2647 462 y Fm(\000)p Fn(c)2745 442 y Ff(0)2771 462 y Fn(z)2824 428 y Ft(.)43 b(F)-8 b(or)30 b(all)g Fp(m)e Fo(\025)g Ft(1)i(one)113 602 y(has)846 618 y Fk(\022)936 698 y Fp(a)989 662 y Fm(0)1121 698 y Fp(b)1164 662 y Fm(0)941 818 y Fp(c)984 782 y Fm(0)1116 818 y Fp(d)1168 782 y Fm(0)1212 618 y Fk(\023)17 b(\022)1392 698 y Ft(0)119 b(1)1392 818 y(1)100 b Fp(m)1646 618 y Fk(\023)1747 758 y Ft(=)1852 618 y Fk(\022)2087 698 y Fp(b)2130 662 y Fm(0)2542 698 y Fp(d)2594 662 y Fm(0)1942 818 y Fp(a)1995 782 y Fm(0)2045 818 y Ft(+)22 b Fp(mb)2274 782 y Fm(0)2402 818 y Fp(c)2445 782 y Fm(0)2495 818 y Ft(+)h Fp(md)2734 782 y Fm(0)2779 618 y Fk(\023)2897 758 y Fp(;)113 1004 y Ft(from)33 b(whic)m(h)h(it)f(follo)m(ws)g(that)840 1280 y Fp(L)908 1228 y Fs(\(3\))908 1309 y Fn(g)950 1289 y Ff(0)1015 1280 y Ft(\()p Fp(T)1126 1239 y Fs(\(3\))1233 1280 y Fp( )t Ft(\)\()p Fp(z)t Ft(\))27 b(=)h(\()p Fp(b)1683 1239 y Fm(0)1733 1280 y Fo(\000)22 b Fp(d)1884 1239 y Fm(0)1912 1280 y Fp(z)t Ft(\))2001 1239 y Fm(\000)p Fs(3)2139 1185 y Fk(X)2125 1399 y Fn(m)p Fm(\025)p Fs(1)2314 1280 y Fp( )2399 1139 y Fk(\022)2499 1212 y Ft(1)p 2485 1257 79 4 v 2485 1348 a Fp(z)2535 1319 y Fm(0)2597 1280 y Fo(\000)h Fp(m)2784 1139 y Fk(\023)2902 1280 y Fp(:)113 1628 y Ft(Since)35 b Fp(z)424 1592 y Fm(0)480 1628 y Fo(2)28 b Fp(V)632 1643 y Fn(\032)678 1628 y Ft(\()p Fp(D)799 1643 y Fm(1)884 1628 y Ft(\),)33 b(as)g(w)m(e)h(ha)m(v)m(e)f (seen)i(during)e(the)h(pro)s(of)f(of)h(Prop)s(osition)e(3.1)h(one)g (has)1124 1788 y Fk(\014)1124 1848 y(\014)1124 1908 y(\014)1124 1968 y(\014)1124 2028 y(\014)1124 2087 y(\014)1171 1898 y(X)1157 2112 y Fn(m)p Fm(\025)p Fs(1)1346 1993 y Fp( )1431 1852 y Fk(\022)1531 1925 y Ft(1)p 1517 1970 V 1517 2061 a Fp(z)1567 2032 y Fm(0)1630 1993 y Fo(\000)22 b Fp(m)1816 1852 y Fk(\023)1890 1788 y(\014)1890 1848 y(\014)1890 1908 y(\014)1890 1968 y(\014)1890 2028 y(\014)1890 2087 y(\014)1951 1993 y Fo(\024)28 b Fp(c)2099 2008 y Fn(\032)2231 1993 y Ft(sup)2162 2081 y Fn(V)2209 2091 y Fj(\032)2250 2081 y Fs(\()p Fn(D)2346 2091 y Ff(1)2419 2081 y Fs(\))2467 1993 y Fo(j)p Fp( )t Fo(j)f Fp(:)113 2360 y Ft(On)32 b(the)e(other)h(hand)g(one)g(has)g Fo(j)p Fp(z)21 b Fo(\000)16 b Fp(b)1535 2324 y Fm(0)1563 2360 y Fp(=d)1665 2324 y Fm(0)1693 2360 y Fo(j)1721 2324 y Fm(\000)p Fs(1)1855 2360 y Fo(\024)28 b Fp(c)2003 2375 y Fn(\032)2080 2360 y Ft(for)j(all)e Fp(z)k Fo(2)28 b Fp(V)2596 2375 y Fn(\032)2642 2360 y Ft(\()p Fp(D)2763 2375 y Fm(1)2848 2360 y Ft(\))i(and)h Fp(b)3151 2324 y Fm(0)3178 2360 y Fp(=d)3280 2324 y Fm(0)3336 2360 y Fo(2)d Ft([0)p Fp(;)17 b Ft(1].)113 2502 y(Th)m(us)35 b(w)m(e)f(get)f Fo(j)p Fp(L)775 2451 y Fs(\(3\))775 2531 y Fn(g)817 2512 y Ff(0)881 2502 y Ft(\()p Fp(T)992 2466 y Fs(\(3\))1099 2502 y Fp( )t Ft(\)\()p Fp(z)t Ft(\))p Fo(j)27 b(\024)j Ft(~)-52 b Fp(c)1538 2517 y Fn(\032)1585 2502 y Ft(\()p Fp(d)1676 2466 y Fm(0)1703 2502 y Ft(\))1742 2466 y Fm(\000)p Fs(3)1877 2502 y Fp(:)33 b Ft(But)f(no)m(w)i(it)f(is)g (enough)h(to)f(recall)g(that)g(\(see)113 2690 y(app)s(endix)38 b(A1\))f(min)916 2713 y Fm(M)1010 2693 y Fi(\()p Fj(r)r Ff(\000)p Fi(1\))1213 2690 y Fp(d)1265 2654 y Fm(0)1328 2690 y Fo(\025)e Fp(C)1535 2579 y Fk(\020)1607 2593 y Fm(p)p 1672 2593 40 4 v 58 x Fs(5+1)p 1607 2667 207 4 v 1690 2724 a(2)1825 2579 y Fk(\021)1884 2600 y Fn(r)r Fm(\000)p Fs(1)2081 2690 y Fp(;)j Ft(and)2345 2615 y Fk(P)2450 2720 y Fm(M)2544 2700 y Fi(\()p Fj(r)r Ff(\000)p Fi(1\))2747 2690 y Fp(d)2799 2654 y Fm(0\000)p Fs(2)2963 2690 y Fo(\024)e Fp(C)42 b(;)37 b Ft(to)g(obtain)113 2834 y(the)d(desired)g(estimate.)2561 b Fd(\003)113 3015 y Fl(R)-5 b(emark)38 b Ft(3.4)70 b(In)36 b(a)g(completely)e(analogous)h (w)m(a)m(y)h(w)m(e)g(ma)m(y)e(pro)m(v)m(e)i(that)g(for)g Fp(\032)31 b Fo(\025)h Ft(0)j(and)i(all)113 3135 y Fp(')28 b Fo(2)g(O)382 3099 y Fs(1)427 3135 y Ft(\()p 466 3054 72 4 v Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\),)922 3471 y(sup)853 3559 y Fn(V)900 3569 y Fj(\032)941 3559 y Fs(\()p Fn(D)1037 3569 y Ff(1)1110 3559 y Fs(\))1158 3471 y Fo(j)p Fp(T)1258 3430 y Fn(r)1302 3471 y Fp(')p Ft(\()p Fp(z)t Ft(\))p Fo(j)27 b(\024)h Fp(c)1698 3430 y Fm(0)1698 3496 y Fn(\032)1761 3301 y Fk( )1852 3321 y Fo(p)p 1935 3321 50 4 v 83 x Ft(5)22 b Fo(\000)h Ft(1)p 1852 3448 305 4 v 1980 3539 a(2)2169 3301 y Fk(!)2248 3321 y Fn(r)2377 3471 y Ft(sup)2308 3559 y Fn(V)2355 3569 y Fj(\032)2396 3559 y Fs(\()p Fn(D)2492 3569 y Ff(1)2566 3559 y Fs(\))2613 3471 y Fo(j)p Fp(')p Ft(\()p Fp(z)t Ft(\))p Fo(j)28 b Fp(:)113 3833 y Fl(R)-5 b(emark)44 b Ft(3.5)83 b(W)-8 b(e)42 b(ma)m(y)f(also)g(consider)i(the)f(Hardy)g (space)h Fp(H)2572 3797 y Fn(p)2618 3833 y Ft(\()p Fp(D)2739 3848 y Fm(1)2823 3833 y Ft(\),)h(1)e Fo(\024)g Fp(p)h(<)f Ft(+)p Fo(1)g Ft(of)113 3953 y(analytic)j(functions)i Fp(')64 b Ft(:)h Fp(D)1251 3968 y Fm(1)1384 3953 y Fo(!)48 b Fq(C)71 b Ft(suc)m(h)47 b(that)e(the)h(subharmonic)g(function)g Fo(j)p Fp(')p Fo(j)3421 3916 y Fn(p)3512 3953 y Ft(has)113 4072 y(a)39 b(harmonic)f(ma)6 b(joran)m(t.)60 b(It)38 b(is)h(an)g(immediate)e(consequence)j(of)g(the)f(Riemann)f(mapping)113 4192 y(theorem)26 b(that)h(this)f(space)i(is)f(isomorphic)e(to)i Fp(H)1949 4156 y Fn(p)1995 4192 y Ft(\()p Fq(D)18 b Ft(\).)48 b(Indeed)28 b(if)f Fp(h)g Ft(maps)f Fp(D)3036 4207 y Fm(1)3147 4192 y Ft(conformally)113 4311 y(on)m(to)34 b Fq(D)58 b Ft(one)33 b(can)h(use)g(the)g(norm)520 4615 y Fo(k)p Fp(')p Fo(k)685 4633 y Fn(H)755 4613 y Fj(p)797 4633 y Fs(\()p Fn(D)893 4643 y Ff(1)966 4633 y Fs(\))1030 4615 y Ft(=)28 b Fo(k)p Fp(')22 b Fo(\016)g Fp(h)1401 4574 y Fm(\000)p Fs(1)1509 4615 y Fo(k)1559 4633 y Fn(H)1629 4613 y Fj(p)1671 4633 y Fs(\()p Fg(D)6 b Fs(\))1816 4615 y Ft(=)1921 4474 y Fk(\022)1995 4479 y(Z)2050 4706 y Fn(@)t(D)2162 4716 y Ff(1)2256 4615 y Fo(j)p Fp(')p Ft(\()p Fp(z)t Ft(\))p Fo(j)2505 4574 y Fn(p)2551 4615 y Fo(j)p Fp(h)2636 4574 y Fm(0)2664 4615 y Ft(\()p Fp(z)t Ft(\))p Fo(jj)p Fp(dz)t Fo(j)2978 4474 y Fk(\023)3051 4495 y Fs(1)p Fn(=p)3222 4615 y Fp(:)113 4916 y Ft(Note)31 b(that)f(since)i Fp(@)6 b(D)948 4931 y Fm(1)1063 4916 y Ft(is)31 b(a)f(recti\014able)h (Jordan)h(curv)m(e)f Fp(h)g Ft(extends)h(to)e(a)h(homeomorphism)113 5036 y(of)36 b Fp(@)6 b(D)370 5051 y Fm(1)490 5036 y Ft(on)m(to)36 b Fq(S)778 5000 y Fs(1)852 5036 y Ft(whic)m(h)h(is)e (conformal)g(almost)f(ev)m(erywhere.)52 b(It)35 b(is)h(immediate)d(to)i (c)m(hec)m(k)113 5156 y(that)40 b(the)g(pro)s(of)h(of)f(Prop)s(osition) f(3.3)g(can)i(b)s(e)f(easily)g(adapted)g(so)h(as)f(to)f(sho)m(w)i(that) f Fp(T)54 b Ft(is)113 5289 y(b)s(ounded)35 b(linear)e(op)s(erator)g(on) g Fp(H)1428 5253 y Fn(p)1474 5289 y Ft(\()p Fp(D)1595 5304 y Fm(1)1680 5289 y Ft(\))g(with)g(sp)s(ectral)g(radius)h Fo(\024)2770 5192 y Fm(p)p 2836 5192 40 4 v 2836 5250 a Fs(5)o Fm(\000)p Fs(1)p 2770 5266 208 4 v 2854 5323 a(2)2989 5289 y Ft(.)313 5440 y Fr(3.2.4)i Ft(W)-8 b(e)36 b(ha)m(v)m(e)g(no)m(w)g(the)g(follo)m(wing)g(imp)s(ortan)m(t)e (Corollary)-8 b(,)34 b(whic)m(h)j(establishes)f(the)113 5559 y(con)m(v)m(ergence)f(of)f(the)f(series)h(\(1.4\).)p eop %%Page: 17 17 17 16 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(17)113 383 y Fr(Corollary)39 b(3.6)185 516 y Fe(1.)50 b(Let)41 b Fp( )k Fo(2)c(O)798 480 y Fs(3)843 516 y Ft(\()p 882 436 72 4 v Fq(C)53 b Fo(n)27 b Ft([0)p Fp(;)17 b Ft(1]\))p Fe(.)66 b(The)42 b(family)d Ft(\()p Fp(L)2024 465 y Fs(\(3\))2024 528 y Fn(g)2131 516 y Fp( )t Ft(\))2239 531 y Fn(g)s Fm(2M)2473 516 y Fe(is)i(summable,)g(uniformly)g(on)313 636 y(compact)23 b(subsets)i(of)p 1141 555 V 24 w Fq(C)30 b Fo(n)s Ft([0)p Fp(;)17 b Ft(1])p Fe(.)39 b(Its)24 b(sum)g(is)g(equal) f(to)2335 561 y Fk(P)2440 666 y Fn(r)r Fm(\025)p Fs(0)2586 636 y Ft(\()p Fp(T)2697 600 y Fs(\(3\))2804 636 y Ft(\))2843 600 y Fn(r)2887 636 y Fp( )t Fe(,)h(will)f(b)s(e)i(denoted)313 718 y Fk(P)418 741 y Fs(\(3\))418 822 y Fm(M)541 793 y Fp( )36 b Fe(and)d(for)g(all)f(compact)g(subset)i Fp(K)39 b Fe(of)p 2065 712 V 33 w Fq(C)46 b Fo(n)21 b Ft([0)p Fp(;)c Ft(1])31 b Fe(there)i(exists)f Fp(")d(>)f Ft(0)k Fe(suc)m(h)i(that)1376 1059 y Ft(sup)1415 1143 y Fn(K)1542 1059 y Fo(j)1587 964 y Fk(X)1731 979 y Fs(\(3\))1731 1119 y Fm(M)1854 1059 y Fp( )t Fo(j)27 b(\024)h Fp(C)2154 1074 y Fn(K)2248 1059 y Ft(sup)2276 1143 y Fn(U)2330 1153 y Fj(")2414 1059 y Fo(j)p Fp( )t Fo(j)f Fp(:)313 1371 y Fe(The)33 b(family)e Ft(\()p Fp(L)928 1319 y Fs(\(3\))928 1383 y Fn(g)1034 1371 y Fp( )t Ft(\))1142 1386 y Fn(g)s Fm(2)p Fn(Z)5 b Fm(\001M)1451 1371 y Fe(is)32 b(summable,)f(uniformly)h (on)h(all)f(domains)f(of)i(the)g(form)313 1491 y Fo(fj)17 b(<)p Fp(e)f(z)t Fo(j)31 b Fp(<)h(A)17 b(;)47 b Fo(j)17 b(=)p Fp(m)f(z)t Fo(j)32 b(\025)g Fp(\016)t Fo(g)j Fe(\(where)h Fp(A)f Fe(and)h Fp(\016)j Fe(are)d(p)s(ositiv)m(e\).)49 b(Its)36 b(sum)f(is)g(equal)g(to)313 1555 y Fk(P)418 1659 y Fg(Z)486 1555 y Fk(P)591 1578 y Fs(\(3\))591 1659 y Fm(M)715 1630 y Fp( )t Fe(,)e(will)g(b)s(e)i(denoted)1548 1555 y Fk(P)1653 1578 y Fs(\(3\))1653 1659 y Fn(Z)5 b Fm(\001M)1851 1630 y Fp( )t Fe(.)46 b(It)34 b(is)g(holomorphic)f(in)h Fq(C)49 b Fo(n)22 b Fq(R)5 b Fe(,)40 b(p)s(erio)s(dic)34 b(of)313 1749 y(p)s(erio)s(d)f Ft(1)g Fe(and)h(b)s(ounded)g(in)g(a)f (neigh)m(b)s(orho)s(o)s(d)g(of)h Fo(\006)p Fp(i)p Fo(1)p Fe(.)185 1869 y(2.)50 b(Let)c Fp(')i Fo(2)h(O)814 1833 y Fs(1)859 1869 y Ft(\()p 898 1788 V Fq(C)56 b Fo(n)30 b Ft([0)p Fp(;)17 b Ft(1]\))p Fe(.)80 b(The)46 b(family)e Ft(\()p Fp(L)2069 1884 y Fn(g)2115 1869 y Fp(')p Ft(\))2219 1884 y Fn(g)s Fm(2M)2458 1869 y Fe(is)i(summable,)h(uniformly)e(on)313 1988 y(compact)34 b(subsets)j(of)p 1175 1908 V 35 w Fq(C)50 b Fo(n)23 b Ft([0)p Fp(;)17 b Ft(1])p Fe(.)48 b(Its)35 b(sum)g(is)g(equal)g(to)2473 1914 y Fk(P)2579 2018 y Fn(r)r Fm(\025)p Fs(0)2741 1988 y Fp(T)2813 1952 y Fn(r)2857 1988 y Fp(')p Fe(,)h(will)e(b)s(e)i(denoted)313 2033 y Fk(P)418 2138 y Fm(M)533 2108 y Fp(')p Fe(.)42 b(The)29 b(function)1252 2033 y Fk(P)1358 2138 y Fg(Z)1409 2108 y Ft(\()1448 2033 y Fk(P)1553 2138 y Fm(M)1668 2108 y Fp(')p Ft(\))f Fe(will)f(b)s(e)h(denoted)2484 2033 y Fk(P)2589 2138 y Fn(Z)5 b Fm(\001M)2787 2108 y Fp(')p Fe(.)42 b(It)28 b(is)g(holomorphic)313 2227 y(in)33 b Fq(C)48 b Fo(n)22 b Fq(R)5 b Fe(,)39 b(p)s(erio)s(dic)33 b(of)g(p)s(erio)s(d)h Ft(1)f Fe(and)g(v)-6 b(anishes)35 b(at)d Fo(\006)p Fp(i)p Fo(1)p Fe(.)45 b(One)34 b(has)911 2489 y Ft(\()950 2394 y Fk(X)975 2608 y Fm(M)1110 2489 y Fp(')p Ft(\))1214 2448 y Fm(00)1293 2489 y Ft(=)1398 2394 y Fk(X)1542 2409 y Fs(\(3\))1542 2549 y Fm(M)1666 2489 y Fp(')1731 2448 y Fm(00)1881 2489 y Fp(;)116 b Ft(\()2080 2394 y Fk(X)2064 2608 y Fn(Z)5 b Fm(\001M)2257 2489 y Fp(')p Ft(\))2361 2448 y Fm(00)2440 2489 y Ft(=)2545 2394 y Fk(X)2689 2409 y Fs(\(3\))2689 2549 y Fn(Z)g Fm(\001M)2887 2489 y Fp(')2952 2448 y Fm(00)3030 2489 y Fp(:)185 2827 y Fe(3.)313 2752 y Fk(P)418 2857 y Fm(M)545 2827 y Fe(\(resp.)833 2752 y Fk(P)938 2775 y Fs(\(3\))938 2856 y Fm(M)1045 2827 y Fe(\))29 b(and)g Ft(\(1)14 b Fo(\000)g Fp(T)g Ft(\))28 b Fe(\(resp.)44 b Ft(\(1)14 b Fo(\000)g Fp(T)2189 2791 y Fs(\(3\))2295 2827 y Ft(\))p Fe(\))29 b(acting)f(on)i Fo(O)2913 2791 y Fs(1)2957 2827 y Ft(\()p 2996 2746 V Fq(C)40 b Fo(n)14 b Ft([0)p Fp(;)j Ft(1]\))26 b Fe(\(resp.)313 2946 y Fo(O)395 2910 y Fs(3)439 2946 y Ft(\()p 478 2866 V Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))p Fe(\))31 b(are)i(the)h(in)m(v)m(erses)g(of)g(one)g(another)f(:)1323 3180 y Ft(\(1)22 b Fo(\000)g Fp(T)14 b Ft(\))1661 3086 y Fk(X)1686 3299 y Fm(M)1832 3180 y Ft(=)1938 3086 y Fk(X)1963 3299 y Fm(M)2082 3180 y Ft(\(1)21 b Fo(\000)i Fp(T)14 b Ft(\))27 b(=)h Fe(id.)1109 3481 y Ft(\(1)22 b Fo(\000)h Fp(T)1392 3440 y Fs(\(3\))1499 3481 y Ft(\))1555 3387 y Fk(X)1698 3401 y Fs(\(3\))1698 3541 y Fm(M)1832 3481 y Ft(=)1938 3387 y Fk(X)2082 3401 y Fs(\(3\))2082 3541 y Fm(M)2188 3481 y Ft(\(1)f Fo(\000)h Fp(T)2471 3440 y Fs(\(3\))2578 3481 y Ft(\))k(=)h Fe(id.)113 3715 y Fl(Pr)-5 b(o)g(of.)174 3848 y Ft(1\))50 b(The)25 b(only)f(non)h (trivial)d(assertions)j(are)g(the)g(summabilit)m(y)c(of)k(the)g (families)e(\()p Fp(L)3249 3797 y Fs(\(3\))3249 3860 y Fn(g)3356 3848 y Fp( )t Ft(\))3464 3863 y Fn(g)s Fm(2M)313 3991 y Ft(and)35 b(\()p Fp(L)615 3939 y Fs(\(3\))615 4002 y Fn(g)722 3991 y Fp( )t Ft(\))830 4006 y Fn(g)s Fm(2)p Fn(Z)5 b Fm(M)1082 3991 y Ft(.)50 b(W)-8 b(riting)35 b Fp(g)f Ft(=)d Fp(g)1770 3955 y Fm(0)1797 3991 y Fp(g)t Ft(\()p Fp(m)p Ft(\),)k(\()p Fp(m)c Fo(\025)h Ft(1)p Fp(;)17 b(g)2490 3955 y Fm(0)2547 3991 y Fo(2)31 b(M)p Ft(\))k(for)h Fp(g)e Fo(2)d(M)p Ft(,)k Fp(g)f Fo(6)p Ft(=)e(id,)313 4110 y(w)m(e)h(ha)m(v)m(e)1168 4265 y Fp(L)1236 4224 y Fs(\(3\))1236 4289 y Fn(g)1343 4265 y Fp( )t Ft(\()p Fp(z)t Ft(\))27 b(=)i(\()p Fp(b)1755 4224 y Fm(0)1804 4265 y Fo(\000)23 b Fp(d)1956 4224 y Fm(0)1984 4265 y Fp(z)t Ft(\))2073 4224 y Fm(\000)p Fs(3)2180 4265 y Fp( )2265 4124 y Fk(\022)2365 4197 y Ft(1)p 2350 4242 79 4 v 2350 4333 a Fp(z)2400 4304 y Fm(0)2463 4265 y Fo(\000)g Fp(m)2650 4124 y Fk(\023)2773 4265 y Fp(;)313 4497 y Ft(with)1218 4652 y Fp(g)1270 4611 y Fm(0)1325 4652 y Ft(=)1430 4511 y Fk(\022)1520 4592 y Fp(a)1573 4556 y Fm(0)1705 4592 y Fp(b)1748 4556 y Fm(0)1525 4711 y Fp(c)1568 4675 y Fm(0)1701 4711 y Fp(d)1753 4675 y Fm(0)1797 4511 y Fk(\023)1987 4652 y Fp(;)115 b(z)2180 4611 y Fm(0)2237 4652 y Ft(=)2354 4584 y Fp(d)2406 4548 y Fm(0)2434 4584 y Fp(z)27 b Fo(\000)c Fp(b)2650 4548 y Fm(0)p 2354 4629 325 4 v 2354 4720 a Fp(a)2407 4691 y Fm(0)2457 4720 y Fo(\000)f Fp(c)2599 4691 y Fm(0)2627 4720 y Fp(z)2723 4652 y(:)313 4929 y Ft(No)m(w)45 b(for)h Fp(z)54 b Fo(2)49 b Fp(V)990 4944 y Fn(\032)1037 4929 y Ft(\()p Fp(D)1158 4944 y Fm(1)1242 4929 y Ft(\),)1358 4815 y Fk(\014)1358 4875 y(\014)1358 4934 y(\014)1391 4929 y Fp(L)1459 4878 y Fs(\(3\))1459 4941 y Fn(g)1566 4929 y Fp( )t Ft(\()p Fp(z)t Ft(\))1763 4815 y Fk(\014)1763 4875 y(\014)1763 4934 y(\014)1845 4929 y Ft(=)g Fp(c)2014 4944 y Fn(\032)2077 4849 y Fk(\000)2123 4929 y Fp(md)2262 4893 y Fm(0)2290 4929 y Ft(dist)2451 4849 y Fk(\000)2497 4929 y Ft(z)p Fp(;)17 b Ft([0)p Fp(;)g Ft(1])2787 4849 y Fk(\001)m(\001)2876 4869 y Fm(\000)p Fs(3)2999 4929 y Ft(sup)3149 4954 y Fn(V)3196 4964 y Fj(\032)3238 4954 y Fs(\()p Fn(D)3334 4964 y Ff(1)3407 4954 y Fs(\))3459 4929 y Fo(j)p Fp( )t Fo(j)45 b Fp(;)313 5079 y Ft(and)33 b(the)h(summabilit)m(y)d(assertions)i(follo)m(w)g(\(see)h(pro)s(of)f (of)h(Prop)s(osition)e(3.3\).)174 5198 y(2\))50 b(Again,)26 b(the)g(only)g(non)g(trivial)e(assertion)i(is)g(the)g(summabilit)m(y)d (one,)28 b(whic)m(h)e(is)g(obtained)313 5318 y(from)32 b(the)i(\014rst)f(part)g(b)m(y)h(in)m(tegrating)f(t)m(wice.)174 5438 y(3\))50 b(The)33 b(third)h(part)f(of)g(the)h(corollary)e(is)h (immediate.)3579 5557 y Fd(\003)p eop %%Page: 18 18 18 17 bop 113 144 a Ft(18)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y Fr(4.)51 b(Boundary)39 b(b)s(eha)m(viour)e(of)1449 308 y Fk(P)1554 413 y Fm(M)1670 383 y Fp(')313 532 y Ft(In)24 b(this)f(section)i(w)m(e)f(will)e(study)i(the)g(b)s(eha)m (viour)g(of)2235 458 y Fk(P)2340 562 y Fm(M)2456 532 y Fp(')p Ft(\()p Fp(z)t Ft(\))g(when)g Fp(z)29 b Ft(is)23 b(close)i(to)e([0)p Fp(;)17 b Ft(1].)113 652 y(Our)42 b(main)e(to)s(ol)g(for)h(this)g(study)g(will)f(b)s(e)i(the)f(complex)f (con)m(tin)m(ued)i(fraction)f(in)m(tro)s(duced)113 771 y(in)c(the)g(previous)g(section.)55 b(In)37 b(this)g(section)g(and)h (in)e(the)h(next)g(one)g(w)m(e)h(will)d(for)i(shortness)113 891 y(often)d(denote)g Fp(c)g Ft(or)f Fp(C)40 b Ft(v)-6 b(arious)33 b(p)s(ositiv)m(e)g(univ)m(ersal)g(constan)m(ts.)113 1129 y Fr(4.1)39 b(Decomp)s(osition)f(in)m(to)f(principal)f(and)i (residual)e(terms)313 1278 y(4.1.1)28 b Ft(W)-8 b(e)28 b(b)s(egin)g(our)g(study)g(of)f(the)i(b)s(oundary)f(b)s(eha)m(viour)f (of)2727 1204 y Fk(P)2833 1308 y Fm(M)2948 1278 y Fp(')p Ft(\()p Fp(z)t Ft(\))h(considering)113 1398 y(the)34 b(case)g Fp(z)k Ft(is)33 b(close)g(to)g(0.)113 1577 y Fr(Prop)s(osition)39 b(4.1)185 1696 y Fe(1.)50 b(Let)i Fp(I)66 b Ft(=)59 b([)p Fp(\015)836 1711 y Fs(0)880 1696 y Fp(;)17 b(\015)977 1711 y Fs(1)1021 1696 y Ft(])p Fe(,)56 b Fp(\015)1185 1711 y Fs(0)1288 1696 y Fp(>)j Fo(\000)p Ft(1)p Fe(.)100 b(There)53 b(exists)e 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y(\014)1970 5445 y(\014)1970 5505 y(\014)1970 5564 y(\014)2031 5470 y Fo(\024)28 b Fp(c)p Fo(j)p Fp(z)t Fo(j)17 b Ft(sup)2308 5553 y Fn(D)2373 5563 y Ff(1)2468 5470 y Fo(j)p Fp(')p Fo(j)28 b Fp(:)p eop %%Page: 19 19 19 18 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(19)113 383 y(In)40 b(the)g(case)g Fp(m)e(<)g Ft(3)p Fo(j)p Fp(z)t Fo(j)1035 347 y Fm(\000)p Fs(1)1181 383 y Ft(w)m(e)i(separate)g(the)g(three)g(terms)e(constituting)h Fp(L)3084 401 y Fn(g)s Fs(\()p Fn(m)p Fs(\))3279 383 y Ft(~)-65 b Fp(')p Ft(\()p Fp(z)t Ft(\))39 b(and)113 503 y(w)m(e)34 b(obtain)606 715 y Fo(j)16 b Ft(~)-66 b Fp(')699 674 y Fm(0)727 715 y Ft(\()p Fo(\000)p Fp(m)p Ft(\))p Fo(j)28 b(\024)g Fp(cm)1260 674 y Fm(\000)p Fs(2)1384 715 y Ft(sup)1390 799 y Fn(D)1455 809 y Ff(1)1551 715 y Fo(j)16 b Ft(~)-66 b Fp(')p Fo(j)99 b Fp(;)116 b Fo(j)p Fp(z)20 b Ft(~)-66 b Fp(')p Ft(\()p Fo(\000)p Fp(m)p Ft(\))p Fo(j)28 b(\024)g Fp(cm)2591 674 y Fm(\000)p Fs(1)2699 715 y Fo(j)p Fp(z)t Fo(j)17 b Ft(sup)2827 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2721 y(\014)210 2627 y Fp(T)14 b(')p Ft(\()p Fp(z)t Ft(\))23 b(+)f Fp(z)t(')729 2486 y Fk(\022)815 2559 y Ft(1)p 815 2604 51 4 v 815 2695 a Fp(z)900 2627 y Fo(\000)g Ft(1)1049 2486 y Fk(\023)1145 2627 y Fo(\000)1259 2532 y Fk(X)1244 2746 y Fn(m)p Fm(\025)p Fs(1)1434 2627 y Fp(')1499 2586 y Fm(0)1527 2627 y Ft(\()p Fo(\000)p Fp(m)p Ft(\))1769 2422 y Fk(\014)1769 2482 y(\014)1769 2542 y(\014)1769 2602 y(\014)1769 2662 y(\014)1769 2721 y(\014)1830 2627 y Fo(\024)28 b Fp(c)p Fo(j)p Fp(z)f Fo(\000)c Ft(1)p Fo(j)17 b Ft(sup)2279 2710 y Fn(D)2344 2720 y Ff(1)2439 2627 y Fo(j)p Fp(')p Fo(j)27 b Fp(;)837 b Ft(\(4)p Fp(:)p Ft(3\))420 2821 y Fk(\014)420 2881 y(\014)420 2941 y(\014)420 3000 y(\014)420 3060 y(\014)453 2901 y(X)478 3114 y Fm(M)614 2995 y Fp(')p Ft(\()p Fp(z)t Ft(\))22 b Fo(\000)h Fp(')p Ft(\()p Fp(z)t Ft(\))f(+)h Fp(z)t(')1376 2855 y Fk(\022)1462 2928 y Ft(1)p 1462 2972 V 1462 3064 a Fp(z)1546 2995 y Fo(\000)g Ft(1)1696 2855 y Fk(\023)1769 2821 y(\014)1769 2881 y(\014)1769 2941 y(\014)1769 3000 y(\014)1769 3060 y(\014)1830 2995 y Fo(\024)28 b Fp(c)p Fo(j)p Fp(z)f Fo(\000)c Ft(1)p Fo(j)p Ft(\(1)e(+)h(log)c Fo(j)p Fp(z)26 b Fo(\000)d Ft(1)p Fo(j)2890 2954 y Fm(\000)p Fs(1)2996 2995 y Ft(\))17 b(sup)3058 3079 y Fn(D)3123 3089 y Ff(1)3218 2995 y Fo(j)p Fp(')p Fo(j)27 b Fp(:)58 b Ft(\(4)p Fp(:)p Ft(4\))113 3365 y Fl(Pr)-5 b(o)g(of.)34 b Ft(F)-8 b(or)31 b Fp(z)i Fo(2)28 b Fp(D)843 3380 y Fs(1)920 3365 y Ft(w)m(e)33 b(ha)m(v)m(e)1307 3325 y Fs(1)p 1307 3342 41 4 v 1307 3399 a Fn(z)1380 3365 y Fo(\000)21 b Ft(1)27 b Fo(2)h Fp(D)1731 3380 y Fs(0)1776 3365 y Ft(.)44 b(Moreo)m(v)m(er,)33 b(there)g(exists)f Fp(c)c(>)g Ft(0)k(suc)m(h)i(that)d(for)113 3484 y Fp(m)e Fo(\025)f Ft(2)33 b(and)g Fp(w)k Ft(on)c(the)h(segmen)m (t)f(with)g(endp)s(oin)m(ts)h(1)22 b Fo(\000)h Fp(m)33 b Ft(and)h(1)p Fp(=z)27 b Fo(\000)22 b Fp(m)34 b Ft(w)m(e)g(ha)m(v)m(e) 1332 3721 y Fo(j)p Fp(')1425 3679 y Fm(0)1452 3721 y Ft(\()p Fo(\000)p Fp(w)s Ft(\))p Fo(j)28 b(\024)g Fp(cm)1972 3679 y Fm(\000)p Fs(2)2096 3721 y Ft(sup)2102 3804 y Fn(D)2167 3814 y Ff(1)2263 3721 y Fo(j)p Fp(')p Fo(j)f Fp(:)1013 b Ft(\(4)p Fp(:)p Ft(5\))113 4019 y(Since)340 4290 y Fp(T)14 b(')p Ft(\()p Fp(z)t Ft(\))23 b(+)f Fp(z)t(')859 4149 y Fk(\022)946 4223 y Ft(1)p 945 4267 51 4 v 945 4358 a Fp(z)1030 4290 y Fo(\000)h Ft(1)1180 4149 y Fk(\023)1275 4290 y Fo(\000)1389 4195 y Fk(X)1375 4409 y Fn(m)p Fm(\025)p Fs(1)1564 4290 y Fp(')1629 4249 y Fm(0)1657 4290 y Ft(\()p Fo(\000)p Fp(m)p Ft(\))28 b(=)g Fp(z)2114 4195 y Fk(X)2100 4409 y Fn(m)p Fm(\025)p Fs(2)2289 4149 y Fk(\024)2341 4290 y Fp(')p Ft(\(1)22 b Fo(\000)h Fp(m)p Ft(\))f Fo(\000)h Fp(')2947 4149 y Fk(\022)3032 4223 y Ft(1)p 3032 4267 V 3032 4358 a Fp(z)3117 4290 y Fo(\000)f Fp(m)3303 4149 y Fk(\023)q(\025)113 4633 y Ft(from)33 b(\(4.5\))f(one)i(deduces)h (easily)d(\(4.3\).)313 4782 y(Let)49 b(~)-66 b Fp(')28 b Ft(=)689 4708 y Fk(P)794 4812 y Fm(M)910 4782 y Fp(')p Ft(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)50 b(~)-66 b Fp(')p Ft(\()p Fp(z)t Ft(\))28 b(=)g Fp(')p Ft(\()p Fp(z)t Ft(\))23 b(+)f Fp(T)30 b Ft(~)-66 b Fp(')p Ft(\()p Fp(z)t Ft(\))28 b Fp(;)33 b Ft(and)275 5083 y(~)-66 b Fp(')q Ft(\()p Fp(z)t Ft(\))27 b(=)i Fp(')p Ft(\()p Fp(z)t Ft(\))22 b Fo(\000)h Fp(z)t(')1033 4943 y Fk(\022)1119 5016 y Ft(1)p 1119 5061 V 1119 5152 a Fp(z)1203 5083 y Fo(\000)g Ft(1)1353 4943 y Fk(\023)1448 5083 y Ft(+)g Fp(T)30 b Ft(~)-66 b Fp(')p Ft(\()p Fp(z)t Ft(\))22 b(+)h Fp(z)e Ft(~)-67 b Fp(')2068 4943 y Fk(\022)2153 5016 y Ft(1)p 2153 5061 V 2153 5152 a Fp(z)2238 5083 y Fo(\000)22 b Ft(1)2387 4943 y Fk(\023)2483 5083 y Fo(\000)2597 4989 y Fk(X)2582 5203 y Fn(m)p Fm(\025)p Fs(1)2787 5083 y Ft(~)-66 b Fp(')2836 5042 y Fm(0)2865 5083 y Ft(\()p Fo(\000)p Fp(m)p Ft(\))475 5470 y(+)23 b Fp(z)642 5269 y Fk(2)642 5449 y(4)708 5470 y Fp(')790 5329 y Fk(\022)876 5402 y Ft(1)p 875 5447 V 875 5538 a Fp(z)960 5470 y Fo(\000)g Ft(1)1110 5329 y Fk(\023)1205 5470 y Fo(\000)39 b Ft(~)-66 b Fp(')1386 5329 y Fk(\022)1472 5402 y Ft(1)p 1472 5447 V 1472 5538 a Fp(z)1557 5470 y Fo(\000)22 b Ft(1)1706 5329 y Fk(\023)1801 5470 y Ft(+)1915 5375 y Fk(X)1901 5589 y Fn(m)p Fm(\025)p Fs(1)2106 5470 y Ft(~)-66 b Fp(')2155 5429 y Fm(0)2183 5470 y Ft(\()p Fo(\000)p Fp(m)p Ft(\))2425 5269 y Fk(3)2425 5449 y(5)2514 5470 y Ft(+)23 b(\(1)f Fo(\000)g Fp(z)t Ft(\))2945 5375 y Fk(X)2930 5589 y Fn(m)p Fm(\025)p Fs(1)3136 5470 y Ft(~)-66 b Fp(')3185 5429 y Fm(0)3213 5470 y Ft(\()p Fo(\000)p Fp(m)p Ft(\))28 b Fp(:)p eop %%Page: 20 20 20 19 bop 113 144 a Ft(20)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y(By)33 b(\(4.3\))f(one)h(has)666 531 y Fk(\014)666 590 y(\014)666 650 y(\014)666 710 y(\014)666 770 y(\014)666 829 y(\014)699 735 y Fp(T)d Ft(~)-66 b Fp(')p Ft(\()p Fp(z)t Ft(\))23 b(+)f Fp(z)f Ft(~)-67 b Fp(')1219 594 y Fk(\022)1305 667 y Ft(1)p 1304 712 51 4 v 1304 803 a Fp(z)1389 735 y Fo(\000)23 b Ft(1)1539 594 y Fk(\023)1634 735 y Fo(\000)1748 640 y Fk(X)1733 854 y Fn(m)p Fm(\025)p Fs(1)1939 735 y Ft(~)-66 b Fp(')1988 694 y Fm(0)2016 735 y Ft(\()p Fo(\000)p Fp(m)p Ft(\))2258 531 y Fk(\014)2258 590 y(\014)2258 650 y(\014)2258 710 y(\014)2258 770 y(\014)2258 829 y(\014)2319 735 y Fo(\024)28 b Fp(c)p Fo(j)p Fp(z)f Fo(\000)c Ft(1)p Fo(j)17 b Ft(sup)2768 818 y Fn(D)2833 828 y Ff(1)2928 735 y Fo(j)f Ft(~)-66 b Fp(')p Fo(j)27 b Fp(:)113 1086 y Ft(W)-8 b(e)34 b(also)f(ha)m(v)m(e)g (:)44 b(sup)936 1111 y Fn(D)1001 1121 y Ff(1)1096 1086 y Fo(j)16 b Ft(~)-66 b Fp(')p Fo(j)27 b(\024)h Fp(c)17 b Ft(sup)1559 1111 y Fn(D)1624 1121 y Ff(1)1718 1086 y Fo(j)p Fp(')p Fo(j)33 b Ft(and)h(b)m(y)f(\(4.2\))283 1243 y Fk(\014)283 1303 y(\014)283 1363 y(\014)283 1423 y(\014)283 1482 y(\014)283 1542 y(\014)316 1448 y Fp(')p Ft(\()432 1380 y(1)p 432 1425 V 432 1516 a Fp(z)517 1448 y Fo(\000)22 b Ft(1\))g Fo(\000)39 b Ft(~)-66 b Fp(')p Ft(\()943 1380 y(1)p 943 1425 V 943 1516 a Fp(z)1027 1448 y Fo(\000)23 b Ft(1\))f(+)1352 1353 y Fk(X)1337 1567 y Fn(m)p Fm(\025)p Fs(1)1543 1448 y Ft(~)-66 b Fp(')1592 1406 y Fm(0)1620 1448 y Ft(\()p Fo(\000)p Fp(m)p Ft(\))1862 1243 y Fk(\014)1862 1303 y(\014)1862 1363 y(\014)1862 1423 y(\014)1862 1482 y(\014)1862 1542 y(\014)1923 1448 y Fo(\024)28 b Fp(c)p Fo(j)p Fp(z)f Fo(\000)c Ft(1)p Fo(j)p Ft(\(1)e(+)h(log)c Fo(j)p Fp(z)26 b Fo(\000)d Ft(1)p Fo(j)2983 1406 y Fm(\000)p Fs(1)3089 1448 y Ft(\))17 b(sup)3151 1531 y Fn(D)3216 1541 y Ff(1)3311 1448 y Fo(j)p Fp(')p Fo(j)27 b Fp(:)113 1839 y Ft(As)258 1725 y Fk(\014)258 1784 y(\014)258 1844 y(\014)291 1764 y(P)396 1869 y Fn(m)p Fm(\025)p Fs(1)606 1839 y Ft(~)-66 b Fp(')655 1803 y Fm(0)684 1839 y Ft(\()p Fo(\000)p Fp(m)p Ft(\))926 1725 y Fk(\014)926 1784 y(\014)926 1844 y(\014)987 1839 y Fo(\024)28 b Fp(c)17 b Ft(sup)1302 1863 y Fn(D)1367 1873 y Ff(1)1461 1839 y Fo(j)f Ft(~)-66 b Fp(')p Fo(j)27 b(\024)i Fp(c)17 b Ft(sup)1924 1863 y Fn(D)1989 1873 y Ff(1)2084 1839 y Fo(j)p Fp(')p Fo(j)30 b Fp(;)g Ft(w)m(e)h(get)f(the)h(second)h (inequalit)m(y)-8 b(.)41 b Fd(\003)113 2041 y Fl(R)-5 b(emark)34 b Ft(4.3)63 b(It)31 b(is)h(easy)g(to)f(c)m(hec)m(k)i(that)e (the)i(estimates)d(in)i(Prop)s(osition)f(4.2)g(are)h(v)-6 b(alid)31 b(if)h Fp(z)113 2161 y Ft(is)i(suc)m(h)g(that)f Fp(z)27 b Fo(\000)c Ft(1)k Fo(2)h Ft(\001.)113 2397 y Fr(4.2)39 b(Boundary)f(b)s(eha)m(viour)f(and)h(con)m(tin)m(ued)d (fraction)313 2545 y Ft(Let)d Fp(k)f Fo(\025)d Ft(1)k(and)h Fp(m)1040 2560 y Fs(1)1085 2545 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)1394 2560 y Fn(k)1475 2545 y Ft(b)s(e)33 b(in)m(tegers)g Fo(\025)28 b Ft(1.)44 b(W)-8 b(e)33 b(denote)g(b)m(y)f Fp(D)s Ft(\()p Fp(m)3044 2560 y Fs(1)3089 2545 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)3398 2560 y Fn(k)3447 2545 y Ft(\))32 b(the)113 2665 y(set)i(of)f Fp(z)428 2680 y Fs(0)501 2665 y Fo(2)28 b Fp(D)37 b Ft(for)c(whic)m(h)h (the)g(complex)e(con)m(tin)m(ued)j(fraction)e(is)1170 2897 y Fp(z)1220 2855 y Fm(\000)p Fs(1)1216 2925 y Fn(i)1356 2897 y Ft(=)28 b Fp(m)1548 2912 y Fn(i)p Fs(+1)1705 2897 y Ft(+)22 b Fp(z)1850 2912 y Fn(i)p Fs(+1)2012 2897 y Fp(;)100 b Ft(0)27 b Fo(\024)h Fp(i)g(<)g(k)j(;)113 3129 y Ft(with)i Fp(z)386 3144 y Fn(i)447 3129 y Fo(2)c Fp(D)36 b Ft(for)e(0)27 b Fo(\024)h Fp(i)g(<)g(k)37 b Ft(and)c Fp(z)1490 3144 y Fn(k)1567 3129 y Fo(2)28 b Ft(\001.)44 b(In)34 b(the)f(follo)m(wing)g(w)m(e)h(set)f(for)h(0)27 b Fp(<)h(i)g Fo(\024)h Fp(k)1411 3416 y(")1457 3431 y Fn(i)1518 3416 y Ft(=)1623 3275 y Fk(\032)1715 3358 y Ft(0)99 b(if)33 b Fp(m)2042 3373 y Fn(i)2104 3358 y Ft(=)28 b(1)f Fp(;)1715 3478 y Ft(1)99 b(if)33 b Fp(m)2042 3493 y Fn(i)2104 3478 y Fp(>)28 b Ft(1)f Fp(:)113 3780 y Fr(Prop)s(osition) 39 b(4.4)57 b Fe(F)-8 b(or)33 b Fp(')28 b Fo(2)g(O)1384 3744 y Fs(1)1429 3780 y Ft(\()p 1468 3699 72 4 v Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))31 b Fe(w)m(e)j(ha)m(v)m(e)f(in)h Fp(D)s Ft(\()p Fp(m)2609 3795 y Fs(1)2654 3780 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)2963 3795 y Fn(k)3012 3780 y Ft(\))200 4017 y Fp(T)272 3976 y Fn(k)321 4017 y Fp(')p Ft(\()p Fp(z)471 4032 y Fs(0)516 4017 y Ft(\))28 b(=)g(\()p Fp(p)777 4032 y Fn(k)r Fm(\000)p Fs(1)950 4017 y Fo(\000)22 b Fp(q)1093 4032 y Fn(k)r Fm(\000)p 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Fp(R)1126 4505 y Fs([)p Fn(k)r Fs(])1219 4542 y Ft(\()p Fp(')p Ft(\))g Fe(is)g(holomorphic)g(in)g(in)m(t)16 b Fp(D)s Ft(\()p Fp(m)2556 4557 y Fs(1)2602 4542 y Fp(;)h(:)g(:)g(:)d(;)j (m)2911 4557 y Fn(k)2960 4542 y Ft(\))p Fe(,)41 b(con)m(tin)m(uous)h (in)113 4661 y Fp(D)s Ft(\()p Fp(m)324 4676 y Fs(1)370 4661 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)679 4676 y Fn(k)727 4661 y Ft(\))33 b Fe(and)h(satis\014es)g(there)1015 5007 y Fo(j)p Fp(R)1120 4966 y Fs([)p Fn(k)r Fs(])1213 5007 y Ft(\()p Fp(')p Ft(\)\()p Fp(z)1441 5022 y Fs(0)1485 5007 y Ft(\))p Fo(j)27 b(\024)h Fp(ck)1799 4836 y Fk( )1890 4857 y Fo(p)p 1973 4857 50 4 v 82 x Ft(5)22 b(+)h(1)p 1890 4984 305 4 v 2017 5075 a(2)2206 4836 y Fk(!)2285 4857 y Fm(\000)p Fn(k)2413 5007 y Ft(sup)2419 5090 y Fn(D)2484 5100 y Ff(1)2580 5007 y Fo(j)p Fp(')p Fo(j)k Fp(:)696 b Ft(\(4)p Fp(:)p Ft(7\))113 5352 y Fl(Pr)-5 b(o)g(of.)34 b Ft(One)h(has)391 5584 y Fp(T)14 b(')p Ft(\()p Fp(z)613 5599 y Fs(0)658 5584 y Ft(\))28 b(=)g Fo(\000)p Fp(z)953 5599 y Fs(0)998 5584 y Ft([)p Fp(')p Ft(\()p Fp(z)1176 5599 y Fs(1)1221 5584 y Ft(\))22 b(+)g Fp(')p Ft(\()p Fp(z)1531 5599 y Fs(1)1599 5584 y Fo(\000)g Ft(1\))g(+)g Fp(")1954 5599 y Fs(1)2000 5584 y Fp(')p Ft(\()p Fp(z)2150 5599 y Fs(1)2217 5584 y Ft(+)g(1\)])g(+)g Fp(R)2631 5543 y Fs(\()p Fn(m)2733 5553 y Fi(1)2771 5543 y Fs(\))2808 5584 y Ft(\()p Fp(')p Ft(\)\()p Fp(z)3036 5599 y Fs(0)3080 5584 y Ft(\))27 b Fp(;)278 b Ft(\(4)p Fp(:)p Ft(8\))p eop %%Page: 21 21 21 20 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(21)113 383 y(with)261 630 y Fp(R)338 588 y Fs(\()p Fn(m)440 598 y Fi(1)478 588 y Fs(\))514 630 y Ft(\()p Fp(')p Ft(\)\()p Fp(z)742 645 y Fs(0)786 630 y Ft(\))28 b(=)972 535 y Fk(X)958 749 y Fn(m)p Fm(\025)p Fs(1)1147 630 y Fp(')1212 588 y Fm(0)1240 630 y Ft(\()p Fo(\000)p Fp(m)p Ft(\))23 b Fo(\000)g Fp(z)1651 645 y Fs(0)1948 535 y Fk(X)1712 753 y Fn(m)p Fm(\025)p Fs(1)14 b Fn(;)f Fm(j)p Fn(m)p Fm(\000)p Fn(m)2164 763 y Fi(1)2202 753 y Fm(j)p Fn(>)p Fs(1)2328 630 y Ft([)p Fp(')p Ft(\()p Fp(z)2506 645 y Fs(1)2572 630 y Ft(+)23 b Fp(m)2759 645 y Fs(1)2826 630 y Fo(\000)g Fp(m)p Ft(\))f Fo(\000)h Fp(')p Ft(\()p Fo(\000)p Fp(m)p Ft(\)])847 925 y(+)g Fp(z)993 940 y Fs(0)1290 831 y Fk(X)1055 1049 y Fn(m)p Fm(\025)p Fs(1)13 b Fn(;)g Fm(j)p Fn(m)p Fm(\000)p Fn(m)1506 1059 y Fi(1)1544 1049 y Fm(j\024)p Fs(1)1687 925 y Fp(')p Ft(\()p Fo(\000)p Fp(m)p Ft(\))28 b Fp(:)113 1307 y Ft(F)-8 b(or)39 b Fp(z)344 1322 y Fs(0)425 1307 y Fo(2)e Fp(D)s Ft(\()p Fp(m)739 1322 y Fs(1)785 1307 y Ft(\),)i(one)g(easily)f(c)m (hec)m(ks)i(that)e(:)55 b Fo(j)p Fp(R)2081 1271 y Fs(\()p Fn(m)2183 1281 y Fi(1)2221 1271 y Fs(\))2257 1307 y Ft(\()p Fp(')p Ft(\)\()p Fp(z)2485 1322 y Fs(0)2529 1307 y Ft(\))p Fo(j)36 b(\024)i Fp(c)17 b Ft(sup)2956 1331 y Fn(D)3021 1341 y Ff(1)3116 1307 y Fo(j)p Fp(')p Fo(j)38 b Ft(and)h(that)113 1439 y Fp(R)190 1403 y Fs(\()p Fn(m)292 1413 y Fi(1)331 1403 y Fs(\))367 1439 y Ft(\()p Fp(')p Ft(\))32 b(is)i(holomorphic)e (in)h(the)h(neigh)m(b)s(orho)s(o)s(d)g(of)f Fp(D)s Ft(\()p Fp(m)2451 1454 y Fs(1)2496 1439 y Ft(\).)313 1563 y(Iterating)f (\(4.8\))g Fp(k)k Ft(times)d(w)m(e)g(ha)m(v)m(e)593 1913 y Fp(T)665 1872 y Fn(k)714 1913 y Fp(')p Ft(\()p Fp(z)864 1928 y Fs(0)909 1913 y Ft(\))27 b(=)1080 1788 y Fn(k)r Fm(\000)p Fs(1)1090 1818 y Fk(Y)1089 2030 y Fn(i)p Fs(=0)1226 1913 y Ft(\()p Fo(\000)p Fp(z)1388 1928 y Fn(i)1422 1913 y Ft(\)[)p Fp(')p Ft(\()p Fp(z)1639 1928 y Fn(k)1688 1913 y Ft(\))22 b(+)g Fp(')p Ft(\()p Fp(z)1998 1928 y Fn(k)2070 1913 y Fo(\000)g Ft(1\))g(+)h Fp(")2426 1928 y Fn(k)2475 1913 y Fp(')p Ft(\()p Fp(z)2625 1928 y Fn(k)2696 1913 y Ft(+)g(1\)])970 2266 y(+)1069 2141 y Fn(k)r Fm(\000)p Fs(1)1070 2171 y Fk(X)1073 2383 y Fn(j)t Fs(=1)1215 2266 y Ft([)1243 2136 y Fn(j)t Fm(\000)p Fs(1)1249 2171 y Fk(Y)1248 2383 y Fn(i)p Fs(=0)1382 2266 y Ft(\()p Fo(\000)p Fp(z)1544 2281 y Fn(i)1578 2266 y Ft(\)]\()p Fp(T)1756 2224 y Fn(k)r Fm(\000)p Fn(j)1903 2266 y Fp(')p Ft(\()p Fp(z)2053 2281 y Fn(j)2117 2266 y Fo(\000)g Ft(1\))f(+)g Fp(")2473 2281 y Fn(j)2516 2266 y Fp(T)2588 2224 y Fn(k)r Fm(\000)p Fn(j)2736 2266 y Fp(')p Ft(\()p Fp(z)2886 2281 y Fn(j)2951 2266 y Ft(+)g(1\)\))970 2632 y(+)1119 2507 y Fn(k)1069 2537 y Fk(X)1072 2749 y Fn(j)t Fs(=1)1213 2632 y Ft([)1241 2503 y Fn(j)t Fm(\000)p Fs(2)1247 2537 y Fk(Y)1246 2749 y Fn(i)p Fs(=0)1380 2632 y Ft(\()p Fo(\000)p Fp(z)1542 2647 y Fn(i)1575 2632 y Ft(\)])p Fp(R)1719 2591 y Fs(\()p Fn(m)1821 2601 y Fj(j)1857 2591 y Fs(\))1893 2632 y Ft(\()p Fp(T)2004 2591 y Fn(k)r Fm(\000)p Fn(j)2153 2632 y Fp(')p Ft(\)\()p Fp(z)2342 2647 y Fn(j)t Fm(\000)p Fs(1)2485 2632 y Ft(\))28 b Fp(:)3452 2276 y Ft(\(4)p Fp(:)p Ft(9\))113 3013 y(W)-8 b(e)42 b(ha)m(v)m(e)f(here)h(:)841 2939 y Fk(Q)935 2963 y Fn(l)p Fm(\000)p Fs(1)935 3043 y(0)1067 3013 y Ft(\()p Fo(\000)p Fp(z)1229 3028 y Fn(i)1263 3013 y Ft(\))e(=)h Fp(p)1510 3028 y Fn(l)p Fm(\000)p Fs(1)1670 3013 y Fo(\000)27 b Fp(q)1818 3028 y Fn(l)p Fm(\000)p Fs(1)1951 3013 y Fp(z)1997 3028 y Fs(0)2083 3013 y Fp(;)40 b Ft(and,)j Fo(j)p Fp(p)2460 3028 y Fn(l)p Fm(\000)p Fs(1)2620 3013 y Fo(\000)27 b Fp(q)2768 3028 y Fn(l)p Fm(\000)p Fs(1)2901 3013 y Fp(z)2947 3028 y Fs(0)2992 3013 y Fo(j)40 b(\024)h Fp(cq)3269 2971 y Fm(\000)p Fs(1)3265 3043 y Fn(l)3417 3013 y Fp(:)f Ft(The)113 3133 y(function)922 3341 y Fp(R)999 3289 y Fs([)p Fn(k)r Fs(])998 3367 y(0)1092 3341 y Ft(\()p Fp(z)1177 3356 y Fs(0)1222 3341 y Ft(\))27 b(=)1443 3216 y Fn(k)1393 3246 y Fk(X)1396 3458 y Fn(j)t Fs(=1)1537 3341 y Ft([)1565 3211 y Fn(j)t Fm(\000)p Fs(2)1571 3246 y Fk(Y)1570 3458 y Fn(i)p Fs(=0)1704 3341 y Ft(\()p Fo(\000)p Fp(z)1866 3356 y Fn(i)1900 3341 y Ft(\)])p Fp(R)2044 3300 y Fs(\()p Fn(m)2146 3310 y Fj(j)2182 3300 y Fs(\))2218 3341 y Ft(\()p Fp(T)2329 3300 y Fn(k)r Fm(\000)p Fn(j)2477 3341 y Fp(')p Ft(\)\()p Fp(z)2666 3356 y Fn(j)t Fm(\000)p Fs(1)2810 3341 y Ft(\))553 b(\(4)p Fp(:)p Ft(10\))113 3649 y(is)34 b(holomorphic)e(in)h(a)g(neigh) m(b)s(orho)s(o)s(d)h(of)f Fp(D)s Ft(\()p Fp(m)1933 3664 y Fs(1)1979 3649 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)2288 3664 y Fn(k)2336 3649 y Ft(\))33 b(and)h(satis\014es)g(there)553 4014 y Fo(j)p Fp(R)658 3962 y Fs([)p Fn(k)r Fs(])657 4040 y(0)751 4014 y Ft(\()p Fp(z)836 4029 y Fs(0)881 4014 y Ft(\))p Fo(j)27 b(\024)h Fp(c)1190 3889 y Fn(k)1140 3919 y Fk(X)1143 4131 y Fn(j)t Fs(=1)1301 4014 y Fp(q)1349 3971 y Fm(\000)p Fs(1)1345 4041 y Fn(j)t Fm(\000)p Fs(1)1506 4014 y Ft(sup)1512 4097 y Fn(D)1577 4107 y Ff(1)1672 4014 y Fo(j)p Fp(T)1772 3972 y Fn(k)r Fm(\000)p Fn(j)1920 4014 y Fp(')p Fo(j)g(\024)g Fp(ck)2261 3843 y Fk( )2351 3864 y Fo(p)p 2434 3864 50 4 v 82 x Ft(5)22 b(+)h(1)p 2351 3991 305 4 v 2479 4082 a(2)2668 3843 y Fk(!)2747 3864 y Fm(\000)p Fn(k)2875 4014 y Ft(sup)2881 4097 y Fn(D)2946 4107 y Ff(1)3041 4014 y Fo(j)p Fp(')p Fo(j)k Fp(:)113 4366 y Ft(The)34 b(function)928 4571 y Fp(R)1005 4519 y Fs([)p Fn(k)r Fs(])1004 4598 y(1)1099 4571 y Ft(\()p Fp(z)1184 4586 y Fs(0)1229 4571 y Ft(\))27 b(=)1400 4447 y Fn(k)r Fm(\000)p Fs(1)1401 4476 y 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Fs(1)2204 5118 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)2513 5133 y Fn(k)2562 5118 y Ft(\))33 b(and)g(satis\014es)h(there)518 5483 y Fo(j)p Fp(R)623 5431 y Fs([)p Fn(k)r Fs(])622 5510 y(1)716 5483 y Ft(\()p Fp(z)801 5498 y Fs(0)846 5483 y Ft(\))p Fo(j)27 b(\024)h Fp(c)1105 5359 y Fn(k)r Fm(\000)p Fs(1)1106 5388 y Fk(X)1109 5601 y Fn(j)t Fs(=1)1267 5483 y Fp(q)1315 5440 y Fm(\000)p Fs(1)1311 5511 y Fn(j)1439 5483 y Ft(sup)1445 5567 y Fn(D)1510 5577 y Ff(1)1606 5483 y Fo(j)p Fp(T)1706 5442 y Fn(k)r Fm(\000)p Fn(j)t Fm(\000)p Fs(1)1956 5483 y Fp(')p Fo(j)f(\024)h Fp(ck)2296 5313 y Fk( )2387 5333 y Fo(p)p 2470 5333 50 4 v 83 x Ft(5)22 b(+)g(1)p 2387 5460 305 4 v 2514 5551 a(2)2703 5313 y Fk(!)2782 5333 y Fm(\000)p Fn(k)2910 5483 y Ft(sup)2916 5567 y Fn(D)2981 5577 y Ff(1)3076 5483 y Fo(j)p Fp(')p Fo(j)28 b Fp(:)p eop %%Page: 22 22 22 21 bop 113 144 a Ft(22)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y(The)34 b(function)884 576 y Fp(R)961 524 y Fs([)p Fn(k)r Fs(])960 603 y(2)1055 576 y Ft(\()p Fp(z)1140 591 y Fs(0)1184 576 y Ft(\))28 b(=)1356 451 y Fn(k)r Fm(\000)p Fs(2)1357 481 y Fk(X)1360 694 y Fn(j)t Fs(=1)1502 576 y Ft(\()p Fp(p)1591 591 y Fn(j)t Fm(\000)p Fs(1)1757 576 y Fo(\000)23 b Fp(q)1901 591 y Fn(j)t Fm(\000)p Fs(1)2045 576 y Fp(z)2091 591 y Fs(0)2136 576 y Ft(\))p Fp(")2221 591 y Fn(j)2263 576 y Fp(T)2335 535 y Fn(k)r Fm(\000)p Fn(j)2483 576 y Fp(')p Ft(\()p Fp(z)2633 591 y Fn(j)2698 576 y Ft(+)f(1\))516 b(\(4)p Fp(:)p Ft(12\))113 866 y(is)46 b(holomorphic)e(in)h(a)g(neigh)m(b)s(orho)s(o)s(d)h(of)f Fp(D)s Ft(\()p Fp(m)2005 881 y Fs(1)2050 866 y Fp(;)17 b(:)g(:)g(:)e(;)i(m)2360 881 y Fn(k)2408 866 y Ft(\))45 b(and)h(satis\014es,)j(according)c(to)113 985 y(Prop)s(ositions)33 b(4.1)g(and)g(4.2)518 1319 y Fo(j)p Fp(R)623 1267 y Fs([)p Fn(k)r Fs(])622 1345 y(2)716 1319 y Ft(\()p Fp(z)801 1334 y Fs(0)846 1319 y Ft(\))p Fo(j)27 b(\024)h Fp(c)1105 1194 y Fn(k)r Fm(\000)p Fs(2)1106 1224 y Fk(X)1109 1436 y Fn(j)t Fs(=1)1267 1319 y Fp(q)1315 1276 y Fm(\000)p Fs(1)1311 1347 y Fn(j)1439 1319 y Ft(sup)1445 1403 y Fn(D)1510 1413 y Ff(1)1606 1319 y Fo(j)p Fp(T)1706 1278 y Fn(k)r Fm(\000)p Fn(j)t Fm(\000)p Fs(2)1956 1319 y Fp(')p Fo(j)f(\024)h Fp(ck)2296 1149 y Fk( )2387 1169 y Fo(p)p 2470 1169 50 4 v 82 x Ft(5)22 b(+)g(1)p 2387 1296 305 4 v 2514 1387 a(2)2703 1149 y Fk(!)2782 1169 y Fm(\000)p Fn(k)2910 1319 y Ft(sup)2916 1403 y Fn(D)2981 1413 y Ff(1)3076 1319 y Fo(j)p Fp(')p Fo(j)28 b Fp(:)113 1651 y Ft(Finally)-8 b(,)25 b(w)m(e)f(apply)g(Prop)s(osition)f(4.2)h (to)g(\()p Fp(p)1748 1666 y Fn(k)r Fm(\000)p Fs(2)1903 1651 y Fo(\000)t Fp(q)2028 1666 y Fn(k)r Fm(\000)p Fs(2)2180 1651 y Fp(z)2226 1666 y Fs(0)2271 1651 y Ft(\))p Fp(")2356 1666 y Fn(k)r Fm(\000)p Fs(1)2507 1651 y Fp(T)14 b(')p Ft(\()p Fp(z)2729 1666 y Fn(k)r Fm(\000)p Fs(1)2884 1651 y Ft(+)t(1\))24 b(and)h(summing)113 1770 y(up)34 b(the)g(di\013eren)m (t)g(con)m(tributions)g(to)f(\(4.9\))f(w)m(e)h(get)g(the)h(desired)h (result.)695 b Fd(\003)113 2032 y Fr(4.3)39 b Fp(H)389 1996 y Fn(p)435 2032 y Fr({estimates)113 2180 y(4.3.1)53 b Ft(Let)f(1)58 b Fp(<)h(p)g(<)g Ft(+)p Fo(1)p Ft(.)99 b(W)-8 b(e)52 b(consider)h(the)f(space)h Fp(H)2554 2144 y Fn(p)2600 2180 y Ft(\()p Fq(H)2710 2144 y Fm(\006)2783 2180 y Ft(\))f(of)g(functions)h Fp(F)72 b Fo(2)113 2300 y(O)195 2263 y Fs(1)240 2300 y Ft(\()p 279 2219 72 4 v Fq(C)52 b Fo(n)25 b Ft([0)p Fp(;)17 b Ft(1]\))37 b(whose)j (restrictions)f(to)f(b)s(oth)h Fq(H)1991 2263 y Fs(+)2102 2300 y Ft(and)g Fq(H)2373 2263 y Fm(\000)2485 2300 y Ft(b)s(elong)g(to)f(the)h(Hardy)g(space)113 2419 y Fp(H)204 2383 y Fn(p)284 2419 y Ft(of)33 b(these)h(half{spaces,)g(endo)m(w)m(ed) h(with)e(the)h(norm)1000 2645 y Fo(k)p Fp(F)14 b Fo(k)1178 2664 y Fn(H)1248 2644 y Fj(p)1290 2664 y Fs(\()p Fg(H)1369 2644 y Ff(\006)1434 2664 y Fs(\))1498 2645 y Ft(=)28 b Fo(k)p Fp(F)42 b Fo(j)1787 2664 y Fg(H)1835 2644 y Fi(+)1929 2645 y Fo(k)1979 2660 y Fn(H)2049 2640 y Fj(p)2119 2645 y Ft(+)22 b Fo(k)p Fp(F)42 b Fo(j)2402 2664 y Fg(H)2450 2644 y Ff(\000)2547 2645 y Fo(k)2597 2660 y Fn(H)2667 2640 y Fj(p)2742 2645 y Fp(:)632 b Ft(\(4)p Fp(:)p Ft(13\))113 2871 y(See)60 b(references)h([Du,)k(St])59 b(for)g(details.)120 b(Then,)66 b(it)58 b(is)h(a)f(classical)h(result)g([Ga])f(that)113 2990 y Fp(F)d Fo(2)41 b Fp(H)430 2954 y Fn(p)476 2990 y Ft(\()p Fq(H)587 2954 y Fm(\006)660 2990 y Ft(\))g(if)g(and)h(only)e (if)h(the)h(asso)s(ciated)f(h)m(yp)s(erfunction)i Fp(u)e Ft(b)s(elongs)g(to)g Fp(L)3334 2954 y Fn(p)3380 2990 y Ft(\([0)p Fp(;)17 b Ft(1]\))113 3110 y(and)34 b(that)f(the)h(corresp) s(ondence)h(is)e(an)h(isomorphism)d(of)j(Banac)m(h)f(spaces.)313 3243 y(But)f(w)m(e)h(kno)m(w)g([MMY])g(that)f Fp(T)47 b Ft(acting)33 b(on)g Fp(L)2087 3207 y Fn(p)2133 3243 y Ft(\([0)p Fp(;)17 b Ft(1]\))30 b(has)j(sp)s(ectral)g(radius)h Fo(\024)3410 3146 y Fm(p)p 3475 3146 40 4 v 58 x Fs(5)p Fm(\000)p Fs(1)p 3410 3220 208 4 v 3494 3278 a(2)3629 3243 y Ft(.)113 3363 y(Consequen)m(tly)g(the)g(same)e(is)h(true)h(for)g Fp(T)47 b Ft(acting)33 b(on)g Fp(H)2249 3327 y Fn(p)2295 3363 y Ft(\()p Fq(H)2405 3327 y Fm(\006)2478 3363 y Ft(\).)313 3511 y Fr(4.3.2)42 b Ft(F)-8 b(or)40 b(the)i(case)g Fp(p)g Ft(=)f Fo(1)g Ft(and)h(on)g(the)g(larger)f(domain)p 2708 3430 72 4 v 40 w Fq(C)54 b Fo(n)27 b Ft([0)p Fp(;)17 b Ft(1])40 b(w)m(e)h(ha)m(v)m(e)h(the)113 3630 y(follo)m(wing)113 3807 y Fr(Prop)s(osition)28 b(4.5)58 b Fe(The)25 b(restriction)f(of)g Fp(T)39 b Fe(to)23 b Fp(H)1994 3771 y Fm(1)2079 3807 y Ft(\()p 2118 3726 V Fq(C)30 b Fo(n)t Ft([0)p Fp(;)17 b Ft(1]\))t Fo(\\)t(O)2644 3771 y Fs(1)2687 3807 y Ft(\()p 2726 3726 V Fq(C)30 b Fo(n)t Ft([0)p Fp(;)17 b Ft(1]\))22 b Fe(is)i(a)g(b)s(ounded)113 3946 y(op)s(erator)33 b(on)h(this)f(space) h(with)f(sp)s(ectral)g(radius)h Fo(\024)2132 3849 y Fm(p)p 2198 3849 40 4 v 2198 3907 a Fs(5)o Fm(\000)p Fs(1)p 2132 3923 208 4 v 2216 3980 a(2)2351 3946 y Fe(.)113 4164 y Fl(Pr)-5 b(o)g(of.)29 b Ft(Let)e Fp(k)k Fo(\025)d Ft(1,)g Fp(')g Fo(2)g Fp(H)1150 4128 y Fm(1)1234 4164 y Ft(\()p 1273 4083 72 4 v Fq(C)36 b Fo(n)10 b Ft([0)p Fp(;)17 b Ft(1]\))10 b Fo(\\)g(O)1823 4128 y Fs(1)1867 4164 y Ft(\()p 1906 4083 V Fq(C)36 b Fo(n)10 b Ft([0)p Fp(;)17 b Ft(1]\))25 b(and)j Fp(z)2547 4179 y Fs(0)2620 4164 y Fo(2)p 2714 4083 V 28 w Fq(C)36 b Fo(n)10 b Ft([0)p Fp(;)17 b Ft(1].)40 b(W)-8 b(e)28 b(estimate)113 4284 y Fo(j)p Fp(T)213 4247 y Fn(k)262 4284 y Fp(')p Ft(\()p Fp(z)412 4299 y Fs(0)457 4284 y Ft(\))p Fo(j)33 b Ft(in)g(v)-6 b(arious)33 b(cases.)158 4403 y(\(i\))49 b(If)33 b Fp(z)458 4418 y Fs(0)531 4403 y Fo(2)28 b Fp(D)707 4418 y Fm(1)825 4403 y Ft(w)m(e)34 b(ha)m(v)m(e)1205 4742 y Fo(j)p Fp(T)1305 4701 y Fn(k)1354 4742 y Fp(')p Ft(\()p Fp(z)1504 4757 y Fs(0)1549 4742 y Ft(\))p Fo(j)27 b(\024)h Fp(c)1808 4572 y Fk( )1899 4593 y Fo(p)p 1982 4593 50 4 v 82 x Ft(5)22 b(+)g(1)p 1899 4719 305 4 v 2026 4811 a(2)2215 4572 y Fk(!)2294 4593 y Fm(\000)p Fn(k)2422 4742 y Ft(sup)2428 4826 y Fn(D)2493 4836 y Ff(1)2588 4742 y Fo(j)p Fp(')p Fo(j)28 b Fp(;)313 5052 y Ft(according)33 b(to)g(Remark)f(3.4.)130 5172 y(\(ii\))49 b(If)37 b Fp(z)462 5187 y Fs(0)542 5172 y Fo(2)e Fp(D)725 5187 y Fs(0)795 5172 y Fo([)25 b Fp(H)969 5187 y Fs(0)1038 5172 y Fo([)h Fp(H)1221 5136 y Fm(0)1213 5197 y Fs(0)1294 5172 y Ft(w)m(e)38 b(ha)m(v)m(e,)h(according)e(to)g (Prop)s(osition)f(4.1)h(and)h(to)f(Remark)313 5291 y(3.4)854 5500 y Fo(j)p Fp(T)954 5458 y Fn(k)1003 5500 y Fp(')p Ft(\()p Fp(z)1153 5515 y Fs(0)1197 5500 y Ft(\))p Fo(j)28 b(\024)g Fp(c)17 b Ft(sup)1462 5583 y Fn(D)1527 5593 y Ff(1)1623 5500 y Fo(j)p Fp(T)1723 5458 y Fn(k)r Fm(\000)p Fs(1)1874 5500 y Fp(')p Fo(j)27 b(\024)h Fp(c)2159 5329 y Fk( )2250 5350 y Fo(p)p 2333 5350 50 4 v 82 x Ft(5)22 b(+)h(1)p 2250 5477 305 4 v 2377 5568 a(2)2566 5329 y Fk(!)2645 5350 y Fm(\000)p Fn(k)2773 5500 y Ft(sup)2779 5583 y Fn(D)2844 5593 y Ff(1)2940 5500 y Fo(j)p Fp(')p Fo(j)k Fp(:)p eop %%Page: 23 23 23 22 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(23)102 383 y(\(iii\))49 b(if)33 b Fp(z)450 398 y Fs(0)523 383 y Fo(2)28 b Fp(D)699 398 y Fs(1)777 383 y Ft(w)m(e)34 b(ha)m(v)m(e,)f(according)h(to)f(Prop)s(osition)f(4.2)913 640 y Fo(j)p Fp(T)1013 599 y Fn(k)1061 640 y Fp(')p Ft(\()p Fp(z)1211 655 y Fs(0)1256 640 y Ft(\))p Fo(j)27 b(\024)i Fp(c)1516 500 y Fk(\024)1568 640 y Ft(sup)1574 724 y Fn(D)1639 734 y Ff(1)1734 640 y Fo(j)p Fp(T)1834 599 y Fn(k)r Fm(\000)p Fs(1)1985 640 y Fp(')p Fo(j)22 b Ft(+)2200 496 y Fk(\014)2200 556 y(\014)2200 615 y(\014)2200 675 y(\014)2233 640 y Fp(T)2305 599 y Fn(k)r Fm(\000)p Fs(1)2456 640 y Fp(')2538 500 y Fk(\022)2644 573 y Ft(1)p 2623 617 92 4 v 2623 709 a Fp(z)2669 724 y Fs(0)2748 640 y Fo(\000)h Ft(1)2898 500 y Fk(\023)2971 496 y(\014)2971 556 y(\014)2971 615 y(\014)2971 675 y(\014)3004 500 y(\025)313 906 y Ft(whic)m(h)44 b(giv)m(es,)i(for)e Fp(k)k Ft(=)e(1)e(:)65 b Fo(j)p Fp(T)14 b(')p Ft(\()p Fp(z)1716 921 y Fs(0)1760 906 y Ft(\))p Fo(j)45 b(\024)h Fp(c)p Fo(k)p Fp(')p Fo(k)2203 921 y Fn(H)2273 901 y Ff(1)2395 906 y Ft(and)e(for)g Fp(k)k(>)e Ft(1,)g(according)e(to)313 1026 y(Prop)s(osition)32 b(4.1)g(:)854 1331 y Fo(j)p Fp(T)954 1290 y Fn(k)1003 1331 y Fp(')p Ft(\()p Fp(z)1153 1346 y Fs(0)1197 1331 y Ft(\))p Fo(j)c(\024)g Fp(c)17 b Ft(sup)1462 1415 y Fn(D)1527 1425 y Ff(1)1623 1331 y Fo(j)p Fp(T)1723 1290 y Fn(k)r Fm(\000)p Fs(2)1874 1331 y Fp(')p Fo(j)27 b(\024)h Fp(c)2159 1161 y Fk( )2250 1182 y Fo(p)p 2333 1182 50 4 v 82 x Ft(5)22 b(+)h(1)p 2250 1308 305 4 v 2377 1400 a(2)2566 1161 y Fk(!)2645 1182 y Fm(\000)p Fn(k)2773 1331 y Ft(sup)2779 1415 y Fn(D)2844 1425 y Ff(1)2940 1331 y Fo(j)p Fp(')p Fo(j)k Fp(:)105 1631 y Ft(\(iv\))49 b(If)27 b Fp(z)452 1646 y Fs(0)525 1631 y Fo(2)h Fp(D)j Ft(has)d(con)m(tin)m(ued)h(fraction)e Fp(z)1764 1589 y Fm(\000)p Fs(1)1760 1659 y Fn(i)1900 1631 y Ft(=)h Fp(m)2092 1646 y Fn(i)p Fs(+1)2237 1631 y Ft(+)10 b Fp(z)2370 1646 y Fn(i)p Fs(+1)2532 1631 y Ft(with)28 b(0)f Fo(\024)h Fp(i)g(<)g(l)r Ft(,)g Fp(z)3237 1646 y Fn(l)3296 1631 y Fo(2)g Ft(\001)10 b Fo(n)g Fp(D)s Ft(,)313 1751 y Fp(l)29 b(<)f(k)s Ft(,)33 b(w)m(e)h(apply)f(Prop)s(osition)f(4.4)h(to)f(write) 372 1948 y Fp(T)444 1907 y Fn(k)493 1948 y Fp(')p Ft(\()p Fp(z)643 1963 y Fs(0)688 1948 y Ft(\))27 b(=)h(\()p Fp(p)948 1963 y Fn(l)p Fm(\000)p Fs(1)1103 1948 y Fo(\000)22 b Fp(q)1246 1963 y Fn(l)p Fm(\000)p Fs(1)1379 1948 y Fp(z)1425 1963 y Fs(0)1470 1948 y Ft(\)[)p Fp(T)1609 1907 y Fn(k)r Fm(\000)p Fn(l)1746 1948 y Fp(')p Ft(\()p Fp(z)1896 1963 y Fn(l)1927 1948 y Ft(\))g(+)g Fp(T)2159 1907 y Fn(k)r Fm(\000)p Fn(l)2296 1948 y Fp(')p Ft(\()p Fp(z)2446 1963 y Fn(l)2499 1948 y Fo(\000)h Ft(1\))e(+)i Fp(")2855 1963 y Fn(l)2886 1948 y Fp(T)2958 1907 y Fn(k)r Fm(\000)p Fn(l)3095 1948 y Fp(')p Ft(\()p Fp(z)3245 1963 y Fn(l)3298 1948 y Ft(+)g(1\)])580 2167 y Fo(\000)g Ft(\()p Fp(p)769 2182 y Fn(l)p Fm(\000)p Fs(2)923 2167 y Fo(\000)g Fp(q)1067 2182 y Fn(l)p Fm(\000)p Fs(2)1200 2167 y Fp(z)1246 2182 y Fs(0)1291 2167 y Ft(\)\(1)e(+)i Fp(z)1586 2182 y Fn(l)p Fm(\000)p Fs(1)1719 2167 y Ft(\))p Fp(")1804 2182 y Fn(l)p Fm(\000)p Fs(1)1937 2167 y Fp(T)2009 2126 y Fn(k)r Fm(\000)p Fn(l)2146 2167 y Fp(')2228 2026 y Fk(\022)2360 2099 y Fo(\000)p Fp(z)2483 2114 y Fn(l)p Fm(\000)p Fs(1)p 2313 2144 351 4 v 2313 2235 a Ft(1)f(+)g Fp(z)2530 2250 y Fn(l)p Fm(\000)p Fs(1)2675 2026 y Fk(\023)2771 2167 y Ft(+)g Fp(R)2947 2126 y Fs([)p Fn(l)p Fs(])3022 2167 y Ft(\()p Fp(T)3133 2126 y Fn(k)r Fm(\000)p Fn(l)3270 2167 y Fp(')p Ft(\)\()p Fp(z)3459 2182 y Fs(0)3504 2167 y Ft(\))27 b Fp(:)313 2462 y Ft(Here,)32 b(w)m(e)h(ha)m(v)m(e)g Fp(z)996 2477 y Fn(l)1027 2462 y Ft(,)f Fp(z)1133 2477 y Fn(l)1185 2462 y Fo(\000)21 b Ft(1)32 b(and)1617 2408 y Fo(\000)p Fp(z)1740 2423 y Fn(l)p Fm(\000)p Fs(1)p 1570 2439 V 1570 2507 a Ft(1)22 b(+)h Fp(z)1788 2522 y Fn(l)p Fm(\000)p Fs(1)1965 2462 y Ft(in)33 b Fp(D)2163 2477 y Fm(1)2268 2462 y Fo([)21 b Fp(D)2437 2477 y Fs(0)2503 2462 y Fo([)g Fp(H)2673 2477 y Fs(0)2739 2462 y Fo([)g Fp(H)2917 2425 y Fm(0)2909 2486 y Fs(0)2986 2462 y Ft(hence)34 b(the)f(v)-6 b(alue)313 2625 y(of)48 b Fp(T)513 2589 y Fn(k)r Fm(\000)p Fn(l)650 2625 y Fp(')g Ft(at)g(these)h(p)s(oin)m(ts) f(is)g(in)g(absolute)g(v)-6 b(alue)48 b(less)h(than)f Fp(c)17 b Ft(sup)3066 2649 y Fn(D)3131 2659 y Ff(1)3225 2625 y Fo(j)p Fp(T)3325 2589 y Fn(k)r Fm(\000)p Fn(l)p Fm(\000)p Fs(1)3564 2625 y Fp(')p Fo(j)313 2744 y Ft(\(Prop)s(osition) 25 b(4.1\).)41 b(W)-8 b(e)26 b(also)h(ha)m(v)m(e,)g(according)g(to)f (Prop)s(osition)g(4.2)g(and)h(Prop)s(osition)313 2864 y(4.1)820 3018 y Fo(j)p Fp(T)920 2977 y Fn(k)r Fm(\000)p Fn(l)1056 3018 y Fp(')p Ft(\()p Fp(z)1206 3033 y Fn(l)1259 3018 y Ft(+)c(1\))p Fo(j)k(\024)1608 2878 y Fk(\032)1699 2954 y Fp(c)p Fo(k)p Fp(')p Fo(k)1907 2969 y Fn(H)1977 2949 y Ff(1)2600 2954 y Ft(if)33 b Fp(k)e Ft(=)d Fp(l)c Ft(+)e(1)28 b Fp(;)1699 3073 y(c)17 b Ft(sup)1909 3098 y Fn(D)1974 3108 y Ff(1)2069 3073 y Fo(j)p Fp(T)2169 3037 y Fn(k)r Fm(\000)p Fn(l)p Fm(\000)p Fs(2)2407 3073 y Fp(')p Fo(j)100 b Ft(if)33 b Fp(k)e(>)d(l)c Ft(+)e(1)28 b Fp(:)313 3243 y Ft(On)34 b(the)f(other)h(hand)g(w)m(e)f(ha)m(v)m(e) 1073 3529 y Fo(j)p Fp(p)1151 3544 y Fn(l)p Fm(\000)p Fs(2)1305 3529 y Fo(\000)23 b Fp(q)1449 3544 y Fn(l)p Fm(\000)p Fs(2)1582 3529 y Fp(z)1628 3544 y Fs(0)1673 3529 y Fo(j)k(\024)h Fp(cq)1924 3487 y Fm(\000)p Fs(1)1920 3559 y Fn(l)p Fm(\000)p Fs(1)2081 3529 y Fo(\024)g Fp(c)2246 3359 y Fk( )2337 3379 y Fo(p)p 2420 3379 50 4 v 83 x Ft(5)22 b(+)g(1)p 2337 3506 305 4 v 2464 3598 a(2)2653 3359 y Fk(!)2732 3380 y Fm(\000)p Fn(l)2869 3529 y Fp(;)1073 3892 y Fo(j)p Fp(p)1151 3907 y Fn(l)p Fm(\000)p Fs(1)1305 3892 y Fo(\000)h Fp(q)1449 3907 y Fn(l)p Fm(\000)p Fs(1)1582 3892 y Fp(z)1628 3907 y Fs(0)1673 3892 y Fo(j)k(\024)h Fp(cq)1924 3849 y Fm(\000)p Fs(1)1920 3922 y Fn(l)2059 3892 y Fo(\024)g Fp(c)2224 3721 y Fk( )2315 3742 y Fo(p)p 2398 3742 50 4 v 82 x Ft(5)22 b(+)g(1)p 2315 3869 305 4 v 2442 3960 a(2)2631 3721 y Fk(!)2710 3742 y Fm(\000)p Fn(l)2847 3892 y Fp(;)313 4193 y Ft(and)40 b(th)m(us)g(from)f(the)h (estimate)e(of)i Fp(R)1760 4156 y Fs([)p Fn(l)p Fs(])1874 4193 y Ft(in)g(Prop)s(osition)e(4.4)h(and)h(from)f(Remark)f(3.4)313 4312 y(w)m(e)33 b(ha)m(v)m(e)817 4634 y Fo(j)p Fp(T)917 4593 y Fn(k)966 4634 y Fp(')p Ft(\()p Fp(z)1116 4649 y Fs(0)1161 4634 y Ft(\))p Fo(j)27 b(\024)1360 4400 y Fk(8)1360 4490 y(>)1360 4520 y(<)1360 4699 y(>)1360 4729 y(:)1465 4540 y Fp(c)17 b(l)1573 4429 y Fk(\020)1644 4442 y Fm(p)p 1710 4442 40 4 v 1710 4500 a Fs(5+1)p 1644 4517 207 4 v 1728 4574 a(2)1862 4429 y Fk(\021)1922 4450 y Fm(\000)p Fn(l)2031 4540 y Fo(k)p Fp(')p Fo(k)2196 4555 y Fn(H)2266 4535 y Ff(1)2603 4540 y Ft(if)33 b Fp(k)e Ft(=)d Fp(l)c Ft(+)e(1)28 b Fp(;)1465 4753 y(c)17 b(k)1596 4642 y Fk(\020)1668 4655 y Fm(p)p 1733 4655 40 4 v 58 x Fs(5+1)p 1668 4730 207 4 v 1751 4787 a(2)1886 4642 y Fk(\021)1945 4663 y Fm(\000)p Fn(k)2073 4753 y Ft(sup)2223 4777 y Fn(D)2288 4787 y Ff(1)2383 4753 y Fo(j)p Fp(')p Fo(j)99 b Ft(if)33 b Fp(k)e(>)d(l)c Ft(+)e(1)28 b Fp(:)133 4994 y Ft(\(v\))49 b(If)35 b Fp(z)460 5009 y Fs(0)536 4994 y Fo(2)c Fp(D)38 b Ft(has)d(con)m(tin)m(ued)i(fraction)e Fp(z)1808 4951 y Fm(\000)p Fs(1)1804 5022 y Fn(i)1946 4994 y Ft(=)c Fp(m)2141 5009 y Fn(i)p Fs(+1)2299 4994 y Ft(+)24 b Fp(z)2446 5009 y Fn(i)p Fs(+1)2615 4994 y Ft(with)35 b(0)30 b Fo(\024)h Fp(i)g(<)g(k)s Ft(,)k Fp(z)3369 5009 y Fn(k)3449 4994 y Fo(2)c Ft(\001,)313 5113 y(w)m(e)i(again)g (apply)g(Prop)s(osition)f(4.4)h(to)g(get)402 5310 y Fp(T)474 5269 y Fn(k)523 5310 y Fp(')p Ft(\()p Fp(z)673 5325 y Fs(0)718 5310 y Ft(\))28 b(=)g(\()p Fp(p)979 5325 y Fn(k)r Fm(\000)p Fs(1)1152 5310 y Fo(\000)22 b Fp(q)1295 5325 y Fn(k)r Fm(\000)p Fs(1)1447 5310 y Fp(z)1493 5325 y Fs(0)1538 5310 y Ft(\)[)p Fp(')p Ft(\()p Fp(z)1755 5325 y Fn(k)1804 5310 y Ft(\))g(+)g Fp(')p Ft(\()p Fp(z)2114 5325 y Fn(k)2186 5310 y Fo(\000)g Ft(1\))g(+)h Fp(")2542 5325 y Fn(k)2591 5310 y Fp(')p Ft(\()p Fp(z)2741 5325 y Fn(k)2812 5310 y Ft(+)g(1\)])779 5529 y Fo(\000)g Ft(\()p Fp(p)968 5544 y Fn(k)r Fm(\000)p Fs(2)1141 5529 y Fo(\000)f Fp(q)1284 5544 y Fn(k)r Fm(\000)p Fs(2)1436 5529 y Fp(z)1482 5544 y Fs(0)1527 5529 y Ft(\)\(1)f(+)i Fp(z)1822 5544 y Fn(k)r Fm(\000)p Fs(1)1973 5529 y Ft(\))p Fp(")2058 5544 y Fn(k)r Fm(\000)p Fs(1)2209 5529 y Fp(')2291 5389 y Fk(\022)2365 5529 y Fo(\000)2540 5462 y Fp(z)2586 5477 y Fn(k)r Fm(\000)p Fs(1)p 2454 5507 369 4 v 2454 5598 a Ft(1)f(+)h Fp(z)2672 5613 y Fn(k)r Fm(\000)p Fs(1)2835 5389 y Fk(\023)2930 5529 y Ft(+)g Fp(R)3107 5488 y Fs([)p Fn(k)r Fs(])3201 5529 y Ft(\()p Fp(')p Ft(\)\()p Fp(z)3429 5544 y Fs(0)3473 5529 y Ft(\))k Fp(;)p eop %%Page: 24 24 24 23 bop 113 144 a Ft(24)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)313 383 y(whic)m(h)34 b(giv)m(es)841 697 y Fo(j)p Fp(T)941 656 y Fn(k)990 697 y Fp(')p Ft(\()p Fp(z)1140 712 y Fs(0)1185 697 y Ft(\))p Fo(j)27 b(\024)h Fp(cq)1475 655 y Fm(\000)p Fs(1)1471 728 y Fn(k)r Fm(\000)p Fs(1)1622 697 y Fo(k)p Fp(')p Fo(k)1787 712 y Fn(H)1857 692 y Ff(1)1958 697 y Ft(+)22 b Fp(ck)2172 527 y Fk( )2263 548 y Fo(p)p 2346 548 50 4 v 82 x Ft(5)g(+)g(1)p 2263 675 305 4 v 2390 766 a(2)2579 527 y Fk(!)2658 548 y Fm(\000)p Fn(k)2786 697 y Ft(sup)2792 781 y Fn(D)2857 791 y Ff(1)2953 697 y Fo(j)p Fp(')p Fo(j)27 b Fp(:)313 994 y Ft(Collecting)32 b(the)i(estimates)e(obtained)i(in)f(\(i\){\(v\))e(w)m(e)j(get)f(the)g (desired)i(result.)247 b Fd(\003)313 1192 y Fr(4.3.3)37 b Ft(Let)f(us)h(no)m(w)g(consider)h(the)f(conformally)e(in)m(v)-6 b(arian)m(t)36 b(v)m(ersion)h(of)g Fp(H)3192 1156 y Fn(p)3237 1192 y Ft(,)g(1)c Fp(<)h(p)f(<)113 1332 y Ft(+)p Fo(1)p Ft(.)80 b(The)46 b(application)f Fp(w)50 b Fo(7!)d Fp(z)53 b Ft(=)1646 1283 y Fs(\()p Fn(w)r Fs(+1\))1869 1253 y Fi(2)p 1646 1309 262 4 v 1727 1366 a Fs(4)p Fn(w)1964 1332 y Ft(is)45 b(a)g(conformal)g(represen)m(tation)h(of)f Fq(D)71 b Ft(on)113 1451 y Fq(C)56 b Fo(n)29 b Ft([0)p Fp(;)17 b Ft(1])h(;)49 b(letting)43 b Fp(w)48 b Ft(=)e Fp(e)1204 1415 y Fs(2)p Fn(\031)s(i\022)1366 1451 y Ft(,)h(one)d (\014nds)h Fp(z)50 b Ft(=)c(cos)2235 1415 y Fs(2)2297 1451 y Fp(\022)g Ft(th)m(us)f Fo(j)p Fp(dz)t Fo(j)g Ft(=)h(2)p Fo(j)17 b Ft(sin)f Fp(\022)s Fo(jj)h Ft(cos)e Fp(\022)s Fo(j)p Fp(d\022)s Ft(,)113 1571 y(i.e.)102 b Fp(d\022)62 b Ft(=)799 1532 y Fn(dx)p 653 1548 378 4 v 653 1628 a Fs(2)693 1560 y Fo(p)p 776 1560 255 4 v 68 x Fn(x)p Fs(\(1)p Fm(\000)p Fn(x)p Fs(\))1043 1571 y Ft(.)102 b(W)-8 b(e)53 b(can)f(therefore)i(iden)m(tify)f Fp(L)2453 1535 y Fn(p)2499 1571 y Ft(\()p Fq(T)2601 1535 y Fs(1)2649 1571 y Ft(\))f(with)g(the)h (direct)g(sum)113 1753 y(of)62 b(t)m(w)m(o)f(copies)h(of)f Fp(L)1002 1717 y Fn(p)1048 1753 y Ft(\([0)p Fp(;)17 b Ft(1])p Fp(;)1487 1714 y Fn(dx)p 1342 1730 378 4 v 1342 1810 a Fs(2)1382 1742 y Fo(p)p 1465 1742 255 4 v 68 x Fn(x)p Fs(\(1)p Fm(\000)p Fn(x)p Fs(\))1731 1753 y Ft(\))61 b(\(one)h(for)f(eac)m(h)h(side)g(of)f([0)p Fp(;)17 b Ft(1]\).)126 b(Let)61 b(no)m(w)113 1936 y Fp(')28 b Fo(2)g(O)382 1900 y Fs(1)427 1936 y Ft(\()p 466 1856 72 4 v Fq(C)45 b Fo(n)19 b Ft([0)p Fp(;)e Ft(1]\))g Fo(\\)j Fp(H)1060 1900 y Fn(p)1106 1936 y Ft(\()p 1145 1856 V Fq(C)44 b Fo(n)19 b Ft([0)p Fp(;)e Ft(1]\))30 b(whic)m(h)j(w)m(e)f(will)f(assume) g(to)h(b)s(e)g Fl(r)-5 b(e)g(al)p Ft(.)43 b(The)33 b(asso)s(ciated)113 2056 y(h)m(yp)s(erfunction)39 b Fp(u)d Ft(is)h Fp(u)d Ft(=)g Fo(=)p Fp(m)17 b(')p Ft(\()p Fp(x)24 b Ft(+)h Fp(i)p Ft(0\),)37 b(and)g Fp(u)d Fo(2)g Fp(L)2260 2020 y Fn(p)2305 2056 y Ft(\([0)p Fp(;)17 b Ft(1])p Fp(;)2745 2017 y Fn(dx)p 2600 2033 378 4 v 2600 2113 a Fs(2)2640 2045 y Fo(p)p 2722 2045 255 4 v 2722 2113 a Fn(x)p Fs(\(1)p Fm(\000)p Fn(x)p Fs(\))2989 2056 y Ft(\).)54 b(W)-8 b(e)37 b(ha)m(v)m(e)g(the)113 2220 y(follo)m(wing)113 2392 y Fr(Prop)s(osition)62 b(4.6)c Fe(F)-8 b(or)53 b Fp(p)61 b Fo(\025)h Ft(1)53 b Fe(the)h(op)s(erator)f Fp(T)68 b Fe(de\014nes)55 b(a)e(b)s(ounded)i(op)s(erator)f(on)113 2525 y Fp(L)181 2489 y Fn(p)227 2525 y Ft(\([0)p Fp(;)17 b Ft(1])p Fp(;)667 2486 y Fn(dx)p 521 2502 378 4 v 521 2582 a Fs(2)561 2514 y Fo(p)p 644 2514 255 4 v 68 x Fn(x)p Fs(\(1)p Fm(\000)p Fn(x)p Fs(\))911 2525 y Ft(\))33 b Fe(and)g(its)g(sp)s(ectral)h(radius)f(is)h Fo(\024)2207 2428 y Fm(p)p 2272 2428 40 4 v 58 x Fs(5)p Fm(\000)p Fs(1)p 2207 2502 208 4 v 2291 2560 a(2)2426 2525 y Fe(.)113 2783 y Fl(Pr)-5 b(o)g(of.)34 b Ft(One)h(has)e Fp(T)14 b(u)p Ft(\()p Fp(x)p Ft(\))28 b(=)g Fp(xu)1327 2703 y Fk(\000)1387 2744 y Fs(1)p 1384 2760 46 4 v 1384 2818 a Fn(x)1463 2783 y Fo(\000)23 b Fp(m)1650 2703 y Fk(\001)1729 2783 y Ft(if)1898 2744 y Fs(1)p 1832 2760 172 4 v 1832 2818 a Fn(m)p Fs(+1)2044 2783 y Fo(\024)28 b Fp(x)f Fo(\024)2366 2744 y Fs(1)p 2351 2760 71 4 v 2351 2818 a Fn(m)2433 2783 y Ft(,)33 b(th)m(us)174 2950 y Fk(Z)273 2975 y Fs(1)p Fn(=m)229 3177 y Fs(1)p Fn(=)p Fs(\()p Fn(m)p Fs(+1\))565 3086 y Fo(j)p Fp(T)14 b(u)p Ft(\()p Fp(x)p Ft(\))p Fo(j)885 3045 y Fn(p)1144 3019 y Fp(dx)p 942 3063 513 4 v 942 3168 a Ft(2)992 3083 y Fk(p)p 1091 3083 363 4 v 1091 3168 a Fp(x)p Ft(\(1)22 b Fo(\000)h Fp(x)p Ft(\))1494 3086 y Fo(\024)28 b Fp(m)1686 3045 y Fs(1)p Fn(=)p Fs(2)p Fm(\000)p Fn(p)1932 2950 y Fk(Z)2031 2975 y Fs(1)p Fn(=m)1987 3177 y Fs(1)p Fn(=)p Fs(\()p Fn(m)p Fs(+1\))2323 2942 y Fk(\014)2323 3001 y(\014)2323 3061 y(\014)2323 3121 y(\014)2356 3086 y Fp(u)2430 2946 y Fk(\022)2519 3019 y Ft(1)p 2515 3063 57 4 v 2515 3154 a Fp(x)2606 3086 y Fo(\000)23 b Fp(m)2793 2946 y Fk(\023)2867 2942 y(\014)2867 3001 y(\014)2867 3061 y(\014)2867 3121 y(\014)2900 2966 y Fn(p)3148 3019 y Fp(dx)p 2975 3063 456 4 v 2975 3168 a Ft(2)3025 3083 y Fk(p)p 3124 3083 307 4 v 3124 3168 a Ft(\(1)f Fo(\000)g Fp(x)p Ft(\))1494 3411 y Fo(\024)28 b Fp(m)1686 3370 y Fs(1)p Fn(=)p Fs(2)p Fm(\000)p Fn(p)1932 3275 y Fk(Z)2031 3300 y Fs(1)1987 3501 y(0)2093 3411 y Fo(j)p Fp(u)p Ft(\()p Fp(s)p Ft(\))p Fo(j)2331 3370 y Fn(p)2392 3270 y Fk(\022)2465 3411 y Ft(1)22 b Fo(\000)2752 3343 y Ft(1)p 2649 3388 256 4 v 2649 3479 a Fp(s)g Ft(+)g Fp(m)2917 3270 y Fk(\023)2990 3291 y Fm(\000)p Fs(1)p Fn(=)p Fs(2)3346 3343 y Fp(ds)p 3206 3388 379 4 v 3206 3479 a Ft(\()p Fp(s)g Ft(+)g Fp(m)p Ft(\))3539 3450 y Fs(2)1494 3701 y Fo(\024)28 b Fp(m)1686 3660 y Fs(1)p Fn(=)p Fs(2)p Fm(\000)p Fn(p)1932 3565 y Fk(Z)2031 3590 y Fs(1)1987 3792 y(0)2093 3701 y Fo(j)p Fp(u)p Ft(\()p Fp(s)p Ft(\))p Fo(j)2331 3660 y Fn(p)2375 3701 y Ft(\()p Fp(s)22 b Ft(+)h Fp(m)p Ft(\))2709 3660 y Fm(\000)p Fs(3)p Fn(=)p Fs(2)2924 3634 y Fp(ds)p 2908 3678 130 4 v 2908 3698 a Fo(p)p 2991 3698 47 4 v 72 x Fp(s)3078 3701 y(:)113 3960 y Ft(Summing)32 b(o)m(v)m(er)i Fp(m)28 b Fo(\025)g Ft(1)33 b(w)m(e)h(ha)m(v)m(e)510 4308 y Fo(k)p Fp(T)14 b(u)p Fo(k)739 4349 y Fn(L)794 4329 y Fj(p)835 4349 y Fs(\([0)p Fn(;)p Fs(1])p Fn(;dx=)p Fs(2)1207 4282 y Fo(p)p 1288 4282 255 4 v 1288 4349 a Fn(x)p Fs(\(1)p Fm(\000)p Fn(x)p Fs(\))q(\))1606 4308 y Fo(\024)1712 4108 y Fk(2)1712 4287 y(4)1792 4213 y(X)1778 4427 y Fn(m)p Fm(\025)p Fs(1)1967 4308 y Fp(m)2054 4267 y Fm(\000)p Fn(p)p Fm(\000)p Fs(1)2265 4108 y Fk(3)2265 4287 y(5)2332 4128 y Fs(1)p Fn(=p)2475 4308 y Fo(k)p Fp(u)p Fo(k)2632 4331 y Fn(L)2687 4311 y Fj(p)2728 4331 y Fs(\([0)p Fn(;)p Fs(1])p Fn(;dx=)3060 4280 y Fm(p)p 3124 4280 46 4 v 3124 4331 a Fn(x)p Fs(\))3233 4308 y Fp(;)113 4628 y Ft(whic)m(h)34 b(pro)m(v)m(es)g(the)g(\014rst)g (assertion.)313 4748 y(As)c(far)h(as)g(the)f(sp)s(ectral)h(radius)g(is) g(concerned,)h(le)f(us)g(consider)g(an)g(in)m(teger)g Fp(k)g(>)d Ft(1)i(and)113 4867 y(the)35 b(set)f Fo(I)7 b Ft(\()p Fp(k)s Ft(\))34 b(of)g(the)g(in)m(terv)-6 b(als)34 b(of)g(de\014nition)g(of)g(the)g(branc)m(hes)i(of)e Fp(A)2802 4831 y Fn(k)2851 4867 y Ft(.)46 b(F)-8 b(or)33 b Fp(I)j Fo(2)29 b(I)7 b Ft(\()p Fp(k)s Ft(\))34 b(one)113 4987 y(has)g(\(w)m(e)f(denote)h(with)f Fp(x)1081 5002 y Fn(I)1160 4987 y Ft(the)h(cen)m(ter)g(of)g Fp(I)8 b Ft(\))500 5110 y Fk(Z)555 5336 y Fn(I)618 5245 y Fo(j)p Fp(T)718 5204 y Fn(k)766 5245 y Fp(u)p Ft(\()p Fp(x)p Ft(\))p Fo(j)986 5204 y Fn(p)1245 5178 y Fp(dx)p 1043 5222 513 4 v 1043 5327 a Ft(2)1093 5242 y Fk(p)p 1193 5242 363 4 v 85 x Fp(x)p Ft(\(1)22 b Fo(\000)g Fp(x)p Ft(\))1595 5245 y(=)1701 5110 y Fk(Z)1756 5336 y Fn(I)1818 5245 y Fp(\014)1874 5260 y Fn(k)r Fm(\000)p Fs(1)2026 5245 y Ft(\()p Fp(x)p Ft(\))2161 5204 y Fn(p)2206 5245 y Fo(j)p Fp(u)p Ft(\()p Fp(A)2405 5204 y Fn(k)2453 5245 y Fp(x)p Ft(\))p Fo(j)2577 5204 y Fn(p)2836 5178 y Fp(dx)p 2635 5222 513 4 v 2635 5327 a Ft(2)2685 5242 y Fk(p)p 2784 5242 363 4 v 2784 5327 a Fp(x)p Ft(\(1)g Fo(\000)g Fp(x)p Ft(\))1595 5533 y Fo(\024)29 b Fp(c)p Fo(j)p Fp(I)8 b Fo(j)1852 5492 y Fn(p)1913 5533 y Ft([)p Fp(x)1998 5548 y Fn(I)2044 5533 y Ft(\(1)21 b Fo(\000)i Fp(x)2311 5548 y Fn(I)2357 5533 y Ft(\)])2423 5483 y Fm(\000)p Fs(1)p Fn(=)p Fs(2)2627 5398 y Fk(Z)2682 5624 y Fn(I)2745 5533 y Fo(j)p Fp(u)p Ft(\()p Fp(A)2944 5492 y Fn(k)2992 5533 y Fp(x)p Ft(\))p Fo(j)3116 5492 y Fn(p)3161 5533 y Fp(dx)p eop %%Page: 25 25 25 24 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(25)113 386 y(since)51 b Fp(c)417 350 y Fm(\000)p Fs(1)540 386 y Ft([)p Fp(x)625 401 y Fn(I)671 386 y Ft(\(1)21 b Fo(\000)i Fp(x)938 401 y Fn(I)984 386 y Ft(\)])1050 336 y Fm(\000)p Fs(1)p Fn(=)p Fs(2)1292 386 y Fo(\024)55 b Ft([)o Fp(x)p Ft(\(1)22 b Fo(\000)h Fp(x)p Ft(\)])1842 336 y Fm(\000)p Fs(1)p Fn(=)p Fs(2)2084 386 y Fo(\024)55 b Fp(c)17 b Ft([)o Fp(x)2360 401 y Fn(I)2406 386 y Ft(\(1)22 b Fo(\000)g Fp(x)2673 401 y Fn(I)2719 386 y Ft(\)])2785 336 y Fm(\000)p Fs(1)p Fn(=)p Fs(2)3022 386 y Ft(for)50 b(all)e Fp(x)55 b Fo(2)f Fp(I)8 b Ft(,)113 505 y Fp(I)36 b Fo(2)28 b(I)7 b Ft(\()p Fp(k)s Ft(\))33 b(and)h Fp(k)c(>)f Ft(1)19 b(;)33 b(also)g Fp(c)1269 469 y Fm(\000)p Fs(1)1376 505 y Fo(j)p Fp(I)8 b Fo(j)27 b(\024)h Fp(\014)1672 520 y Fn(k)r Fm(\000)p Fs(1)1823 505 y Ft(\()p Fp(x)p Ft(\))f Fo(\024)i Fp(c)p Fo(j)p Fp(I)8 b Fo(j)32 b Ft(for)h(all)g Fp(x)27 b Fo(2)h Fp(I)8 b Ft(.)44 b(F)-8 b(rom)32 b(this)h(w)m(e)h(get) 509 612 y Fk(Z)564 838 y Fn(I)627 747 y Fo(j)p Fp(T)727 706 y Fn(k)775 747 y Fp(u)p Ft(\()p Fp(x)p Ft(\))p Fo(j)995 706 y Fn(p)1254 680 y Fp(dx)p 1053 724 513 4 v 1053 830 a Ft(2)1103 744 y Fk(p)p 1202 744 363 4 v 1202 830 a Fp(x)p Ft(\(1)22 b Fo(\000)g Fp(x)p Ft(\))1605 747 y Fo(\024)28 b Fp(c)p Fo(j)p Fp(I)8 b Fo(j)1861 706 y Fn(p)p Fs(+1)2023 747 y Ft([)p Fp(x)2108 762 y Fn(I)2154 747 y Ft(\(1)21 b Fo(\000)i Fp(x)2421 762 y Fn(I)2467 747 y Ft(\)])2533 697 y Fm(\000)p Fs(1)p Fn(=)p Fs(2)2737 747 y Fo(k)p Fp(u)p Fo(k)2894 699 y Fn(p)2894 781 y(L)2949 761 y Fj(p)2990 781 y Fs(\([0)p Fn(;)p Fs(1]\))3233 747 y Fp(;)113 1025 y Ft(\(since)34 b Fp(A)471 989 y Fn(k)553 1025 y Ft(has)g(b)s(ounded)h(distorsion)e(on)g Fp(I)8 b Ft(\).)313 1144 y(T)-8 b(aking)38 b(the)j(sum)e(o)m(v)m(er)h(all)f (the)h(in)m(terv)-6 b(als)40 b Fp(I)48 b Ft(\(i.e.)63 b(the)40 b(branc)m(hes)i(of)e Fp(A)3111 1108 y Fn(k)3160 1144 y Ft(\),)h(since)f(the)113 1264 y Fp(I)c Fo(2)28 b(I)7 b Ft(\()p Fp(k)s Ft(\))33 b(form)g(a)g(partition)f(\(mo)s(d)15 b(0\))33 b(of)h([0)p Fp(;)17 b Ft(1],)31 b(one)i(obtains)478 1525 y Fo(k)p Fp(T)600 1484 y Fn(k)649 1525 y Fp(u)p Fo(k)756 1477 y Fn(p)756 1568 y(L)811 1548 y Fj(p)852 1568 y Fs(\([0)p Fn(;)p Fs(1])p Fn(;)1189 1541 y Fj(dx)p 1067 1552 321 4 v 1067 1612 a Fi(2)1101 1564 y Fm(p)p 1166 1564 221 4 v 48 x Fj(x)p Fi(\(1)p Ff(\000)p Fj(x)p Fi(\))1399 1568 y Fs(\))1463 1525 y Fo(\024)28 b Fp(c)1628 1390 y Fk(Z)1727 1414 y Fs(1)1683 1616 y(0)1977 1458 y Fp(dx)p 1800 1502 463 4 v 1800 1522 a Fk(p)p 1900 1522 363 4 v 85 x Fp(x)p Ft(\(1)22 b Fo(\000)g Fp(x)p Ft(\))2275 1525 y Fo(k)p Fp(u)p Fo(k)2432 1477 y Fn(p)2432 1559 y(L)2487 1539 y Fj(p)2528 1559 y Fs(\([0)p Fn(;)p Fs(1]\))2760 1385 y Fk(\024)2813 1525 y Ft(max)2828 1594 y Fm(I)5 b Fs(\()p Fn(k)r Fs(\))3015 1525 y Fo(j)p Fp(I)j Fo(j)3123 1385 y Fk(\025)3174 1405 y Fn(p)3265 1525 y Fp(;)113 1787 y Ft(whic)m(h)34 b(pro)m(v)m(es)g(the)g(second)h(part)e(of)g(the)h (prop)s(osition.)1359 b Fd(\003)113 2005 y Fr(Corollary)37 b(4.7)57 b Fe(F)-8 b(or)31 b Fp(p)d(>)g Ft(1)p Fe(,)j(the)g(op)s (erator)g Fp(T)45 b Fe(maps)30 b Fo(O)2305 1969 y Fs(1)2350 2005 y Ft(\()p 2389 1924 72 4 v Fq(C)43 b Fo(n)18 b Ft([0)p Fp(;)f Ft(1]\))f Fo(\\)i Fp(H)2977 1969 y Fn(p)3023 2005 y Ft(\()p 3062 1924 V Fq(C)43 b Fo(n)18 b Ft([0)p Fp(;)f Ft(1]\))29 b Fe(in)m(to)113 2144 y(itself,)k(is)h(b)s(ounded)g(and)g (its)f(sp)s(ectral)g(radius)h(is)f Fo(\024)2113 2046 y Fm(p)p 2179 2046 40 4 v 2179 2104 a Fs(5)p Fm(\000)p Fs(1)p 2113 2121 208 4 v 2197 2178 a(2)113 2350 y Fl(Pr)-5 b(o)g(of.)27 b Ft(Let)e Fp(')j Fo(2)g(O)843 2314 y Fs(1)888 2350 y Ft(\()p 927 2269 72 4 v Fq(C)j Fo(n)6 b Ft([0)p Fp(;)17 b Ft(1]\))6 b Fo(\\)g Fp(H)1469 2314 y Fn(p)1512 2350 y Ft(\()p 1551 2269 V Fq(C)31 b Fo(n)6 b Ft([0)p Fp(;)17 b Ft(1]\))23 b(b)s(e)i(real,)h Fp(u)i Ft(=)g Fo(=)p Fp(m)17 b(')p Ft(\()p Fp(x)6 b Ft(+)g Fp(i)p Ft(0\))24 b(the)i(asso)s(ciated)113 2469 y(h)m(yp)s(erfunction.)49 b(F)-8 b(or)34 b Fp(k)e Fo(\025)e Ft(0)k(the)h(function)g(of)g Fo(O)2008 2433 y Fs(1)2053 2469 y Ft(\()p 2092 2389 V Fq(C)48 b Fo(n)23 b Ft([0)p Fp(;)17 b Ft(1]\))32 b(asso)s(ciated)i(to)g Fp(T)3201 2433 y Fn(k)3250 2469 y Fp(u)h Ft(is)f Fp(T)3515 2433 y Fn(k)3564 2469 y Fp(')p Ft(.)113 2589 y(Using)26 b(the)h(conformal)e(represen)m(tation)i(of)p 1756 2508 V 26 w Fq(C)34 b Fo(n)8 b Ft([0)p Fp(;)17 b Ft(1])24 b(on)m(to)h Fq(D)51 b Ft(and)27 b(the)f(fact)g(that)g(the)g(Hilb)s(ert) 113 2709 y(transform)33 b(is)g(b)s(ounded)i(on)e Fp(L)1284 2672 y Fn(p)1330 2709 y Ft(\()p Fp(d\022)s Ft(\))f(one)i(obtains)f(the) h(desired)g(result.)723 b Fd(\003)313 2901 y Fr(4.3.4)31 b Ft(Here)f(w)m(e)h(consider)h(the)e(op)s(erator)1908 2827 y Fk(P)2014 2931 y Fm(M)2143 2901 y Ft(acting)g(on)g(the)h(space)h Fp(H)3098 2865 y Fs(1)3142 2901 y Ft(\()p 3181 2821 V Fq(C)42 b Fo(n)17 b Ft([0)p Fp(;)g Ft(1]\))g Fo(\\)113 3021 y(O)195 2985 y Fs(1)240 3021 y Ft(\()p 279 2940 V Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\).)113 3189 y Fr(Lemma)39 b(4.8)57 b Fe(F)-8 b(or)33 b(all)g Fp(g)d Fo(2)f(M)j Fe(the)i(restriction)f(to)g Fp(H)2228 3153 y Fs(1)2273 3189 y Ft(\()p 2312 3109 V Fq(C)47 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))k Fo(\\)h(O)2908 3153 y Fs(1)2953 3189 y Ft(\()p 2992 3109 V Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))31 b Fe(of)j Fp(L)3611 3204 y Fn(g)113 3309 y Fe(is)g(a)f(b)s(ounded)h(op)s(erator)f(of)h (this)f(space)h(in)m(to)f(itself.)113 3502 y Fl(Pr)-5 b(o)g(of.)34 b Ft(It)f(is)g(su\016cien)m(t)i(to)e(consider)h(the)g (case)f Fp(g)e Ft(=)d Fp(g)t Ft(\()p Fp(m)p Ft(\),)k Fp(m)c Fo(\025)h Ft(1.)43 b(One)35 b(has)e(then)975 3692 y Fp(L)1043 3710 y Fn(g)s Fs(\()p Fn(m)p Fs(\))1250 3692 y Ft(=)28 b Fp(\033)d Fo(\016)d Fp(\034)1564 3651 y Fm(\000)p Fs(1)1693 3692 y Fo(\016)g Fp(\037)1827 3707 y Fs(1)1894 3692 y Fo(\016)g Fp(\037)2028 3707 y Fs(0)2096 3692 y Fo(\016)f Fp(\034)33 b Fo(\016)22 b Fp(\033)k Fo(\016)c Fp(\023)2506 3707 y Fn(m)2604 3692 y Fo(\016)g Fp(\034)2720 3707 y Fn(m)113 3882 y Ft(where)144 4001 y Fp(\034)188 4016 y Fn(m)313 4001 y Ft(is)34 b(the)h(isomorphism)e Fp(')p Ft(\()p Fp(z)t Ft(\))d Fo(7!)g Fp(')1606 3921 y Fk(\000)1664 3962 y Fs(1)p 1663 3978 41 4 v 1663 4036 a Fn(z)1738 4001 y Fo(\000)23 b Fp(m)1925 3921 y Fk(\001)2005 4001 y Ft(of)35 b Fp(H)2211 3965 y Fs(1)2256 4001 y Ft(\()p 2295 3921 72 4 v Fq(C)49 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))33 b(on)m(to)h Fp(H)3051 3965 y Fs(1)3096 4001 y Ft(\()p 3135 3921 V Fq(C)49 b Fo(n)23 b Ft([1)p Fp(=)p Ft(\()p Fp(m)f Ft(+)313 4121 y(1\))p Fp(;)17 b Ft(1)p Fp(=m)p Ft(]\))h(;)152 4240 y Fp(\023)187 4255 y Fn(m)313 4240 y Ft(is)33 b(the)g(canonical)h(injection)f(of)h Fp(H)1633 4204 y Fs(1)1678 4240 y Ft(\()p 1717 4160 V Fq(C)47 b Fo(n)22 b Ft([1)p Fp(=)p Ft(\()p Fp(m)g Ft(+)h(1\))p Fp(;)17 b Ft(1)p Fp(=m)p Ft(]\))31 b(in)m(to)i Fp(H)2970 4204 y Fs(1)3015 4240 y Ft(\()p 3054 4160 V Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))g(;)202 4360 y Fp(\033)54 b Ft(is)32 b(the)h(b)s(ounded)h(op)s(erator)f Fp(')28 b Fo(7!)f Fp(')21 b Fo(\000)g Fp(')p Ft(\()p Fo(1)p Ft(\))32 b(of)h Fp(H)2274 4324 y Fs(1)2319 4360 y Ft(\()p 2358 4279 V Fq(C)46 b Fo(n)21 b Ft([0)p Fp(;)c Ft(1]\))31 b(in)m(to)h Fp(H)3084 4324 y Fs(1)3129 4360 y Ft(\()p 3168 4279 V Fq(C)47 b Fo(n)20 b Ft([0)p Fp(;)d Ft(1]\))i Fo(\\)313 4480 y(O)395 4443 y Fs(1)439 4480 y Ft(\()p 478 4399 V Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))h(;)208 4629 y Fp(\034)61 b Ft(is)39 b(the)h(isomorphism)e Fp(')g Fo(7!)g Fp(')1492 4644 y Fs(1)1537 4629 y Ft(\()p Fp(w)s Ft(\))f(=)i Fp(')1924 4518 y Fk(\020)1995 4581 y Fs(\()p Fn(w)r Fs(+1\))2218 4551 y Fi(2)p 1995 4606 262 4 v 2076 4663 a Fs(4)p Fn(w)2269 4518 y Fk(\021)2368 4629 y Ft(of)g Fp(H)2578 4593 y Fs(1)2623 4629 y Ft(\()p 2662 4548 72 4 v Fq(C)52 b Fo(n)26 b Ft([0)p Fp(;)17 b Ft(1]\))38 b(on)m(to)h Fp(H)3435 4593 y Fs(1)3480 4629 y Ft(\()p Fq(D)18 b Ft(\),)313 4792 y(whose)47 b(restriction)f(to)h Fp(H)1337 4756 y Fs(1)1381 4792 y Ft(\()p 1420 4712 V Fq(C)57 b Fo(n)31 b Ft([0)p Fp(;)17 b Ft(1]\))29 b Fo(\\)i(O)2052 4756 y Fs(1)2097 4792 y Ft(\()p 2136 4712 V Fq(C)57 b Fo(n)31 b Ft([0)p Fp(;)17 b Ft(1]\))44 b(is)j(an)g(isomorphism)e(on)m (to)313 4912 y Fo(f)p Fp(')428 4927 y Fs(1)500 4912 y Fo(2)28 b Fp(H)685 4876 y Fs(1)730 4912 y Ft(\()p Fq(D)18 b Ft(\))34 b Fp(;)44 b(')1044 4927 y Fs(1)1089 4912 y Ft(\(0\))27 b(=)h(0)p Fo(g)19 b Ft(;)156 5031 y Fp(\037)218 5046 y Fs(0)313 5031 y Ft(is)44 b(the)h(isomorphism)e Fp(')1266 5046 y Fs(1)1311 5031 y Ft(\()p Fp(w)s Ft(\))k Fo(7!)1678 4992 y Fs(1)p 1668 5008 60 4 v 1668 5066 a Fn(w)1739 5031 y Fp(')1804 5046 y Fs(1)1849 5031 y Ft(\()p Fp(w)s Ft(\))d(of)h Fo(f)p Fp(')2285 5046 y Fs(1)2377 5031 y Fo(2)i Fp(H)2581 4995 y Fs(1)2626 5031 y Ft(\()p Fq(D)18 b Ft(\))53 b Fp(;)63 b(')2978 5046 y Fs(1)3023 5031 y Ft(\(0\))46 b(=)h(0)p Fo(g)e Ft(on)m(to)313 5151 y Fp(H)404 5115 y Fs(1)448 5151 y Ft(\()p Fq(D)19 b Ft(\))26 b(;)156 5296 y Fp(\037)218 5311 y Fs(1)313 5296 y Ft(is)k(the)h(m)m (ultiplication)d(op)s(erator)i(b)m(y)g(the)h(function)g Fo(\000)2394 5248 y Fs(\()p Fn(w)r Fs(+1\))2617 5218 y Fi(2)p 2395 5273 262 4 v 2506 5331 a Fs(4)2696 5296 y Fo(2)d Fp(H)2881 5260 y Fm(1)2965 5296 y Ft(\()p Fq(D)19 b Ft(\))36 b(in)m(to)30 b Fp(H)3435 5260 y Fs(1)3480 5296 y Ft(\()p Fq(D)18 b Ft(\).)313 5416 y Fd(\003)113 5584 y Ft(W)-8 b(e)33 b(w)m(an)m(t)g(no)m(w)g(to)f(estimate)f(the)i (norm)f(of)h Fp(L)1865 5599 y Fn(g)1943 5584 y Ft(acting)f(on)h Fp(H)2470 5548 y Fs(1)2515 5584 y Ft(\()p 2554 5504 72 4 v Fq(C)46 b Fo(n)21 b Ft([0)p Fp(;)c Ft(1]\))h Fo(\\)j(O)3144 5548 y Fs(1)3189 5584 y Ft(\()p 3228 5504 V Fq(C)47 b Fo(n)20 b Ft([0)p Fp(;)d Ft(1]\).)p eop %%Page: 26 26 26 25 bop 113 144 a Ft(26)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y Fr(Prop)s(osition)40 b(4.9)58 b Fe(There)35 b(exists)f(a)g(constan)m(t)h Fp(K)h(>)30 b Ft(0)k Fe(suc)m(h)i(that,)e(if)g Fp(g)f Fo(2)d(M)j Fe(is)i(di\013eren)m(t)113 557 y(from)349 416 y Fk(\022)439 497 y Ft(1)99 b(0)439 617 y(0)g(1)655 416 y Fk(\023)728 557 y Fe(,)789 416 y Fk(\022)879 497 y Ft(0)g(1)879 617 y(1)g(1)1095 416 y Fk(\023)1201 557 y Fe(\(th)m(us)34 b Fp(d)28 b(>)g(b)f(>)h Ft(0)p Fe(\))33 b(one)h(has)843 884 y Fo(k)p Fp(L)961 899 y Fn(g)1007 884 y Fo(k)1057 902 y Fn(H)1127 882 y Fi(1)1199 884 y Fo(\024)28 b Fp(K)7 b(d)1448 843 y Fm(\000)p Fs(5)p Fn(=)p Fs(2)1635 884 y Ft([min)o(\()p Fp(b;)17 b(d)k Fo(\000)i Fp(b)p Ft(\)])2239 843 y Fm(\000)p Fs(1)p Fn(=)p Fs(2)2442 884 y Ft(log)q(\(1)e(+)i Fp(d)p Ft(\))k Fp(:)113 1168 y Fl(Pr)-5 b(o)g(of.)33 b Ft(A)f(function)h Fp(')28 b Fo(2)g Fp(H)1184 1132 y Fs(1)1229 1168 y Ft(\()p 1268 1088 72 4 v Fq(C)46 b Fo(n)20 b Ft([0)p Fp(;)d Ft(1]\))30 b(has)i(non)h(tangen)m(tial)f(limits)e(at)i (almost)e(all)h(p)s(oin)m(ts)113 1288 y(of)j(the)f(b)s(oundary)h(and) 836 1602 y Fo(k)p Fp(')p Fo(k)1001 1620 y Fn(H)1071 1600 y Fi(1)1143 1602 y Ft(=)1248 1467 y Fk(Z)1348 1491 y Fs(1)1303 1693 y(0)1421 1535 y Fo(j)p Fp(')p Ft(\()p Fp(x)22 b Ft(+)g Fp(i)p Ft(0\))p Fo(j)p Fp(dx)p 1421 1579 570 4 v 1450 1684 a Ft(2)1500 1599 y Fk(p)p 1599 1599 363 4 v 1599 1684 a Fp(x)p Ft(\(1)g Fo(\000)g Fp(x)p Ft(\))2025 1602 y(+)2124 1467 y Fk(Z)2224 1491 y Fs(1)2180 1693 y(0)2297 1535 y Fo(j)p Fp(')p Ft(\()p Fp(x)g Fo(\000)g Fp(i)p Ft(0\))p Fo(j)p Fp(dx)p 2297 1579 570 4 v 2326 1684 a Ft(2)2376 1599 y Fk(p)p 2475 1599 363 4 v 2475 1684 a Fp(x)p Ft(\(1)g Fo(\000)g Fp(x)p Ft(\))2906 1602 y Fp(:)113 1942 y Ft(On)34 b(the)g(other)f(hand)h(a)g(function)g Fp(')28 b Fo(2)g Fp(H)1712 1906 y Fs(1)1756 1942 y Ft(\()p 1795 1862 72 4 v Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))31 b(v)m(eri\014es)1014 2199 y Fo(j)p Fp(')p Ft(\()p Fp(t)p Ft(\))p Fo(j)c(\024)h Fp(C)7 b Fo(j)p Fp(t)p Fo(j)1551 2158 y Fm(\000)p Fs(1)p Fn(=)p Fs(2)1738 2199 y Fo(k)p Fp(')p Fo(k)1903 2217 y Fn(H)1973 2197 y Fi(1)2073 2199 y Fo(8)p Fp(t)28 b Fo(2)g Ft(\()p Fo(\000)p Ft(1)p Fp(;)17 b Ft(0\))26 b Fp(;)1014 2368 y Fo(j)p Fp(')p Ft(\()p Fp(t)p Ft(\))p Fo(j)h(\024)h Fp(C)7 b Ft(\()p Fp(t)22 b Fo(\000)h Ft(1\))1745 2326 y Fm(\000)p Fs(1)p Fn(=)p Fs(2)1932 2368 y Fo(k)p Fp(')p Fo(k)2097 2385 y Fn(H)2167 2365 y Fi(1)2266 2368 y Fo(8)p Fp(t)28 b Fo(2)g Ft(\(1)p Fp(;)17 b Ft(2\))26 b Fp(;)3402 2279 y Ft(\(4)p Fp(:)p Ft(14\))113 2610 y(as)49 b(one)g(can)f(c)m(hec)m(k)i (directly)e(applying)g(P)m(oisson's)h(in)m(tegral)f(form)m(ula.)88 b(Moreo)m(v)m(er,)53 b(if)48 b(it)113 2730 y(b)s(elongs)32 b(to)f Fp(H)680 2694 y Fs(1)724 2730 y Ft(\()p 763 2649 V Fq(C)44 b Fo(n)18 b Ft([0)p Fp(;)f Ft(1]\))f Fo(\\)i(O)1343 2694 y Fs(1)1388 2730 y Ft(\()p 1427 2649 V Fq(C)44 b Fo(n)18 b Ft([0)p Fp(;)f Ft(1]\))29 b(the)i(same)g(argumen)m(t)g(leads) g(to)g(the)h(estimates)218 2978 y Fo(j)p Fp(')p Ft(\()p Fp(z)t Ft(\))p Fo(j)c(\024)g Fp(C)7 b Fo(j)p Fp(z)t Fo(j)784 2937 y Fm(\000)p Fs(1)891 2978 y Fo(k)p Fp(')p Fo(k)1056 2996 y Fn(H)1126 2976 y Fi(1)1198 2978 y Fp(;)116 b Fo(j)p Fp(')1435 2937 y Fm(0)1463 2978 y Ft(\()p Fp(z)t Ft(\))p Fo(j)27 b(\024)h Fp(C)7 b Fo(j)p Fp(z)t Fo(j)1935 2937 y Fm(\000)p Fs(2)2043 2978 y Fo(k)p Fp(')p Fo(k)2208 2996 y Fn(H)2278 2976 y Fi(1)2350 2978 y Fp(;)116 b Fo(j)p Fp(')2587 2937 y Fm(00)2637 2978 y Ft(\()p Fp(z)t Ft(\))p Fo(j)28 b(\024)g Fp(C)7 b Fo(j)p Fp(z)t Fo(j)3110 2937 y Fm(\000)p Fs(3)3217 2978 y Fo(k)p Fp(')p Fo(k)3382 2996 y Fn(H)3452 2976 y Fi(1)3524 2978 y Fp(;)3402 3098 y Ft(\(4)p Fp(:)p Ft(15\))113 3218 y(for)34 b(all)e Fp(z)38 b Ft(suc)m(h)d(that)e Fo(j)p Fp(z)26 b Fo(\000)d Ft(1)p Fp(=)p Ft(2)p Fo(j)k Fp(>)h Ft(1.)313 3340 y(Giv)m(en)j(these)g (preliminary)f(elemen)m(tary)g(estimates,)g(let)g Fp(')e Fo(2)g Fp(H)2774 3304 y Fs(1)2819 3340 y Ft(\()p 2858 3260 V Fq(C)42 b Fo(n)17 b Ft([0)p Fp(;)g Ft(1]\))f Fo(\\)h(O)3434 3304 y Fs(1)3479 3340 y Ft(\()p 3518 3260 V Fq(C)43 b Fo(n)113 3520 y Ft([0)p Fp(;)17 b Ft(1]\),)44 b Fp(g)49 b Ft(=)645 3379 y Fk(\022)735 3460 y Fp(a)103 b(b)739 3579 y(c)i(d)955 3379 y Fk(\023)1074 3520 y Fo(2)46 b(M)p Ft(,)g(with)d Fp(d)i(>)h(b)f(>)h Ft(0.)75 b(Let)60 b(~)-65 b Fp(')45 b Ft(=)h Fp(L)2692 3535 y Fn(g)2738 3520 y Fp(')20 b Ft(;)65 b(~)-66 b Fp(')44 b Ft(is)g(holomorphic)113 3699 y(outside)c(the)f(in)m(terv)-6 b(al)38 b(with)h(end)h(p)s(oin)m (ts)f Fp(b=d)f Ft(and)h(\()p Fp(a)25 b Ft(+)i Fp(b)p Ft(\))p Fp(=)p Ft(\()p Fp(c)e Ft(+)h Fp(d)p Ft(\).)60 b(W)-8 b(e)39 b(m)m(ust)g(estimate)113 3754 y Fk(R)180 3779 y Fs(1)160 3870 y(0)253 3786 y Fm(j)12 b Fs(~)-52 b Fn(')p Fs(\()p Fn(x)p Fm(\006)p Fn(i)p Fs(0\))p Fm(j)p Fn(dx)p 253 3811 424 4 v 296 3823 a Fo(p)p 379 3823 255 4 v 68 x Fn(x)p Fs(\(1)p Fm(\000)p Fn(x)p Fs(\))725 3834 y Ft(and)36 b(b)m(y)g(symmetry)e(it)h(is)h(enough)h(to)f(consider)52 b(~)-66 b Fp(')q Ft(\()p Fp(x)23 b Ft(+)i Fp(i)p Ft(0\).)52 b(W)-8 b(e)36 b(de\014ne)i(an)113 3998 y(in)m(terv)-6 b(al)33 b Fp(I)517 4013 y Fn(g)597 3998 y Ft(of)g(the)h(follo)m(wing)e (form)h(:)158 4184 y(\(i\))49 b(if)33 b Fp(g)e Ft(=)588 4044 y Fk(\022)678 4124 y Ft(0)118 b(1)678 4244 y(1)99 b Fp(m)931 4044 y Fk(\023)1005 4184 y Ft(,)33 b Fp(m)28 b Fo(\025)g Ft(2,)33 b Fp(I)1441 4199 y Fn(g)1514 4184 y Ft(=)c([0)p Fp(;)17 b Ft(3)p Fp(=)p Ft(2)p Fp(m)p Ft(])h(;)130 4436 y(\(ii\))49 b(if)33 b Fp(g)e Ft(=)588 4296 y Fk(\022)678 4377 y Ft(1)99 b Fp(m)23 b Fo(\000)g Ft(1)678 4496 y(1)185 b Fp(m)1103 4296 y Fk(\023)1176 4436 y Ft(,)33 b Fp(m)28 b Fo(\025)g Ft(2,)33 b Fp(I)1612 4451 y Fn(g)1686 4436 y Ft(=)28 b([1)22 b Fo(\000)g Ft(3)p Fp(=)p Ft(2)p Fp(m;)17 b Ft(1])i(;)102 4632 y(\(iii\))49 b(in)32 b(all)g(the)h(other)f(cases)i (one)f(has)g Fp(c)27 b(>)i(a)e(>)h Ft(0,)2088 4593 y Fn(a)p 2088 4609 44 4 v 2092 4666 a(c)2171 4632 y Ft(=)2291 4593 y Fn(b)p 2288 4609 42 4 v 2288 4666 a(d)2362 4632 y Ft(+)2472 4581 y Fn(")2510 4591 y Fj(g)p 2472 4609 80 4 v 2473 4666 a Fn(cd)2583 4632 y Ft(;)33 b Fp(I)2688 4647 y Fn(g)2766 4632 y Ft(is)g(the)g(in)m(terv)-6 b(al)32 b(whose)313 4772 y(end)i(p)s(oin)m(ts)815 4732 y Fn(b)p 812 4749 42 4 v 812 4806 a(d)888 4772 y Ft(+)999 4721 y Fs(3)p Fn(")1077 4731 y Fj(g)p 999 4749 120 4 v 1000 4806 a Fs(2)p Fn(cd)1164 4772 y Ft(and)1372 4732 y Fn(b)p 1369 4749 42 4 v 1369 4806 a(d)1445 4772 y Fo(\000)1575 4721 y Fn(")1613 4731 y Fj(g)p 1556 4749 117 4 v 1556 4806 a Fs(2)p Fn(cd)1705 4772 y Ft(;)f(clearly)f Fp(I)2128 4787 y Fn(g)2202 4772 y Fo(\032)c Ft([0)p Fp(;)17 b Ft(1].)113 4920 y(W)-8 b(e)31 b(will)e(no)m(w)h(directly)g(estimate)f(the)h(in)m (tegral)g(of)2090 4871 y Fs(~)-53 b Fn(')q Fs(\()p Fn(x)p Fs(+)p Fn(i)p Fs(0\))p Fn(dx)p 2077 4897 376 4 v 2096 4909 a Fo(p)p 2179 4909 255 4 v 67 x Fn(x)p Fs(\(1)p Fm(\000)p Fn(x)p Fs(\))2495 4920 y Ft(on)30 b Fp(I)2674 4935 y Fn(g)2751 4920 y Ft(and)g(on)g([0)p Fp(;)17 b Ft(1])f Fo(n)g Fp(I)3403 4935 y Fn(g)3447 4920 y Ft(.)43 b(W)-8 b(e)113 5083 y(b)s(egin)34 b(with)f(the)g(latter.)313 5206 y(If)g Fp(x)28 b Fo(2)g Ft([0)p Fp(;)17 b Ft(1])j Fo(n)i Fp(I)928 5221 y Fn(g)974 5206 y Ft(,)49 b(~)-66 b Fp(')34 b Ft(is)f(holomorphic)f(in)h(a)h(neigh)m(b)s(orho)s(o)s(d)f (of)h Fp(x)f Ft(and)h(one)f(has)594 5529 y(~)-66 b Fp(')q Ft(\()p Fp(x)p Ft(\))27 b(=)h Fp(c)954 5488 y Fm(\000)p Fs(2)1061 5529 y Ft(\()p Fp(a)21 b Fo(\000)i Fp(cx)p Ft(\))1413 5488 y Fm(\000)p Fs(1)1536 5394 y Fk(Z)1636 5418 y Fs(1)1592 5620 y(0)1681 5529 y Ft(\(1)e Fo(\000)i Fp(t)p Ft(\))p Fp(')2031 5488 y Fm(00)2098 5389 y Fk(\022)2172 5529 y Fo(\000)2261 5462 y Fp(d)p 2261 5507 52 4 v 2265 5598 a(c)2347 5529 y Ft(+)g Fp(")2493 5544 y Fn(g)2731 5462 y Fp(t)p 2552 5507 396 4 v 2552 5598 a(c)p Ft(\()p Fp(a)e Fo(\000)i Fp(cx)p Ft(\))2959 5389 y Fk(\023)3049 5529 y Fp(dt)k(:)p eop %%Page: 27 27 27 26 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(27)113 383 y(W)-8 b(e)34 b(ha)m(v)m(e)1152 544 y Fp(x)k(=)-61 b Fo(2)28 b Fp(I)1374 559 y Fn(g)1448 544 y Fo(\))1739 477 y Fp(")1785 492 y Fn(g)p 1588 522 396 4 v 1588 613 a Fp(c)p Ft(\()p Fp(a)21 b Fo(\000)i Fp(cx)p Ft(\))2022 544 y Fo(2)2116 404 y Fk(\024)2169 544 y Fo(\000)p Ft(2)2308 477 y Fp(d)p 2308 522 52 4 v 2312 613 a(c)2372 544 y(;)2428 477 y Ft(2)p 2428 522 50 4 v 2428 613 a(3)2502 477 y Fp(d)p 2502 522 52 4 v 2506 613 a(c)2566 404 y Fk(\025)3402 544 y Ft(\(4)p Fp(:)p Ft(16\))113 807 y(th)m(us)42 b(the)f(v)-6 b(alues)40 b(of)h(the)g(second)h(deriv)-6 b(ativ)m(e)40 b(in)g(the)h(in)m(tegral)f(are)g Fo(\024)g Fp(C)7 b(c)2998 771 y Fs(3)3044 807 y Fp(d)3096 771 y Fm(\000)p Fs(3)3202 807 y Fo(k)p Fp(')p Fo(k)3367 824 y Fn(H)3437 805 y Fi(1)3482 807 y Ft(,)42 b(b)m(y)113 926 y(the)34 b(third)f(estimate)f(a)h(of)h (\(4.15\),)d(and)j(one)g(has)1163 1172 y Fo(j)16 b Ft(~)-66 b Fp(')o Ft(\()p Fp(x)p Ft(\))p Fo(j)27 b(\024)h Fp(C)7 b(d)1680 1131 y Fm(\000)p Fs(3)1787 1172 y Fo(j)p Fp(x)22 b Fo(\000)h Fp(b=d)p Fo(j)2167 1131 y Fm(\000)p Fs(1)2273 1172 y Fo(k)p Fp(')p Fo(k)2438 1190 y Fn(H)2508 1170 y Fi(1)2580 1172 y Fp(;)113 1418 y Ft(since)34 b Fp(x)23 b Fo(\000)f Fp(a=c)33 b Ft(and)h Fp(x)22 b Fo(\000)g Fp(b=d)33 b Ft(are)g(comparable)g(outside)h Fp(I)2351 1433 y Fn(g)2397 1418 y Ft(.)44 b(W)-8 b(e)33 b(th)m(us)i(obtain)478 1590 y Fk(Z)533 1816 y Fs([0)p Fn(;)p Fs(1])p Fm(n)p Fn(I)759 1826 y Fj(g)833 1658 y Fo(j)16 b Ft(~)-66 b Fp(')p Ft(\()p Fp(x)22 b Fo(\006)g Fp(i)p Ft(0\))p Fo(j)p Fp(dx)p 833 1702 570 4 v 886 1722 a Fk(p)p 986 1722 363 4 v 86 x Fp(x)p Ft(\(1)g Fo(\000)g Fp(x)p Ft(\))1442 1725 y Fo(\024)28 b Fp(C)7 b(d)1677 1684 y Fm(\000)p Fs(3)1784 1725 y Fo(k)p Fp(')p Fo(k)1949 1743 y Fn(H)2019 1723 y Fi(1)2080 1590 y Fk(Z)2136 1816 y Fs([0)p Fn(;)p Fs(1])p Fm(n)p Fn(I)2362 1826 y Fj(g)2776 1658 y Fp(dx)p 2435 1702 790 4 v 2435 1722 a Fk(p)p 2535 1722 363 4 v 86 x Fp(x)p Ft(\(1)21 b Fo(\000)i Fp(x)p Ft(\))2914 1723 y Fk(\014)2914 1783 y(\014)2948 1808 y Fp(x)f Fo(\000)3141 1768 y Fn(b)p 3138 1785 42 4 v 3138 1842 a(d)3192 1723 y Fk(\014)3192 1783 y(\014)3264 1725 y Fp(:)113 2048 y Ft(An)34 b(easy)f(estimate)f(of)h(this)h(last)e(in)m(tegral)h(giv)m (es)298 2210 y Fk(Z)354 2437 y Fs([0)p Fn(;)p Fs(1])p Fm(n)p Fn(I)580 2447 y Fj(g)653 2279 y Fo(j)16 b Ft(~)-66 b Fp(')p Ft(\()p Fp(x)22 b Fo(\006)h Fp(i)p Ft(0\))p Fo(j)p Fp(dx)p 653 2323 570 4 v 707 2343 a Fk(p)p 806 2343 363 4 v 806 2428 a Fp(x)p Ft(\(1)f Fo(\000)h Fp(x)p Ft(\))1262 2346 y Fo(\024)29 b Fp(C)7 b(d)1498 2305 y Fm(\000)p Fs(5)p Fn(=)p Fs(2)1685 2346 y Ft([min)o(\()p Fp(b;)17 b(d)k Fo(\000)h Fp(b)p Ft(\)])2288 2305 y Fm(\000)p Fs(1)p Fn(=)p Fs(2)2492 2346 y Ft(log\(1)g(+)h(min)o(\()p Fp(b;)17 b(d)k Fo(\000)h Fp(b)p Ft(\)\))27 b Fp(:)113 2664 y Ft(F)-8 b(or)57 b(the)g(in)m(tegral)f(inside)h Fp(I)1247 2679 y Fn(g)1350 2664 y Ft(w)m(e)g(distinguish)g(the)g(three) g(di\013eren)m(t)h(con)m(tributions)f(to)129 2784 y(~)-66 b Fp(')28 b Ft(=)g Fp(L)p Ft(\()p Fp(g)t Ft(\))p Fp(')p Ft(.)43 b(First)33 b(of)g(all)g(one)g(has)h(\(applying)f(the)g(second)i (estimate)d(of)h(\(4.15\)\))1212 3030 y Fo(j)p Fp(c)1283 2989 y Fm(\000)p Fs(1)1390 3030 y Fp(')1455 2989 y Fm(0)1483 3030 y Ft(\()p Fo(\000)p Fp(d=c)p Ft(\))p Fo(j)27 b(\024)h Fp(C)7 b(cd)2116 2989 y Fm(\000)p Fs(2)2223 3030 y Fo(k)p Fp(')p Fo(k)2388 3047 y Fn(H)2458 3028 y Fi(1)2530 3030 y Fp(;)113 3276 y Ft(and)702 3294 y Fk(Z)758 3520 y Fn(I)793 3530 y Fj(g)1045 3362 y Fp(dx)p 868 3406 463 4 v 868 3426 a Fk(p)p 968 3426 363 4 v 85 x Fp(x)p Ft(\(1)21 b Fo(\000)i Fp(x)p Ft(\))1370 3429 y Fo(\024)29 b Fp(C)7 b(c)1597 3388 y Fm(\000)p Fs(1)1704 3429 y Fp(d)1756 3388 y Fm(\000)p Fs(1)p Fn(=)p Fs(2)1943 3429 y Ft([min)o(\()p Fp(b;)17 b(d)k Fo(\000)h Fp(b)p Ft(\)])2546 3388 y Fm(\000)p Fs(1)p Fn(=)p Fs(2)2733 3429 y Fo(k)p Fp(')p Fo(k)2898 3447 y Fn(H)2968 3427 y Fi(1)3040 3429 y Fp(;)113 3698 y Ft(th)m(us)655 3727 y Fk(Z)710 3954 y Fn(I)745 3964 y Fj(g)821 3796 y Fo(j)p Fp(c)892 3759 y Fm(\000)p Fs(1)999 3796 y Fp(')1064 3759 y Fm(0)1092 3796 y Ft(\()p Fo(\000)p Fp(d=c)p Ft(\))p Fo(j)p Fp(dx)p 821 3840 708 4 v 943 3860 a Fk(p)p 1043 3860 363 4 v 85 x Fp(x)p Ft(\(1)g Fo(\000)g Fp(x)p Ft(\))1568 3863 y Fo(\024)28 b Fp(C)7 b(d)1803 3822 y Fm(\000)p Fs(5)p Fn(=)p Fs(2)1991 3863 y Ft([min)n(\()p Fp(b;)17 b(d)k Fo(\000)i Fp(b)p Ft(\)])2594 3822 y Fm(\000)p Fs(1)p Fn(=)p Fs(2)2780 3863 y Fo(k)p Fp(')p Fo(k)2945 3881 y Fn(H)3015 3861 y Fi(1)3087 3863 y Fp(:)113 4137 y Ft(F)-8 b(or)33 b Fp(x)28 b Fo(2)g Fp(I)515 4152 y Fn(g)594 4137 y Ft(one)34 b(has)g(applying)e(\(4.15\))g (and)i(\(4.16\))1125 4383 y Fo(j)p Ft(\()p Fp(a)21 b Fo(\000)i Fp(cx)p Ft(\))p Fp(')p Ft(\()p Fo(\000)p Fp(d=c)p Ft(\))p Fo(j)k(\024)h Fp(C)7 b(cd)2203 4342 y Fm(\000)p Fs(2)2310 4383 y Fo(k)p Fp(')p Fo(k)2475 4401 y Fn(H)2545 4381 y Fi(1)2617 4383 y Fp(;)113 4629 y Ft(from)30 b(whic)m(h)i(follo)m (ws)e(the)h(same)f(estimate)f(ab)s(o)m(v)m(e)i(for)g(the)g(second)h (term.)42 b(W)-8 b(e)31 b(are)f(left)h(with)923 4907 y Fp(I)36 b Ft(=)1108 4772 y Fk(Z)1163 4998 y Fn(I)1198 5008 y Fj(g)1274 4840 y Fo(j)p Fp(a)21 b Fo(\000)i Fp(cx)p Fo(jj)p Fp(')p Ft(\()p Fp(y)i Fo(\006)d Fp(i)p Ft(0\))p Fo(j)p 1274 4885 787 4 v 1435 4904 a Fk(p)p 1535 4904 363 4 v 86 x Fp(x)p Ft(\(1)g Fo(\000)g Fp(x)p Ft(\))2072 4907 y Fp(dx)27 b(;)72 b(y)31 b Ft(=)2506 4840 y Fp(dx)22 b Fo(\000)g Fp(b)p 2505 4885 275 4 v 2505 4976 a(a)g Fo(\000)g Fp(cx)2819 4907 y(:)113 5287 y Ft(Inside)33 b Fp(I)447 5302 y Fn(g)525 5287 y Ft(one)g(has)962 5231 y Fo(j)p Fp(a)21 b Fo(\000)i Fp(cx)p Fo(j)p 895 5264 463 4 v 895 5276 a Fk(p)p 995 5276 363 4 v 90 x Fp(x)p Ft(\(1)f Fo(\000)g Fp(x)p Ft(\))1398 5287 y Fo(\024)28 b Fp(C)7 b(d)1633 5251 y Fm(\000)p Fs(1)p Fn(=)p Fs(2)1820 5287 y Ft([min)o(\()p Fp(b;)17 b(d)h Fo(\000)j Fp(b)p Ft(\)])2419 5251 y Fm(\000)p Fs(1)p Fn(=)p Fs(2)2633 5287 y Ft(;)32 b(on)g(the)g(other)g(hand)h(one)113 5532 y(has)298 5452 y Fk(R)345 5567 y Fn(I)380 5577 y Fj(g)444 5532 y Fo(j)p Fp(')p Ft(\()p Fp(y)d Fo(\006)e Fp(i)p Ft(0\))p Fo(j)p Fp(dx)39 b Ft(=)i Fp(c)1220 5496 y Fm(\000)p Fs(2)1343 5452 y Fk(R)1438 5476 y Fo(j)p Fp(')p Ft(\()p Fp(y)25 b Fo(\006)e Fp(i)p Ft(0\))p Fo(j)p Fp(dy)p 1438 5509 561 4 v 1498 5599 a Ft(\()p Fp(y)i Ft(+)e Fp(d=c)p Ft(\))1895 5558 y Fs(2)2051 5532 y Fp(;)40 b Ft(where)i(this)e(in)m (tegral)g(is)h(tak)m(en)g(on)f(the)p eop %%Page: 28 28 28 27 bop 113 144 a Ft(28)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 428 y(complemen)m(t)40 b(in)p 800 347 72 4 v 40 w Fq(R)52 b Ft(of)41 b(the)f(in)m(terv)-6 b(al)1580 317 y Fk(h)1627 428 y Fo(\000)1716 389 y Ft(3)p Fp(d)p 1716 405 102 4 v 1745 462 a(c)1830 428 y(;)17 b Fo(\000)1984 389 y Fp(d)p 1964 405 93 4 v 1964 473 a Ft(3)p Fp(c)2068 317 y Fk(i)2156 428 y Ft(\(if)40 b Fp(c)g(>)g Ft(1)20 b(;)44 b(when)d Fp(g)i Ft(=)3111 287 y Fk(\022)3201 368 y Ft(0)118 b(1)3201 488 y(1)99 b Fp(m)3454 287 y Fk(\023)3568 428 y Ft(or)113 537 y Fk(\022)203 617 y Ft(1)h Fp(m)22 b Fo(\000)h Ft(1)203 737 y(1)186 b Fp(m)628 537 y Fk(\023)702 677 y Ft(,)33 b Fp(m)28 b Fo(\025)g Ft(2,)33 b(the)g(in)m(tegral)g(is)g (tak)m(en)g(from)g Fo(\000)p Fp(m=)p Ft(3)h(to)e(+)p Fo(1)p Ft(\).)313 851 y(One)i(has)g(then)803 992 y Fk(Z)902 1016 y Fs(3)p Fn(=)p Fs(2)858 1218 y Fm(\000)p Fs(1)p Fn(=)p Fs(2)1074 1060 y Fo(j)p Fp(')p Ft(\()p Fp(y)25 b Fo(\006)e Fp(i)p Ft(0\))p Fo(j)p Fp(dy)p 1074 1104 561 4 v 1134 1196 a Ft(\()p Fp(y)i Ft(+)e Fp(d=c)p Ft(\))1531 1167 y Fs(2)1674 1127 y Fo(\024)29 b Fp(C)7 b(c)1901 1086 y Fs(2)1946 1127 y Fp(d)1998 1086 y Fm(\000)p Fs(2)2104 1127 y Fo(k)p Fp(')p Fo(k)2269 1145 y Fn(H)2339 1125 y Fi(1)2411 1127 y Fp(;)758 1303 y Fk(Z)858 1328 y Fm(\000)p Fs(1)p Fn(=)p Fs(2)814 1529 y Fm(\000)p Fn(d=)p Fs(3)p Fn(c)1074 1371 y Fo(j)p Fp(')p Ft(\()p Fp(y)25 b Fo(\006)e Fp(i)p Ft(0\))p Fo(j)p Fp(dy)p 1074 1416 V 1134 1507 a Ft(\()p Fp(y)i Ft(+)e Fp(d=c)p Ft(\))1531 1478 y Fs(2)1674 1439 y Fo(\024)29 b Fp(C)7 b(c)1901 1398 y Fs(2)1946 1439 y Fp(d)1998 1398 y Fm(\000)p Fs(2)2104 1439 y Fo(k)p Fp(')p Fo(k)2269 1456 y Fn(H)2339 1436 y Fi(1)2400 1439 y Ft(log)2546 1298 y Fk(\022)2619 1439 y Ft(1)22 b(+)2803 1371 y Fp(d)p 2803 1416 52 4 v 2807 1507 a(c)2866 1298 y Fk(\023)2984 1439 y Fp(;)113 1768 y Ft(and)34 b(the)g(same)e(for)881 1688 y Fk(R)947 1713 y Fs(3)p Fn(d=c)928 1804 y Fs(3)p Fn(=)p Fs(2)1110 1768 y Ft(.)44 b(Finally)32 b(one)h(has)1003 1941 y Fk(Z)1058 2167 y Fm(j)p Fn(y)s Fm(j)p Fn(>)p Fs(3)p Fn(d=c)1402 2009 y Fo(j)p Fp(')p Ft(\()p Fp(y)25 b Fo(\006)e Fp(i)p Ft(0\))p Fo(j)p Fp(dy)p 1402 2054 561 4 v 1462 2145 a Ft(\()p Fp(y)i Ft(+)e Fp(d=c)p Ft(\))1859 2116 y Fs(2)2002 2076 y Fo(\024)28 b Fp(C)7 b(c)2228 2035 y Fs(2)2274 2076 y Fp(d)2326 2035 y Fm(\000)p Fs(2)2432 2076 y Fo(k)p Fp(')p Fo(k)2597 2094 y Fn(H)2667 2074 y Fi(1)2739 2076 y Fp(:)113 2367 y Ft(Putting)33 b(the)h(v)-6 b(arious)33 b(b)s(ounds)h(together)g(one)f(gets)454 2506 y Fk(Z)510 2733 y Fn(I)545 2743 y Fj(g)620 2575 y Fo(j)16 b Ft(~)-66 b Fp(')p Ft(\()p Fp(x)22 b Fo(\006)g Fp(i)p Ft(0\))p Fo(j)p Fp(dx)p 620 2619 570 4 v 674 2639 a Fk(p)p 773 2639 363 4 v 773 2724 a Fp(x)p Ft(\(1)g Fo(\000)h Fp(x)p Ft(\))1229 2642 y Fo(\024)29 b Fp(C)7 b(d)1465 2601 y Fm(\000)p Fs(5)p Fn(=)p Fs(2)1652 2642 y Ft([min)o(\()p Fp(b;)17 b(d)k Fo(\000)h Fp(b)p Ft(\)])2255 2601 y Fm(\000)p Fs(1)p Fn(=)p Fs(2)2459 2642 y Ft(log\(1)g(+)h Fp(d=c)p Ft(\))p Fo(k)p Fp(')p Fo(k)3147 2660 y Fn(H)3217 2640 y Fi(1)3288 2642 y Fp(:)113 2943 y Ft(T)-8 b(o)31 b(conclude)i(the)f (pro)s(of)g(it)f(is)g(no)m(w)h(enough)h(to)e(note)h(that)f(log)q(\(1)18 b(+)h Fp(d=c)p Ft(\))27 b Fo(\024)i Ft(log\(1)19 b(+)g Fp(d)p Ft(\))31 b(and)113 3062 y(log)q(\(1)22 b(+)g Fp(c)17 b Ft(min)o(\()p Fp(b;)g(d)k Fo(\000)i Fp(b)p Ft(\)\))j Fo(\024)j Ft(log\(1)22 b(+)h Fp(d)1622 3026 y Fs(2)1666 3062 y Ft(\))28 b Fo(\024)g Ft(2)17 b(log\(1)22 b(+)g Fp(d)p Ft(\).)1217 b Fd(\003)113 3294 y Fr(Corollary)45 b(4.10)58 b Fe(The)40 b(series)1373 3219 y Fk(P)1478 3324 y Fm(M)1593 3294 y Fo(k)p Fp(L)1711 3309 y Fn(g)1757 3294 y Fo(k)1807 3312 y Fn(H)1877 3292 y Fi(1)1960 3294 y Fe(is)f(con)m(v)m(ergen)m(t.)2620 3219 y Fk(P)2725 3324 y Fm(M)2862 3294 y Fe(de\014nes)i(a)d(b)s(ounded)113 3414 y(op)s(erator)e(of)g Fp(H)725 3377 y Fs(1)770 3414 y Ft(\()p 809 3333 72 4 v Fq(C)49 b Fo(n)24 b Ft([0)p Fp(;)17 b Ft(1]\))22 b Fo(\\)j(O)1413 3377 y Fs(1)1457 3414 y Ft(\()p 1496 3333 V Fq(C)50 b Fo(n)24 b Ft([0)p Fp(;)17 b Ft(1]\))34 b Fe(in)m(to)i(itself)20 b(;)37 b(moreo)m(v)m(er)e Fp(T)50 b Fe(is)36 b(a)g(con)m(traction)113 3553 y(of)e(this)f(space,)h(its)f(sp)s(ectral)g(radius)h(b)s(eing)g(b)s (ounded)g(ab)s(o)m(v)m(e)g(b)m(y)2636 3455 y Fm(p)p 2701 3455 40 4 v 58 x Fs(5)p Fm(\000)p Fs(1)p 2636 3530 208 4 v 2720 3587 a(2)2855 3553 y Fe(.)113 3756 y Fl(Pr)-5 b(o)g(of.)36 b Ft(Let)e Fp(k)f Fo(\025)e Ft(1.)47 b(W)-8 b(e)35 b(will)f(tak)m(e)g(the)h(sum)f(of)h Fo(k)p Fp(L)2108 3771 y Fn(g)2154 3756 y Fo(k)2204 3774 y Fn(H)2274 3754 y Fi(1)2352 3756 y Ft(on)g(the)g(elemen)m(ts)g Fp(g)d Fo(2)f(M)j Ft(whic)m(h)113 3876 y(are)k(the)f(pro)s(duct)h(of)g (exactly)e Fp(k)k Ft(generators)e Fp(g)t Ft(\()p Fp(m)p Ft(\).)55 b(The)37 b(branc)m(h)i(of)f Fp(A)2916 3840 y Fn(k)3001 3876 y Ft(asso)s(ciated)g(to)f Fp(g)113 4025 y Ft(has)c(domain)643 3915 y Fk(h)702 3986 y Fn(a)p Fs(+)p Fn(b)p 702 4002 140 4 v 703 4060 a(c)p Fs(+)p Fn(d)853 4025 y Fp(;)913 3986 y Fn(b)p 910 4002 42 4 v 910 4060 a(d)963 3915 y Fk(i)1038 4025 y Ft(=)28 b Fp(I)8 b Ft(\()p Fp(g)t Ft(\))30 b(and)j(on)f(this)g(in)m(terv)-6 b(al)32 b(the)g(max)f(and)h(the)h(min)e(of)h Fp(x)p Ft(\(1)20 b Fo(\000)g Fp(x)p Ft(\))113 4184 y(are)34 b(comparable.)43 b(The)34 b(length)f([)p Fp(d)p Ft(\()p Fp(c)22 b Ft(+)g Fp(d)p Ft(\)])1761 4148 y Fm(\000)p Fs(1)1901 4184 y Ft(of)33 b Fp(I)8 b Ft(\()p Fp(g)t Ft(\))32 b(is)h(b)s(ounded)i(b)s (elo)m(w)e(b)m(y)3167 4145 y Fs(1)p 3167 4161 40 4 v 3167 4219 a(2)3219 4184 y Fp(d)3271 4148 y Fm(\000)p Fs(2)3378 4184 y Ft(,)g(th)m(us)476 4480 y Fo(k)p Fp(L)594 4495 y Fn(g)640 4480 y Fo(k)690 4498 y Fn(H)760 4478 y Fi(1)831 4480 y Fo(\024)949 4413 y Fp(K)7 b(d)1093 4377 y Fm(\000)p Fs(3)1216 4413 y Ft(log\(1)22 b(+)h Fp(d)p Ft(\))p 949 4458 697 4 v 1042 4477 a Fk(q)p 1142 4477 411 4 v 1157 4555 a Fn(b)p 1154 4571 42 4 v 1154 4628 a(d)1224 4513 y Fk(\000)1269 4594 y Ft(1)f Fo(\000)1456 4555 y Fn(b)p 1453 4571 V 1453 4628 a(d)1506 4513 y Fk(\001)1685 4480 y Fo(\024)28 b Fp(K)1882 4439 y Fm(0)1910 4480 y Fp(d)1962 4439 y Fm(\000)p Fs(1)2085 4480 y Ft(log)q(\(1)22 b(+)g Fp(d)p Ft(\))2532 4345 y Fk(Z)2587 4571 y Fn(I)5 b Fs(\()p Fn(g)s Fs(\))2942 4413 y Fp(dx)p 2765 4458 463 4 v 2765 4477 a Fk(p)p 2864 4477 363 4 v 2864 4563 a Fp(x)p Ft(\(1)22 b Fo(\000)h Fp(x)p Ft(\))3267 4480 y Fp(:)113 4841 y Ft(When)33 b(w)m(e)g(sum)e(on)h(the)g(elemen)m(ts)g Fp(g)j Ft(considered,)e(the)f(in)m(terv)-6 b(als)32 b Fp(I)8 b Ft(\()p Fp(g)t Ft(\))30 b(form)h(a)h(partition)e(of)113 4960 y([0)p Fp(;)17 b Ft(1])32 b(mo)s(d)15 b(0)33 b(th)m(us)i(w)m(e)e (get)993 5109 y Fk(X)573 5327 y Fn(m)644 5337 y Fi(1)683 5327 y Fn(;:::)o(;m)873 5338 y Fj(k)929 5327 y Fn(;g)s Fs(=)p Fn(g)s Fs(\()p Fn(m)1200 5337 y Fi(1)1237 5327 y Fs(\))p Fm(\001\001\001)o Fn(g)s Fs(\()p Fn(m)1483 5338 y Fj(k)1526 5327 y Fs(\))1574 5203 y Fo(k)p Fp(L)1692 5218 y Fn(g)1738 5203 y Fo(k)1788 5221 y Fn(H)1858 5201 y Fi(1)1930 5203 y Fo(\024)28 b Fp(K)2127 5162 y Fm(00)2273 5203 y Ft(max)2194 5263 y Fn(m)2265 5273 y Fi(1)2304 5263 y Fn(;:::)n(;m)2493 5274 y Fj(k)2536 5203 y Ft([)p Fp(d)2616 5162 y Fm(\000)p Fs(1)2739 5203 y Ft(log)q(\(1)22 b(+)g Fp(d)p Ft(\)])113 5537 y(whic)m(h)34 b(giv)m(es)g(the)f(desired)i (result.)2154 b Fd(\003)p eop %%Page: 29 29 29 28 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(29)113 383 y Fr(4.4)39 b(Real)e(holomorphic)h(functions)f(with)g(b)s (ounded)g(real)g(part)113 530 y Ft(W)-8 b(e)24 b(denote)f(b)m(y)g Fp(E)28 b Ft(the)c(space)f(of)h(functions)g Fp(')k Fo(2)g(O)2024 494 y Fs(1)2068 530 y Ft(\()p 2107 449 72 4 v Fq(C)g Fo(n)q Ft([0)p Fp(;)17 b Ft(1]\))k(whose)i(real)g(part)g(is)f(b)s (ounded,)113 649 y(endo)m(w)m(ed)33 b(with)e(the)h(norm)e Fo(k)p Fp(')p Fo(k)1333 664 y Fn(E)1428 649 y Ft(=)e(sup)p 1683 623 51 4 v 1683 683 a Fg(C)13 b Fm(n)p Fs([0)p Fn(;)p Fs(1])1944 649 y Fo(j)k(<)p Fp(e)g(')p Fo(j)27 b Fp(:)k Ft(W)-8 b(e)32 b(ha)m(v)m(e)f(then)h(for)g Fo(j)17 b(=)p Fp(m)f(z)t Fo(j)28 b(\024)g Ft(1)p Fp(=)p Ft(2)893 937 y Fo(j)17 b(=)p Fp(m)f(')p Ft(\()p Fp(z)t Ft(\))p Fo(j)28 b(\024)1484 870 y Ft(2)p 1479 915 61 4 v 1479 1006 a Fp(\031)1568 937 y Ft(log)q([\(2)1814 850 y Fo(p)p 1896 850 50 4 v 1896 937 a Ft(2)22 b Fo(\000)h Ft(2\))p Fo(j)17 b(=)p Fp(m)f(z)t Fo(j)p Ft(])2483 896 y Fm(\000)p Fs(1)2590 937 y Fo(k)p Fp(')p Fo(k)2755 952 y Fn(E)2849 937 y Fp(;)652 b Ft(\()p Fo(\003)p Ft(\))113 1180 y(as)34 b(one)f(can)h(pro)m(v)m(e)g (from)e(the)i(analogue)f(estimate)f(for)h(functions)i(in)e(the)h(unit)f (disk)g Fq(D)869 1447 y Fo(j)17 b(=)p Fp(m)g Ft(\010\()p Fp(w)s Ft(\))p Fo(j)27 b(\024)1491 1380 y Ft(2)p 1486 1424 61 4 v 1486 1515 a Fp(\031)1625 1447 y Ft(sup)1575 1535 y Fn(#)p Fm(2)p Fs([0)p Fn(;)p Fs(1])1841 1447 y Fo(j)17 b(<)p Fp(e)f Ft(\010\()p Fp(e)2177 1406 y Fs(2)p Fn(\031)s(i#)2348 1447 y Ft(\))p Fo(j)h Ft(log)2588 1380 y(1)22 b(+)g Fo(j)p Fp(w)s Fo(j)p 2588 1424 301 4 v 2588 1515 a Ft(1)g Fo(\000)g(j)p Fp(w)s Fo(j)113 1757 y Ft(applying)33 b(the)h(conformal)e(represen)m(tation)j(of)p 1919 1676 72 4 v 33 w Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1])26 b Fo(!)h Fq(D)19 b Ft(,)39 b Fp(w)31 b Ft(=)d(\()2818 1685 y Fo(p)p 2901 1685 51 4 v 72 x Fp(z)f Fo(\000)3073 1678 y(p)p 3156 1678 223 4 v 79 x Fp(z)g Fo(\000)c Ft(1\))3418 1721 y Fs(2)3462 1757 y Ft(.)113 1931 y Fr(Prop)s(osition)36 b(4.11)58 b Fe(The)31 b(restriction)f(of)h Fp(T)45 b Fe(to)30 b Fp(E)36 b Fe(is)30 b(a)h(b)s(ounded)h(op)s(erator)e(with)g (sp)s(ectral)113 2064 y(radius)k Fo(\024)530 1967 y Fm(p)p 596 1967 40 4 v 596 2025 a Fs(5)o Fm(\000)p Fs(1)p 530 2041 208 4 v 614 2099 a(2)749 2064 y Fe(.)113 2279 y Fl(Pr)-5 b(o)g(of.)34 b Ft(Let)f Fp(')28 b Fo(2)g Fp(E)6 b Ft(,)32 b Fp(k)f Fo(\025)d Ft(1,)k Fp(z)1260 2294 y Fs(0)1333 2279 y Fo(2)p 1427 2198 72 4 v 28 w Fq(C)47 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1].)42 b(W)-8 b(e)33 b(estimate)f Fo(<)p Fp(e)17 b(T)2640 2242 y Fn(k)2689 2279 y Fp(')p Ft(\()p Fp(z)2839 2294 y Fs(0)2884 2279 y Ft(\))32 b(in)h(v)-6 b(arious)33 b(cases.)158 2398 y(\(i\))49 b(If)33 b Fp(z)458 2413 y Fs(0)531 2398 y Fo(2)28 b Fp(D)707 2413 y Fm(1)825 2398 y Ft(w)m(e)34 b(ha)m(v)m(e)858 2727 y Fo(j)17 b(<)p Fp(e)f(T)1109 2685 y Fn(k)1158 2727 y Fp(')p Ft(\()p Fp(z)1308 2742 y Fs(0)1353 2727 y Ft(\))p Fo(j)27 b(\024)h(j)p Fp(T)1652 2685 y Fn(k)1701 2727 y Fp(')p Ft(\()p Fp(z)1851 2742 y Fs(0)1896 2727 y Ft(\))p Fo(j)f(\024)h Fp(c)2155 2556 y Fk( )2246 2577 y Fo(p)p 2329 2577 50 4 v 82 x Ft(5)22 b(+)g(1)p 2246 2704 305 4 v 2373 2795 a(2)2562 2556 y Fk(!)2641 2577 y Fm(\000)p Fn(k)2769 2727 y Ft(sup)2775 2810 y Fn(D)2840 2820 y Ff(1)2936 2727 y Fo(j)p Fp(')p Fo(j)27 b Fp(;)313 3031 y Ft(and,)33 b(on)g(the)h(other)f(hand,)h(for)g(all)e Fp(')c Fo(2)g Fp(E)6 b Ft(,)32 b(sup)2149 3056 y Fn(D)2214 3066 y Ff(1)2309 3031 y Fo(j)p Fp(')p Fo(j)27 b(\024)h Fp(C)7 b Fo(k)p Fp(')p Fo(k)2805 3046 y Fn(E)2900 3031 y Fp(:)130 3151 y Ft(\(ii\))49 b(If)30 b Fp(z)455 3166 y Fs(0)528 3151 y Fo(2)e Fp(D)704 3166 y Fs(0)765 3151 y Fo([)16 b Fp(H)930 3166 y Fs(0)991 3151 y Fo([)g Fp(H)1164 3115 y Fm(0)1156 3175 y Fs(0)1201 3151 y Ft(,)30 b(or)g(if)g Fp(z)1512 3166 y Fs(0)1585 3151 y Fo(2)e Fp(D)1761 3166 y Fs(1)1806 3151 y Ft(,)j Fp(k)f(>)f Ft(1,)h(or)g(if)g Fp(z)2414 3166 y Fs(0)2487 3151 y Fo(2)e Fp(D)33 b Ft(has)e(con)m(tin)m (ued)h(fraction)313 3270 y Fp(z)363 3228 y Fm(\000)p Fs(1)359 3298 y Fn(i)502 3270 y Ft(=)h Fp(m)699 3285 y Fn(i)p Fs(+1)857 3270 y Ft(+)24 b Fp(z)1004 3285 y Fn(i)p Fs(+1)1139 3270 y Ft(,)36 b(0)31 b Fo(\024)i Fp(i)f(<)h(l)r Ft(,)i(with)h Fp(z)1941 3285 y Fn(l)2004 3270 y Fo(2)c Ft(\001)24 b Fo(n)f Fp(D)s Ft(,)37 b Fp(l)c(<)g(k)27 b Fo(\000)d Ft(1,)36 b(w)m(e)g(ha)m(v)m(e)g(obtained)313 3390 y(in)d(the)g(pro)s(of)h(of)f(Prop)s(ostion)g(4.5)f(the)i(estimate) 1169 3713 y Fo(j)p Fp(T)1269 3672 y Fn(k)1318 3713 y Fp(')p Ft(\()p Fp(z)1468 3728 y Fs(0)1513 3713 y Ft(\))p Fo(j)27 b(\024)h Fp(c)17 b(k)1844 3542 y Fk( )1934 3563 y Fo(p)p 2017 3563 50 4 v 82 x Ft(5)22 b(+)h(1)p 1934 3690 305 4 v 2062 3781 a(2)2251 3542 y Fk(!)2330 3563 y Fm(\000)p Fn(k)2458 3713 y Ft(sup)2464 3796 y Fn(D)2529 3806 y Ff(1)2624 3713 y Fo(j)p Fp(')p Fo(j)k Fp(:)102 4017 y Ft(\(iii\))49 b(If)33 b Fp(z)458 4032 y Fs(0)531 4017 y Fo(2)28 b Fp(D)707 4032 y Fs(1)752 4017 y Ft(,)33 b Fp(k)e Ft(=)d(1,)33 b(w)m(e)g(ha)m(v)m(e)h(from)f(Prop)s(osition)f (4.2)1167 4148 y Fk(\014)1167 4207 y(\014)1167 4267 y(\014)1167 4327 y(\014)1200 4292 y Fp(T)14 b(')p Ft(\()p Fp(z)1422 4307 y Fs(0)1467 4292 y Ft(\))22 b(+)h Fp(z)1674 4307 y Fs(0)1719 4292 y Fp(')1801 4152 y Fk(\022)1907 4225 y Ft(1)p 1886 4269 92 4 v 1886 4360 a Fp(z)1932 4375 y Fs(0)2011 4292 y Fo(\000)g Ft(1)2161 4152 y Fk(\023)2234 4148 y(\014)2234 4207 y(\014)2234 4267 y(\014)2234 4327 y(\014)2295 4292 y Fo(\024)28 b Fp(c)17 b Ft(sup)2466 4376 y Fn(D)2531 4386 y Ff(1)2626 4292 y Fo(j)p Fp(')p Fo(j)27 b Fp(;)313 4570 y Ft(w)m(e)33 b(ha)m(v)m(e)1256 4580 y Fk(\014)1256 4640 y(\014)1256 4700 y(\014)1256 4759 y(\014)1289 4725 y Fo(<)p Fp(e)17 b(z)1470 4740 y Fs(0)1532 4725 y Fo(<)p Fp(e)g(')1749 4584 y Fk(\022)1854 4657 y Ft(1)p 1834 4702 V 1834 4793 a Fp(z)1880 4808 y Fs(0)1959 4725 y Fo(\000)22 b Ft(1)2108 4584 y Fk(\023)2182 4580 y(\014)2182 4640 y(\014)2182 4700 y(\014)2182 4759 y(\014)2242 4725 y Fo(\024)29 b Fp(C)7 b Fo(k)p Fp(')p Fo(k)2591 4740 y Fn(E)2686 4725 y Fp(;)313 4957 y Ft(and)1318 4967 y Fk(\014)1318 5027 y(\014)1318 5087 y(\014)1318 5147 y(\014)1351 5112 y Fo(=)p Fp(m)1528 4971 y Fk(\022)1633 5044 y Ft(1)p 1613 5089 V 1613 5180 a Fp(z)1659 5195 y Fs(0)1738 5112 y Fo(\000)23 b Ft(1)1888 4971 y Fk(\023)1961 4967 y(\014)1961 5027 y(\014)1961 5087 y(\014)1961 5147 y(\014)2022 5112 y Fo(\025)28 b Fp(C)2205 5071 y Fm(\000)p Fs(1)2312 5112 y Fo(j)17 b(=)p Fp(m)f(z)2578 5127 y Fs(0)2623 5112 y Fo(j)313 5350 y Ft(hence)34 b(from)f(\()p Fo(\003)p Ft(\))1215 5385 y Fk(\014)1215 5445 y(\014)1215 5505 y(\014)1215 5564 y(\014)1248 5529 y Fo(=)p Fp(m)17 b(z)1470 5544 y Fs(0)1532 5529 y Fo(=)p Fp(m)g(')1790 5389 y Fk(\022)1895 5462 y Ft(1)p 1875 5507 V 1875 5598 a Fp(z)1921 5613 y Fs(0)2000 5529 y Fo(\000)23 b Ft(1)2150 5389 y Fk(\023)2223 5385 y(\014)2223 5445 y(\014)2223 5505 y(\014)2223 5564 y(\014)2284 5529 y Fo(\024)28 b Fp(C)7 b Fo(k)p Fp(')p Fo(k)2632 5544 y Fn(E)2727 5529 y Fp(:)p eop %%Page: 30 30 30 29 bop 113 144 a Ft(30)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)105 383 y(\(iv\))49 b(If)42 b Fp(z)467 398 y Fs(0)554 383 y Fo(2)h Fp(D)j Ft(has)c(con)m(tin)m(ued)i(fraction)e Fp(z)1867 340 y Fm(\000)p Fs(1)1863 411 y Fn(i)2017 383 y Ft(=)h Fp(m)2224 398 y Fn(i)p Fs(+1)2386 383 y Ft(+)29 b Fp(z)2538 398 y Fn(i)p Fs(+1)2672 383 y Ft(,)44 b(0)e Fo(\024)h Fp(i)g(<)g(k)31 b Fo(\000)d Ft(1,)44 b(with)313 503 y Fp(z)359 518 y Fn(k)r Fm(\000)p Fs(1)563 503 y Fo(2)54 b Ft(\001)33 b Fo(n)f Fp(D)s Ft(,)53 b(the)c(only)f(term)g(in)g (the)h(pro)s(of)g(of)g(Prop)s(ostion)f(4.5)g(whic)m(h)i(giv)m(es)313 622 y(some)34 b(trouble)g(is)h(\()p Fp(p)1099 637 y Fn(k)r Fm(\000)p Fs(2)1273 622 y Fo(\000)23 b Fp(q)1417 637 y Fn(k)r Fm(\000)p Fs(2)1569 622 y Fp(z)1615 637 y Fs(0)1660 622 y Ft(\))p Fp(")1745 637 y Fn(k)r Fm(\000)p Fs(2)1896 622 y Fp(T)14 b(')p Ft(\()p Fp(z)2118 637 y Fn(k)r Fm(\000)p Fs(1)2292 622 y Ft(+)24 b(1\))34 b(\(the)g(others)h(are)g(once)g(again) 313 795 y(dominated)27 b(b)m(y)h Fp(C)7 b(k)1083 685 y Fk(\020)1154 698 y Fm(p)p 1220 698 40 4 v 1220 756 a Fs(5)o(+1)p 1154 773 207 4 v 1237 830 a(2)1372 685 y Fk(\021)1432 705 y Fm(\000)p Fn(k)1560 795 y Ft(sup)1710 820 y Fn(D)1775 830 y Ff(1)1869 795 y Fo(j)p Fp(')p Fo(j)p Ft(\).)41 b(In)29 b(fact,)f(from)g(Prop)s(osition)e(4.2,)i(w)m(e)h(are) 313 939 y(ev)m(en)34 b(left)f(with)887 1180 y(\()p Fp(p)976 1195 y Fn(k)r Fm(\000)p Fs(2)1149 1180 y Fo(\000)23 b Fp(q)1293 1195 y Fn(k)r Fm(\000)p Fs(2)1444 1180 y Fp(z)1490 1195 y Fs(0)1535 1180 y Ft(\))p Fp(")1620 1195 y Fn(k)r Fm(\000)p Fs(2)1771 1180 y Ft(\(1)f(+)h Fp(z)2028 1195 y Fn(k)r Fm(\000)p Fs(1)2179 1180 y Ft(\))p Fp(')2300 1039 y Fk(\022)2544 1113 y Ft(1)p 2385 1157 369 4 v 2385 1248 a(1)f(+)g Fp(z)2602 1263 y Fn(k)r Fm(\000)p Fs(1)2788 1180 y Fo(\000)g Ft(1)2937 1039 y Fk(\023)3055 1180 y Fp(:)313 1445 y Ft(W)-8 b(e)32 b(ha)m(v)m(e)g(here)g(\()p Fp(p)1014 1460 y Fn(k)r Fm(\000)p Fs(2)1184 1445 y Fo(\000)20 b Fp(q)1325 1460 y Fn(k)r Fm(\000)p Fs(2)1477 1445 y Fp(z)1523 1460 y Fs(0)1568 1445 y Ft(\)\(1)e(+)i Fp(z)1857 1460 y Fn(k)r Fm(\000)p Fs(1)2008 1445 y Ft(\))27 b(=)h(\()p Fp(p)2268 1460 y Fn(k)r Fm(\000)p Fs(2)2438 1445 y Fo(\000)20 b Fp(q)2579 1460 y Fn(k)r Fm(\000)p Fs(2)2730 1445 y Fp(z)2776 1460 y Fs(0)2821 1445 y Ft(\))f Fo(\000)h Ft(\()p Fp(p)3065 1460 y Fn(k)r Fm(\000)p Fs(1)3235 1445 y Fo(\000)f Fp(q)3375 1460 y Fn(k)r Fm(\000)p Fs(1)3527 1445 y Fp(z)3573 1460 y Fs(0)3618 1445 y Ft(\))313 1565 y(hence)525 1749 y Fo(j<)p Fp(e)p Ft([\()p Fp(p)788 1764 y Fn(k)r Fm(\000)p Fs(2)961 1749 y Fo(\000)k Fp(q)1105 1764 y Fn(k)r Fm(\000)p Fs(2)1256 1749 y Fp(z)1302 1764 y Fs(0)1347 1749 y Ft(\)\(1)f(+)g Fp(z)1642 1764 y Fn(k)r Fm(\000)p Fs(1)1793 1749 y Ft(\)])p Fo(j)27 b(\024)h Fp(C)7 b(q)2146 1706 y Fm(\000)p Fs(1)2142 1779 y Fn(k)r Fm(\000)p Fs(1)484 1914 y Fo(j=)p Fp(m)p Ft([\()p Fp(p)788 1929 y Fn(k)r Fm(\000)p Fs(2)961 1914 y Fo(\000)23 b Fp(q)1105 1929 y Fn(k)r Fm(\000)p Fs(2)1256 1914 y Fp(z)1302 1929 y Fs(0)1347 1914 y Ft(\)\(1)f(+)g Fp(z)1642 1929 y Fn(k)r Fm(\000)p Fs(1)1793 1914 y Ft(\)])p Fo(j)27 b(\024)h Fp(C)7 b(q)2142 1929 y Fn(k)r Fm(\000)p Fs(1)2294 1914 y Fo(j)17 b(=)p Fp(m)g(z)2561 1929 y Fs(0)2606 1914 y Fo(j)27 b(\024)h Fp(C)7 b(q)2892 1872 y Fm(\000)p Fs(1)2888 1944 y Fn(k)r Fm(\000)p Fs(1)3040 1914 y Fo(j)17 b(=)p Fp(m)f(z)3306 1929 y Fn(k)r Fm(\000)p Fs(1)3457 1914 y Fo(j)313 2118 y Ft(and)1099 2128 y Fk(\014)1099 2188 y(\014)1099 2248 y(\014)1099 2307 y(\014)1132 2273 y Fo(=)p Fp(m)1308 2132 y Fk(\022)1553 2205 y Ft(1)p 1393 2250 V 1393 2341 a(1)22 b(+)h Fp(z)1611 2356 y Fn(k)r Fm(\000)p Fs(1)1796 2273 y Fo(\000)g Ft(1)1946 2132 y Fk(\023)2019 2128 y(\014)2019 2188 y(\014)2019 2248 y(\014)2019 2307 y(\014)2080 2273 y Fo(\025)28 b Fp(C)2263 2231 y Fm(\000)p Fs(1)2370 2273 y Fo(j)17 b(=)p Fp(m)f(z)2636 2288 y Fn(k)r Fm(\000)p Fs(1)2788 2273 y Fo(j)27 b Fp(:)313 2507 y Ft(Th)m(us,)34 b(from)e(\()p Fo(\003)p Ft(\))h(w)m(e)g(get)582 2628 y Fk(\014)582 2688 y(\014)582 2747 y(\014)582 2807 y(\014)615 2772 y Fo(<)p Fp(e)q Ft(\()p Fp(p)823 2787 y Fn(k)r Fm(\000)p Fs(2)996 2772 y Fo(\000)22 b Fp(q)1139 2787 y Fn(k)r Fm(\000)p Fs(2)1291 2772 y Fp(z)1337 2787 y Fs(0)1382 2772 y Ft(\)\(1)f(+)i Fp(z)1677 2787 y Fn(k)r Fm(\000)p Fs(1)1828 2772 y Ft(\))p Fp(')1949 2632 y Fk(\022)2194 2705 y Ft(1)p 2034 2749 V 2034 2841 a(1)f(+)h Fp(z)2252 2856 y Fn(k)r Fm(\000)p Fs(1)2437 2772 y Fo(\000)g Ft(1)2587 2632 y Fk(\023)2660 2628 y(\014)2660 2688 y(\014)2660 2747 y(\014)2660 2807 y(\014)2721 2772 y Fo(\024)28 b Fp(C)7 b(q)2952 2730 y Fm(\000)p Fs(1)2948 2802 y Fn(k)r Fm(\000)p Fs(1)3100 2772 y Fo(k)p Fp(')p Fo(k)3265 2787 y Fn(E)3359 2772 y Fp(;)313 3131 y Ft(and)33 b(\014nally)g Fo(j)17 b(<)p Fp(e)f(T)1059 3095 y Fn(k)1108 3131 y Fp(')p Ft(\()p Fp(z)1258 3146 y Fs(0)1303 3131 y Ft(\))p Fo(j)27 b(\024)h Fp(C)7 b(k)1652 2991 y Fk(\022)1738 3005 y Fo(p)p 1821 3005 50 4 v 87 x Ft(5)22 b(+)g(1)p 1738 3108 305 4 v 1865 3177 a(2)2054 2991 y Fk(\023)2127 3011 y Fm(\000)p Fn(k)2255 3131 y Fo(k)p Fp(')p Fo(k)2420 3146 y Fn(E)2515 3131 y Fp(:)133 3323 y Ft(\(v\))49 b(If)34 b Fp(z)459 3338 y Fs(0)533 3323 y Fo(2)29 b Fp(D)37 b Ft(has)d(con)m(tin)m(ued)i (fraction)e Fp(z)1799 3281 y Fm(\000)p Fs(1)1795 3351 y Fn(i)1935 3323 y Ft(=)c Fp(m)2129 3338 y Fn(i)p Fs(+1)2286 3323 y Ft(+)23 b Fp(z)2432 3338 y Fn(i)p Fs(+1)2566 3323 y Ft(,)34 b(0)29 b Fo(\024)g Fp(i)g(<)g(k)s Ft(,)34 b(with)g Fp(z)3373 3338 y Fn(k)3451 3323 y Fo(2)29 b Ft(\001,)313 3443 y(w)m(e)k(apply)g(Prop)s(osition)g(4.4.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)780 3646 y Fo(j)17 b(<)p Fp(e)p Ft(\()p Fp(p)1032 3661 y Fn(k)r Fm(\000)p Fs(1)1205 3646 y Fo(\000)23 b Fp(q)1349 3661 y Fn(k)r Fm(\000)p Fs(1)1500 3646 y Fp(z)1546 3661 y Fs(0)1591 3646 y Ft(\))p Fo(j)k(\024)h Fp(C)7 b(q)1916 3603 y Fm(\000)p Fs(1)1912 3676 y Fn(k)739 3808 y Fo(j)17 b(=)p Fp(m)p Ft(\()p Fp(p)1032 3823 y Fn(k)r Fm(\000)p Fs(1)1205 3808 y Fo(\000)23 b Fp(q)1349 3823 y Fn(k)r Fm(\000)p Fs(1)1500 3808 y Fp(z)1546 3823 y Fs(0)1591 3808 y Ft(\))p Fo(j)k(\024)h Fp(q)1834 3823 y Fn(k)r Fm(\000)p Fs(1)1986 3808 y Fo(j)17 b(=)p Fp(m)f(z)2252 3823 y Fs(0)2297 3808 y Fo(j)28 b(\024)g Fp(C)7 b(q)2580 3823 y Fn(k)r Fm(\000)p Fs(1)2732 3808 y Fp(q)2780 3766 y Fm(\000)p Fs(2)2776 3838 y Fn(k)2887 3808 y Fo(j)17 b(=)p Fp(m)f(z)3153 3823 y Fn(k)3202 3808 y Fo(j)313 4014 y Ft(hence,)34 b(for)g Fp(")28 b Ft(=)g Fo(\000)p Ft(1)p Fp(;)17 b Ft(0)p Fp(;)g Ft(1,)31 b(from)i(\()p Fo(\003)p Ft(\))988 4220 y Fo(j)17 b(<)p Fp(e)p Ft([\()p Fp(p)1268 4235 y Fn(k)r Fm(\000)p Fs(1)1441 4220 y Fo(\000)22 b Fp(q)1584 4235 y Fn(k)r Fm(\000)p Fs(1)1736 4220 y Fp(z)1782 4235 y Fs(0)1827 4220 y Ft(\))p Fp(')p Ft(\()p Fp(z)2016 4235 y Fn(k)2087 4220 y Ft(+)h Fp(")p Ft(\)])p Fo(j)k(\024)h Fp(C)7 b(q)2586 4177 y Fm(\000)p Fs(1)2582 4250 y Fn(k)2693 4220 y Fo(k)p Fp(')p Fo(k)2858 4235 y Fn(E)2953 4220 y Fp(:)313 4426 y Ft(W)-8 b(e)33 b(deal)h(similarly)c (with)949 4685 y Fo(<)p Fp(e)1084 4545 y Fk(\024)1136 4685 y Ft(\()p Fp(p)1225 4700 y Fn(k)r Fm(\000)p Fs(2)1398 4685 y Fo(\000)23 b Fp(q)1542 4700 y Fn(k)r Fm(\000)p Fs(2)1693 4685 y Fp(z)1739 4700 y Fs(0)1784 4685 y Ft(\)\(1)f(+)g Fp(z)2079 4700 y Fn(k)r Fm(\000)p Fs(1)2231 4685 y Ft(\))p Fp(')2352 4545 y Fk(\022)2425 4685 y Fo(\000)2600 4618 y Fp(z)2646 4633 y Fn(k)r Fm(\000)p Fs(1)p 2514 4663 369 4 v 2514 4754 a Ft(1)g(+)h Fp(z)2732 4769 y Fn(k)r Fm(\000)p Fs(1)2895 4545 y Fk(\023\025)313 4951 y Ft(\(see)33 b(\(iv\))f(ab)s(o)m(v)m(e\))h(and)h(conclude)h(that)1112 5270 y Fo(j)17 b(<)p Fp(e)f(T)1363 5229 y Fn(k)1412 5270 y Fp(')p Ft(\()p Fp(z)1562 5285 y Fs(0)1607 5270 y Ft(\))p Fo(j)27 b(\024)h Fp(C)7 b(k)1956 5100 y Fk( )2047 5120 y Fo(p)p 2130 5120 50 4 v 83 x Ft(5)22 b(+)h(1)p 2047 5247 305 4 v 2174 5338 a(2)2363 5100 y Fk(!)2442 5120 y Fm(\000)p Fn(k)2570 5270 y Fo(k)p Fp(')p Fo(k)2735 5285 y Fn(E)2830 5270 y Fp(:)3579 5558 y Fd(\003)p eop %%Page: 31 31 31 30 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(31)113 383 y Fr(5.)51 b(The)37 b(complex)i(Brjuno)f(function)313 646 y Ft(In)k(this)f(Section)h(w)m(e)h(in)m(tro)s(duce)f(and)g(study)g (the)g(complex)f(Brjuno)h(function.)70 b(Pre-)113 766 y(liminarly)41 b(w)m(e)j(need)g(some)f(further)h(results)g(on)g(the)f (monoid)f Fo(M)h Ft(and)h(on)f(the)h(algebraic)113 885 y(prop)s(erties)34 b(of)696 811 y Fk(P)801 915 y Fm(M)933 885 y Ft(and)1126 811 y Fk(P)1232 915 y Fn(Z)5 b Fm(M)1389 885 y Ft(.)313 1031 y Fr(5.0.1)33 b Ft(Let)h(us)g(recall)f(that)g(one)g (has)1039 1266 y Fp(Z)7 b Fo(M)28 b Ft(=)g Fp(Z)7 b Fo(M)1579 1125 y Fk(\022)1668 1206 y Ft(1)100 b(1)1668 1326 y(0)g(1)1884 1125 y Fk(\023)1980 1266 y Fo(t)22 b Fp(Z)7 b Fo(M)2280 1125 y Fk(\022)2370 1206 y Ft(0)99 b(1)2370 1326 y(1)g(1)2585 1125 y Fk(\023)2703 1266 y Fp(:)113 1526 y Ft(More)34 b(precisely)-8 b(,)33 b(if)g(one)h(denotes)g Fo(M)1552 1490 y Fm(\003)1625 1526 y Ft(=)29 b Fo(M)21 b(n)h(f)p Ft(1)p Fo(g)p Ft(,)33 b(one)g(has)810 1786 y Fo(M)947 1646 y Fk(\022)1036 1726 y Ft(1)99 b(1)1036 1846 y(0)g(1)1252 1646 y Fk(\023)1347 1786 y Fo(t)23 b(M)1573 1646 y Fk(\022)1662 1726 y Ft(0)99 b(1)1662 1846 y(1)g(1)1878 1646 y Fk(\023)1979 1786 y Ft(=)28 b Fo(M)2204 1745 y Fm(\003)2271 1786 y Fo(t)2360 1646 y Fk(\032\022)2525 1726 y Ft(1)99 b(1)2525 1846 y(0)g(1)2741 1646 y Fk(\023\033)2933 1786 y Fp(:)313 2086 y Fr(5.0.2)33 b Ft(Let)h Fp(')28 b Fo(2)g(O)1029 2049 y Fs(1)1074 2086 y Ft(\()p 1113 2005 72 4 v Fq(C)47 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\).)42 b(Let)706 2435 y Fp(')771 2450 y Fs(1)843 2435 y Ft(=)948 2205 y Fk(0)948 2380 y(B)948 2444 y(@)1036 2435 y Fp(L)1104 2405 y Fk(\022)1190 2486 y Ft(0)100 b(1)1190 2605 y(1)g(1)1403 2405 y Fk(\023)1504 2435 y Ft(+)22 b Fp(L)1671 2405 y Fk(\022)1758 2486 y Ft(1)99 b(1)1758 2605 y(0)g(1)1971 2405 y Fk(\023)2049 2205 y(1)2049 2380 y(C)2049 2444 y(A)2153 2435 y Fp(')28 b Fo(2)g(O)2422 2394 y Fs(1)2467 2435 y Ft(\()p 2506 2355 V Fq(C)47 b Fo(n)22 b Ft([1)p Fp(=)p Ft(2)p Fp(;)17 b Ft(2]\))26 b Fp(:)387 b Ft(\(5)p Fp(:)p Ft(1\))113 2819 y(The)29 b(family)c(\()p Fp(L)718 2834 y Fn(g)764 2819 y Fp(')p Ft(\))868 2834 y Fn(g)s Fm(2M)1090 2819 y Ft(is)i(uniformly)g(summable)f(on)i(compact)f(subsets)i(of)p 3071 2739 V 28 w Fq(C)37 b Fo(n)11 b Ft([0)p Fp(;)17 b Ft(1],)27 b(th)m(us)1014 2949 y Fk(X)1039 3162 y Fm(M)1174 3043 y Fp(')p Ft(\()p Fp(z)t Ft(\))c Fo(\000)f Fp(')p Ft(\()p Fp(z)t Ft(\))28 b(=)1816 2949 y Fk(X)1841 3162 y Fm(M)1976 3043 y Fp(')2041 3058 y Fs(1)2086 3043 y Ft(\()p Fp(z)t Ft(\))22 b Fo(\000)h Fp(')p Ft(\()p Fp(z)k Fo(\000)22 b Ft(1\))28 b Fp(:)113 3352 y Ft(On)43 b(the)g(other)f(hand) 996 3277 y Fk(P)1002 3441 y Fm(M)1118 3352 y Fp(')1183 3367 y Fs(1)1270 3352 y Ft(=)h Fp(')1455 3367 y Fs(1)1528 3352 y Ft(+)1634 3277 y Fk(P)1640 3441 y Fm(M)1739 3352 y Ft(\()p Fp(T)14 b(')1915 3367 y Fs(1)1960 3352 y Ft(\))42 b Fp(;)g Ft(and)g Fp(T)14 b(')2450 3367 y Fs(1)2538 3352 y Fo(2)43 b(O)2729 3315 y Fs(1)2773 3352 y Ft(\()p 2812 3271 V Fq(C)54 b Fo(n)28 b Ft([0)p Fp(;)17 b Ft(1]\).)69 b(W)-8 b(e)42 b(can)113 3530 y(therefore)35 b(conclude)f(that)1197 3635 y Fk(X)1192 3849 y Fn(Z)5 b Fm(M)1362 3730 y Fp(')p Ft(\()p Fp(z)t Ft(\))28 b(=)1688 3635 y Fk(X)1730 3849 y Fn(Z)1832 3730 y Ft([)p Fp(')1925 3745 y Fs(1)1992 3730 y Ft(+)2091 3635 y Fk(X)2116 3849 y Fm(M)2235 3730 y Ft(\()p Fp(T)14 b(')2411 3745 y Fs(1)2456 3730 y Ft(\)])27 b Fp(;)874 b Ft(\(5)p Fp(:)p Ft(2\))113 4038 y(where)42 b(the)f(de\014nition)g(of)1159 3964 y Fk(P)1264 4068 y Fn(Z)1345 4038 y Fp(')1410 4053 y Fs(1)1455 4038 y Ft(,)h Fp(')1590 4053 y Fs(1)1674 4038 y Fo(2)f(O)1863 4002 y Fs(1)1908 4038 y Ft(\()p 1947 3958 V Fq(C)52 b Fo(n)27 b Ft([1)p Fp(=)p Ft(2)p Fp(;)17 b Ft(2]\),)40 b(is)g(obtained)h(extending)g(the)113 4158 y(one)34 b(giv)m(en)f(for)h (functions)g(in)f Fo(O)1336 4122 y Fs(1)1381 4158 y Ft(\()p 1420 4077 V Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\).)113 4303 y Fl(R)-5 b(emark)36 b Ft(5.1)65 b(One)35 b(has)661 4538 y Fp(')726 4553 y Fs(1)770 4538 y Ft(\()p Fp(z)t Ft(\))28 b(=)g Fo(\000)p Fp(z)1176 4398 y Fk(\024)1229 4538 y Fp(')1311 4398 y Fk(\022)1396 4471 y Ft(1)p 1396 4515 51 4 v 1396 4607 a Fp(z)1481 4538 y Fo(\000)22 b Ft(1)1630 4398 y Fk(\023)1726 4538 y Fo(\000)g Fp(')p Ft(\()p Fo(\000)p Ft(1\))2095 4398 y Fk(\025)2170 4538 y Ft(+)h Fp(')2335 4497 y Fm(0)2363 4538 y Ft(\()p Fo(\000)p Ft(1\))f(+)g Fp(')p Ft(\()p Fp(z)27 b Fo(\000)c Ft(1\))k Fp(;)342 b Ft(\(5)p Fp(:)p Ft(3\))113 4793 y(from)33 b(whic)m(h)h(it)f(follo)m(ws)g(that)580 5028 y Fp(z)t(')695 5043 y Fs(1)757 4888 y Fk(\022)843 4961 y Ft(1)p 842 5005 V 842 5096 a Fp(z)905 4888 y Fk(\023)1006 5028 y Ft(=)28 b Fo(\000)p Ft([)p Fp(')p Ft(\()p Fp(z)f Fo(\000)c Ft(1\))e Fo(\000)i Fp(')p Ft(\()p Fo(\000)p Ft(1\)])f(+)g Fp(z)t(')2237 4987 y Fm(0)2266 5028 y Ft(\()p Fo(\000)p Ft(1\))g(+)h Fp(z)t(')2725 4888 y Fk(\022)2811 4961 y Ft(1)p 2811 5005 V 2811 5096 a Fp(z)2895 5028 y Fo(\000)g Ft(1)3045 4888 y Fk(\023)3163 5028 y Fp(;)113 5283 y Ft(and)878 5437 y Fp(')943 5452 y Fs(1)988 5437 y Ft(\()p Fp(z)t Ft(\))g(+)f Fp(z)t(')1353 5452 y Fs(1)1415 5297 y Fk(\022)1501 5370 y Ft(1)p 1501 5414 V 1501 5505 a Fp(z)1563 5297 y Fk(\023)1664 5437 y Ft(=)28 b(\(1)22 b(+)h Fp(z)t Ft(\)\()p Fp(')p Ft(\()p Fo(\000)p Ft(1\))f(+)h Fp(')2565 5396 y Fm(0)2593 5437 y Ft(\()p Fo(\000)p Ft(1\)\))k Fp(:)p eop %%Page: 32 32 32 31 bop 113 144 a Ft(32)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y Fr(5.1)39 b(The)e(dilogarithm)313 542 y(5.1.1)c Ft(Let)h(us)g(de\014ne)1396 711 y Fp(')1461 726 y Fs(0)1506 711 y Ft(\()p Fp(z)t Ft(\))28 b(=)g Fo(\000)1862 644 y Ft(1)p 1856 689 61 4 v 1856 780 a Fp(\031)1929 711 y Ft(Li)2019 726 y Fs(2)2080 571 y Fk(\022)2166 644 y Ft(1)p 2166 689 51 4 v 2166 780 a Fp(z)2228 571 y Fk(\023)2346 711 y Fp(;)1078 b Ft(\(5)p Fp(:)p Ft(4\))113 975 y(where)55 b(the)g(dilogarithm)c(is)j(tak)m(en)g(with)g(its)g(principal)g(branc)m (h)h(in)f Fq(C)62 b Fo(n)36 b Ft([1)p Fp(;)17 b Ft(+)p Fo(1)p Ft(])52 b(\(see)113 1095 y(App)s(endix)47 b(3)g(for)g(a)f(short) h(summary)e(of)i(the)g(prop)s(erties)g(of)g(the)g(dilogarithm)e(and)i ([O])113 1214 y(and)30 b(references)i(therein)e(for)g(more)e (details\).)42 b(The)30 b(function)g Fp(')2529 1229 y Fs(0)2604 1214 y Ft(b)s(elongs)f(to)g Fo(O)3157 1178 y Fs(1)3202 1214 y Ft(\()p 3241 1134 72 4 v Fq(C)40 b Fo(n)14 b Ft([0)p Fp(;)j Ft(1]\).)113 1334 y(It)33 b(is)f(real)h(on)g (the)g(real)f(axis)g(outside)i([0)p Fp(;)17 b Ft(1])31 b(and)i(its)f(only)g(singular)h(p)s(oin)m(ts)g(are)g(0)f(and)h(1.)44 b(It)113 1453 y(is)34 b(b)s(ounded)g(outside)g(of)g(an)m(y)f(neigh)m(b) s(orho)s(o)s(d)g(of)h(0)f(and)1043 1749 y Fo(=)p Fp(m)17 b(')1284 1764 y Fs(0)1329 1749 y Ft(\()p Fp(x)22 b Fo(\006)g Fp(i)p Ft(0\))28 b(=)g Fo(\006)17 b Ft(log)2057 1681 y(1)p 2053 1726 57 4 v 2053 1817 a Fp(x)2150 1749 y(;)72 b Ft(0)27 b Fp(<)h(x)g Fo(\024)g Ft(1)f Fp(;)725 b Ft(\(5)p Fp(:)p Ft(5\))113 2042 y(th)m(us)29 b(the)g(relation)e(with)h(the)h (real)f(Brjuno)g(function)h Fp(B)k Ft(is)28 b(clear)g(:)42 b Fp(B)5 b Ft(\()p Fp(x)p Ft(\))27 b(=)h([\(1)12 b Fo(\000)g Fp(T)i Ft(\))3329 2006 y Fm(\000)p Fs(1)3435 2042 y Fp(f)d Ft(]\()p Fp(x)p Ft(\))113 2162 y(with)33 b Fp(f)11 b Ft(\()p Fp(x)p Ft(\))27 b(=)667 2087 y Fk(P)772 2192 y Fn(n)p Fm(2)p Fg(Z)943 2162 y Fo(=)p Fp(m)17 b(')1184 2177 y Fs(0)1229 2162 y Ft(\()p Fp(x)22 b Ft(+)h Fp(i)p Ft(0)f Fo(\000)g Fp(n)p Ft(\).)313 2321 y Fr(5.1.2)33 b Ft(Let)h(us)g(no)m(w)f(consider)704 2700 y Fp(')769 2715 y Fs(1)842 2700 y Ft(=)947 2469 y Fk(2)947 2645 y(6)947 2708 y(4)1013 2700 y Fp(L)1081 2669 y Fk(\022)1168 2750 y Ft(1)99 b(1)1168 2869 y(0)g(1)1381 2669 y Fk(\023)1481 2700 y Ft(+)23 b Fp(L)1649 2669 y Fk(\022)1736 2750 y Ft(0)99 b(1)1736 2869 y(1)g(1)1948 2669 y Fk(\023)2027 2469 y(3)2027 2645 y(7)2027 2708 y(5)2110 2700 y Fp(')2175 2715 y Fs(0)2247 2700 y Fo(2)28 b(O)2423 2658 y Fs(1)2468 2700 y Ft(\()p 2507 2619 72 4 v Fq(C)48 b Fo(n)22 b Ft([1)p Fp(=)p Ft(2)p Fp(;)17 b Ft(2]\))25 b Fp(:)386 b Ft(\(5)p Fp(:)p Ft(6\))113 3133 y(Since)35 b Fp(')439 3148 y Fs(0)483 3133 y Ft(\()p Fo(\000)p Ft(1\))28 b(=)g Fp(\031)t(=)p Ft(12)k(and)i Fp(')1323 3097 y Fm(0)1323 3157 y Fs(0)1368 3133 y Ft(\()p Fo(\000)p Ft(1\))27 b(=)1722 3093 y Fs(1)p 1717 3110 50 4 v 1717 3167 a Fn(\031)1795 3133 y Ft(log)17 b(2)33 b(one)h(has)596 3456 y Fp(')661 3471 y Fs(1)705 3456 y Ft(\()p Fp(z)t Ft(\))28 b(=)984 3388 y(1)p 978 3433 61 4 v 978 3524 a Fp(\031)1067 3315 y Fk(\024)1120 3456 y Fp(z)t Ft(Li)1261 3471 y Fs(2)1322 3315 y Fk(\022)1493 3388 y Fp(z)p 1407 3433 223 4 v 1407 3524 a Ft(1)22 b Fo(\000)h Fp(z)1641 3315 y Fk(\023)1737 3456 y Fo(\000)f Ft(Li)1926 3471 y Fs(2)1988 3315 y Fk(\022)2159 3388 y Ft(1)p 2073 3433 V 2073 3524 a Fp(z)27 b Fo(\000)22 b Ft(1)2307 3315 y Fk(\023)q(\025)2455 3456 y Ft(+)2586 3388 y Fp(\031)p 2567 3433 100 4 v 2567 3524 a Ft(12)2678 3456 y Fp(z)27 b Ft(+)2868 3388 y(1)p 2863 3433 61 4 v 2863 3524 a Fp(\031)2952 3456 y Ft(log)17 b(2)p Fp(:)277 b Ft(\(5)p Fp(:)p Ft(7\))113 3769 y(The)36 b(function)g Fp(')777 3784 y Fs(1)856 3769 y Ft(is)f(real.)49 b(It)34 b(admits)g(as)h(unique)h(singularities)e(the)h(p)s(oin)m(ts)g(1)p Fp(=)p Ft(2,)g(1)g(and)g(2)113 3889 y(and)d(has)f(t)m(w)m(o)f(cuts)i (along)e(\(1)p Fp(=)p Ft(2)p Fp(;)17 b Ft(1\))29 b(and)i(\(1)p Fp(;)17 b Ft(2\).)41 b(It)31 b(can)g(b)s(e)g(con)m(tin)m(uously)g (extended)h(to)e(1)p Fp(=)p Ft(2)113 4008 y(and)k(2)f(and)h(it)e(is)i (b)s(ounded)g(outside)g(an)m(y)f(neigh)m(b)s(orho)s(o)s(d)h(of)g(1.)43 b(Moreo)m(v)m(er)836 4324 y Fo(=)p Fp(m)17 b(')1077 4339 y Fs(1)1122 4324 y Ft(\()p Fp(x)22 b Fo(\006)g Fp(i)p Ft(0\))28 b(=)1595 4184 y Fk(\032)1686 4247 y Fo(\006)p Fp(x)17 b Ft(log)2046 4208 y Fn(x)p 1995 4224 148 4 v 1995 4281 a Fs(1)p Fm(\000)p Fn(x)2253 4247 y Ft(if)34 b(1)p Fp(=)p Ft(2)27 b Fo(\024)h Fp(x)g(<)g Ft(1)f Fp(;)1686 4384 y Fo(\006)17 b Ft(log)1992 4344 y Fs(1)p 1938 4361 V 1938 4418 a Fn(x)p Fm(\000)p Fs(1)2253 4384 y Ft(if)34 b(1)27 b Fp(<)h(x)g Fo(\024)g Ft(2)g Fp(:)3452 4324 y Ft(\(5)p Fp(:)p Ft(8\))113 4645 y(Note)33 b(that)g(if)g(1)p Fp(=)p Ft(2)27 b Fo(\024)i Fp(x)e(<)h Ft(1)718 4946 y Fp(x)17 b Ft(log)1035 4879 y Fp(x)p 949 4923 229 4 v 949 5014 a Ft(1)22 b Fo(\000)h Fp(x)1217 4946 y Ft(=)28 b(log)1569 4879 y(1)p 1479 4923 V 1479 5014 a(1)22 b Fo(\000)h Fp(x)1742 4946 y Ft(+)g Fp(x)17 b Ft(log)g Fp(x)22 b Ft(+)g(\(1)g Fo(\000)h Fp(x)p Ft(\))17 b(log\(1)k Fo(\000)i Fp(x)p Ft(\))k Fp(:)113 5236 y Ft(One)35 b(also)e(has)901 5405 y Fp(')966 5420 y Fs(1)1011 5405 y Ft(\()p Fp(z)t Ft(\))23 b(+)f Fp(z)t(')1376 5420 y Fs(1)1438 5265 y Fk(\022)1524 5338 y Ft(1)p 1524 5382 51 4 v 1524 5473 a Fp(z)1586 5265 y Fk(\023)1687 5405 y Ft(=)28 b(\(1)22 b(+)h Fp(z)t Ft(\))2109 5265 y Fk(\022)2214 5338 y Fp(\031)p 2194 5382 100 4 v 2194 5473 a Ft(12)2328 5405 y(+)2445 5338 y(1)p 2439 5382 61 4 v 2439 5473 a Fp(\031)2528 5405 y Ft(log)18 b(2)2724 5265 y Fk(\023)2841 5405 y Fp(:)583 b Ft(\(5)p Fp(:)p Ft(9\))p eop %%Page: 33 33 33 32 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(33)113 383 y Fr(Lemma)38 b(5.2)58 b Fe(The)34 b(function)f Fp(')1382 398 y Fs(1)1427 383 y Ft(\()p Fp(z)t Ft(\))22 b(+)g Fp(i)17 b Ft(log)q(\(1)k Fo(\000)h Fp(z)t Ft(\))33 b Fe(is)g(con)m(tin)m(uous)i(on)p 2924 299 144 4 v 33 w Fq(H)2995 354 y Fs(+)3100 383 y Fe(and)f(its)f(v)-6 b(alue)113 503 y(at)33 b Ft(1)g Fe(is)435 463 y Fs(1)p 430 480 50 4 v 430 537 a Fn(\031)508 503 y Ft(log)17 b(2)22 b(+)837 463 y Fs(7)p Fn(\031)p 837 480 89 4 v 842 537 a Fs(12)938 503 y Fe(.)113 725 y Fl(Pr)-5 b(o)g(of.)34 b Ft(Applying)f(\(A3.7\))f(to)h(\(5.7\))f(t)m(wice)h(one)h(gets)581 1035 y Fp(')646 1050 y Fs(1)691 1035 y Ft(\()p Fp(z)t Ft(\))28 b(=)969 967 y(1)p 964 1012 61 4 v 964 1103 a Fp(\031)1053 1035 y Ft(log)17 b(2)22 b Fo(\000)1381 967 y Fp(\031)p 1381 1012 V 1386 1103 a Ft(6)1476 1035 y(+)1587 967 y Fp(\031)p 1587 1012 V 1592 1103 a Ft(4)1660 1035 y Fp(z)27 b Ft(+)1849 967 y(1)p 1844 1012 V 1844 1103 a Fp(\031)1933 894 y Fk(\024)1986 1035 y Ft(Li)2076 1050 y Fs(2)2120 1035 y Ft(\()p Fp(z)g Fo(\000)c Ft(1\))e Fo(\000)i Fp(z)t Ft(Li)2683 1050 y Fs(2)2744 894 y Fk(\022)2829 967 y Ft(1)f Fo(\000)h Fp(z)p 2829 1012 223 4 v 2915 1103 a(z)3063 894 y Fk(\023)q(\025)841 1314 y Ft(+)983 1246 y(1)p 953 1291 111 4 v 953 1382 a(2)p Fp(\031)1091 1173 y Fk(\024)1144 1314 y Ft(log)1273 1269 y Fs(2)1317 1314 y Ft(\(1)f Fo(\000)h Fp(z)t Ft(\))f Fo(\000)h Fp(z)e Ft(log)1935 1269 y Fs(2)2008 1246 y Ft(1)h Fo(\000)h Fp(z)p 2008 1291 223 4 v 2055 1382 a Fo(\000)p Fp(z)2242 1173 y Fk(\025)113 1633 y Ft(In)32 b(this)g(expression)h(the)f (function)g(Li)1556 1648 y Fs(2)1601 1633 y Ft(\()p Fp(z)23 b Fo(\000)d Ft(1\))e Fo(\000)i Fp(z)t Ft(Li)2151 1648 y Fs(2)2212 1552 y Fk(\000)2270 1593 y Fs(1)p Fm(\000)p Fn(z)p 2270 1610 143 4 v 2321 1667 a(z)2424 1552 y Fk(\001)2502 1633 y Ft(is)31 b(regular)h(and)g(v)-6 b(anishing)32 b(at)113 1752 y Fp(z)h Ft(=)28 b(1.)44 b(Moreo)m(v)m(er)289 2024 y(log)418 1979 y Fs(2)462 2024 y Ft(\(1)22 b Fo(\000)h Fp(z)t Ft(\))f Fo(\000)h Fp(z)e Ft(log)1080 1979 y Fs(2)1153 1956 y Ft(1)h Fo(\000)h Fp(z)p 1153 2001 223 4 v 1200 2092 a Fo(\000)p Fp(z)1415 2024 y Ft(=)28 b(log)1649 1979 y Fs(2)1694 2024 y Ft(\(1)21 b Fo(\000)i Fp(z)t Ft(\))g Fo(\000)f Ft(log)2244 1979 y Fs(2)2317 1956 y Ft(1)g Fo(\000)h Fp(z)p 2317 2001 V 2364 2092 a Fo(\000)p Fp(z)2574 2024 y Ft(+)f(\(1)g Fo(\000)g Fp(z)t Ft(\))17 b(log)3118 1979 y Fs(2)3191 1956 y Ft(1)22 b Fo(\000)h Fp(z)p 3191 2001 V 3239 2092 a Fo(\000)p Fp(z)3453 2024 y(;)113 2329 y Ft(where)34 b(\(1)22 b Fo(\000)h Fp(z)t Ft(\))17 b(log)847 2285 y Fs(2)920 2290 y(1)p Fm(\000)p Fn(z)p 920 2307 143 4 v 940 2364 a Fm(\000)p Fn(z)1108 2329 y Ft(v)-6 b(anishes)34 b(at)f Fp(z)f Ft(=)c(1,)33 b(and)555 2643 y(log)684 2599 y Fs(2)728 2643 y Ft(\(1)22 b Fo(\000)h Fp(z)t Ft(\))f Fo(\000)h Ft(log)1279 2599 y Fs(2)1352 2576 y Ft(1)f Fo(\000)g Fp(z)p 1352 2621 223 4 v 1399 2712 a Fo(\000)p Fp(z)1614 2643 y Ft(=)28 b Fo(\000)17 b Ft(log)1942 2599 y Fs(2)1986 2643 y Ft(\()p Fo(\000)p Fp(z)t Ft(\))23 b(+)g(2)17 b(log\()p Fo(\000)p Fp(z)t Ft(\))g(log)q(\(1)22 b Fo(\000)g Fp(z)t Ft(\))28 b Fp(:)113 2952 y Ft(In)34 b(a)f(neigh)m(b)s(orho)s(o)s(d)h(of)f(1)g (in)p 1259 2868 144 4 v 33 w Fq(H)1331 2923 y Fs(+)1436 2952 y Ft(one)h(has)g(log\()p Fo(\000)p Fp(z)t Ft(\))23 b(+)g Fp(i\031)31 b Ft(=)d(O)17 b(\()p Fo(j)p Fp(z)27 b Fo(\000)22 b Ft(1)p Fo(j)p Ft(\))33 b(th)m(us)299 3261 y(log)427 3216 y Fs(2)472 3261 y Ft(\(1)22 b Fo(\000)g Fp(z)t Ft(\))h Fo(\000)g Ft(log)1022 3216 y Fs(2)1096 3193 y Ft(1)f Fo(\000)g Fp(z)p 1096 3238 223 4 v 1143 3329 a Fo(\000)p Fp(z)1358 3261 y Ft(=)28 b Fp(\031)1524 3220 y Fs(2)1590 3261 y Fo(\000)23 b Ft(2)p Fp(i\031)d Ft(log\(1)i Fo(\000)h Fp(z)t Ft(\))f(+)h(O)2512 3120 y Fk(\022)2585 3261 y Fo(j)p Fp(z)j Fo(\000)d Ft(1)p Fo(j)17 b Ft(log)3150 3193 y(1)p 3036 3238 278 4 v 3036 3329 a Fo(j)p Fp(z)27 b Fo(\000)c Ft(1)p Fo(j)3326 3120 y Fk(\023)3444 3261 y Fp(:)3579 3562 y Fd(\003)313 3752 y Ft(This)33 b(Lemma)f(leads)h(to)g(the)h(follo)m(wing)e(imp)s(ortan)m (t)113 3942 y Fr(Corollary)37 b(5.3)57 b Fe(The)32 b(real)f(part)g(of)h Fp(')1610 3957 y Fs(1)1686 3942 y Fe(is)f(b)s(ounded)i(in)p 2303 3861 72 4 v 31 w Fq(C)44 b Fo(n)18 b Ft([1)p Fp(=)p Ft(2)p Fp(;)f Ft(2])p Fe(.)41 b(It)31 b(has)h(an)f(extension)113 4061 y(to)i(a)g(con)m(tin)m(uous)i(function)f(on)p 1345 3981 V 33 w Fq(C)49 b Fo(n)21 b(f)p Ft(1)p Fo(g)33 b Fe(and)1150 4357 y Ft(lim)1108 4427 y Fn(x)p Fm(!)p Fs(1)1272 4407 y Ff(\006)1348 4357 y Fo(<)p Fp(e)17 b(')1548 4372 y Fs(1)1592 4357 y Ft(\()p Fp(x)p Ft(\))28 b(=)1877 4290 y(1)p 1872 4334 61 4 v 1872 4426 a Fp(\031)1961 4357 y Ft(log)17 b(2)22 b(+)2309 4290 y Fp(\031)p 2289 4334 100 4 v 2289 4426 a Ft(12)2423 4357 y Fo(\007)2535 4290 y Fp(\031)p 2535 4334 61 4 v 2540 4426 a Ft(2)2635 4357 y Fp(:)739 b Ft(\(5)p Fp(:)p Ft(10\))313 4683 y(This)42 b(Corollary)e(is)i(the)h(motiv)-6 b(ation)39 b(for)k(using)f Fp(')2293 4698 y Fs(1)2380 4683 y Ft(instead)h(of)f Fp(')2920 4698 y Fs(0)3007 4683 y Ft(as)g(the)g(starting)113 4802 y(p)s(oin)m(t)d(of)f(the)h(construction)g(of)g(the)g(complex)e(Brjuno)i (function.)61 b(Equation)38 b(\(5.2\))f(sho)m(ws)113 4922 y(that)c(this)h(leads)f(to)g(the)g(same)g(result.)p eop %%Page: 34 34 34 33 bop 113 144 a Ft(34)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y Fr(5.2)39 b(A)f(natural)e(compacti\014cation)g(of) j Fq(H)1881 347 y Fs(+)313 619 y Ft(By)33 b(Lemma)f(5.2)h(ab)s(o)m(v)m (e,)g Fo(<)p Fp(e)17 b(')1497 634 y Fs(1)1576 619 y Ft(extends)34 b(con)m(tin)m(uously)h(to)p 2641 535 144 4 v 33 w Fq(H)2713 590 y Fs(+)2807 619 y Fo(n)23 b(f)p Ft(1)p Fo(g)33 b Ft(with)g(limits)f(at)113 738 y(1)e(along)g(ra)m(ys.)43 b(This)30 b(means)g(that)g Fo(<)p Fp(e)17 b(')1644 753 y Fs(1)1719 738 y Ft(extends)31 b(con)m(tin)m(uously)g(to)f(the)h (compacti\014cation)113 858 y(of)h Fq(H)296 822 y Fs(+)399 858 y Ft(obtained)g(from)p 1040 774 V 30 w Fq(H)1111 829 y Fs(+)1215 858 y Ft(b)m(y)f(blo)m(wing)g(out)f(1)h(in)m(to)g(a)g (semi{circle)f(\(corresp)s(onding)i(to)f(all)113 977 y(ra)m(ys)i(in)h Fq(H)512 941 y Fs(+)617 977 y Ft(whic)m(h)g(end)g(in)g (1\).)43 b(If)34 b(w)m(e)f(w)m(an)m(t)h(to)e(obtain)h(a)g(similar)e (result)j(for)f(the)h(complex)113 1097 y(Brjuno)g(function)837 1022 y Fk(P)943 1127 y Fn(Z)5 b Fm(M)1117 1097 y Fp(')1182 1112 y Fs(0)1260 1097 y Ft(w)m(e)34 b(ha)m(v)m(e)f(to)g(do)g(the)h (same)f(thing)g(at)g(ev)m(ery)g(p)s(oin)m(t)g(of)p 3336 1016 78 4 v 33 w Fq(Q)18 b Ft(.)313 1256 y Fr(5.2.1)33 b Ft(W)-8 b(e)34 b(will)e(consider)1028 1485 y(^)1014 1511 y Fq(H)1085 1469 y Fs(+)1185 1511 y Ft(=)c Fq(H)1362 1469 y Fs(+)1456 1511 y Fo(t)23 b Ft(\()p Fq(R)32 b Fo(n)22 b Fq(Q)12 b Ft(\))28 b Fo(t)1976 1400 y Fk(\020)p 2036 1430 V 2036 1511 a Fq(Q)39 b Fo(\002)2235 1400 y Fk(h)2282 1511 y Fo(\000)2371 1443 y Fp(\031)p 2371 1488 61 4 v 2376 1579 a Ft(2)2444 1511 y Fp(;)17 b Ft(+)2578 1443 y Fp(\031)p 2578 1488 V 2583 1579 a Ft(2)2650 1400 y Fk(i\021)3452 1511 y Ft(\(5)p Fp(:)p Ft(6\))113 1812 y(\(where)p 460 1732 78 4 v 54 w Fq(Q)78 b Ft(=)62 b Fq(Q)53 b Fo([)35 b(f1g)p Ft(\))53 b(equipp)s(ed)h(with)e(the)i(top)s (ology)d(de\014ned)56 b(b)m(y)d(the)g(follo)m(wing)113 1942 y(fundamen)m(tal)34 b(system)e(of)i(neigh)m(b)s(orho)s(o)s(ds)g (at)f(an)m(y)g(p)s(oin)m(t)g Fp(z)f Fo(2)2554 1916 y Ft(^)2540 1942 y Fq(H)2612 1906 y Fs(+)2717 1942 y Ft(:)174 2067 y(a\))50 b(if)31 b Fp(z)448 2082 y Fs(0)521 2067 y Fo(2)d Fq(H)686 2030 y Fs(+)790 2067 y Ft(a)i(fundamen)m(tal)i (system)e(of)i(neigh)m(b)s(orho)s(o)s(ds)g(is)f(giv)m(en)g(b)m(y)g Fo(fj)p Fp(z)23 b Fo(\000)18 b Fp(z)3327 2082 y Fs(0)3372 2067 y Fo(j)28 b Fp(<)g(")p Fo(g)p Ft(,)313 2186 y(0)f Fp(<)h(")g(<)h Fo(=)p Fp(m)17 b(z)897 2201 y Fs(0)962 2186 y Ft(;)169 2311 y(b\))50 b(if)40 b Fp(\013)475 2326 y Fs(0)560 2311 y Fo(2)g Fq(R)f Fo(n)26 b Fq(Q)12 b Ft(,)48 b(a)41 b(fundamen)m(tal)g(system)f(of)h(neigh)m(b)s(orho)s(o)s(ds)g(is) g(giv)m(en)f(b)m(y)h(the)g(sets)313 2430 y(\()p Fp(")28 b(>)g Ft(0\))852 2565 y Fp(V)910 2580 y Fn(")953 2565 y Ft(\()p Fp(\013)1056 2580 y Fs(0)1100 2565 y Ft(\))f(=)h Fo(f)p Fp(z)33 b Fo(2)28 b Fq(H)1565 2524 y Fs(+)1665 2565 y Fp(;)44 b Fo(j)p Fp(z)26 b Fo(\000)d Fp(\013)2001 2580 y Fs(0)2045 2565 y Fo(j)28 b Fp(<)g(")p Fo(g)1161 2715 y([)22 b(f)p Fp(\013)28 b Fo(2)g Fq(R)33 b Fo(n)22 b Fq(Q)45 b Fp(;)f Fo(j)p Fp(\013)22 b Fo(\000)h Fp(\013)2106 2730 y Fs(0)2150 2715 y Fo(j)k Fp(<)i(")p Fo(g)1161 2904 y([)1249 2793 y Fk(n)1316 2904 y Ft(\()p Fp(\013;)17 b(\022)s Ft(\))26 b Fo(2)i Fq(Q)39 b Fo(\002)1872 2793 y Fk(h)1919 2904 y Fo(\000)2008 2836 y Fp(\031)p 2008 2881 61 4 v 2013 2972 a Ft(2)2081 2904 y Fp(;)17 b Ft(+)2215 2836 y Fp(\031)p 2215 2881 V 2220 2972 a Ft(2)2287 2793 y Fk(i)2378 2904 y Fp(;)44 b Fo(j)p Fp(\013)22 b Fo(\000)h Fp(\013)2728 2919 y Fs(0)2772 2904 y Fo(j)k Fp(<)i(")2979 2793 y Fk(o)3090 2904 y Ft(;)180 3147 y(c\))50 b(if)62 b Fp(\013)497 3162 y Fs(0)619 3147 y Fo(2)77 b Fq(Q)j Ft(a)62 b(fundamen)m(tal)h(system)f(of)h(neigh)m(b)s(orho)s(o)s(ds)h (of)f(\()p Fp(\013)3063 3162 y Fs(0)3107 3147 y Fp(;)17 b(\031)t(=)p Ft(2\))60 b(\(resp.)313 3266 y(\()p Fp(\013)416 3281 y Fs(0)460 3266 y Fp(;)17 b Fo(\000)p Fp(\031)t(=)p Ft(2\)\))31 b(is)i(giv)m(en)h(b)m(y)f(\(0)27 b Fp(<)h(")g(<)h(\031)t Ft(\))620 3521 y Fp(V)678 3536 y Fn(")720 3521 y Ft(\()p Fp(\013)823 3536 y Fs(0)868 3521 y Fp(;)17 b(\031)t(=)p Ft(2\))25 b(=)k Fo(f)p Fp(z)j Fo(2)c Fq(H)1537 3480 y Fs(+)1637 3521 y Fp(;)44 b Fo(j)p Fp(z)27 b Fo(\000)22 b Fp(\013)1973 3536 y Fs(0)2018 3521 y Fo(j)27 b Fp(<)h(")g(;)44 b Ft(0)28 b Fp(<)g Ft(arg)q(\()p Fp(z)e Fo(\000)d Fp(\013)2922 3536 y Fs(0)2967 3521 y Ft(\))k Fp(<)h(")p Fo(g)1133 3670 y([)22 b(f)p Fp(\013)28 b Fo(2)g Fq(R)33 b Fo(n)22 b Fq(Q)45 b Fp(;)f Ft(0)28 b Fp(<)g(\013)22 b Fo(\000)h Fp(\013)2233 3685 y Fs(0)2305 3670 y Fp(<)28 b(")p Fo(g)1133 3859 y([)1221 3749 y Fk(n)1288 3859 y Ft(\()p Fp(\013;)17 b(\022)s Ft(\))26 b Fo(2)i Fq(Q)40 b Fo(\002)1844 3749 y Fk(h)1891 3859 y Fo(\000)1980 3792 y Fp(\031)p 1980 3836 V 1985 3928 a Ft(2)2053 3859 y Fp(;)17 b Ft(+)2187 3792 y Fp(\031)p 2187 3836 V 2192 3928 a Ft(2)2259 3749 y Fk(i)2350 3859 y Fp(;)44 b Ft(0)28 b Fp(<)g(\013)22 b Fo(\000)h Fp(\013)2855 3874 y Fs(0)2927 3859 y Fp(<)28 b(")3078 3749 y Fk(o)1133 4052 y Fo([)22 b(f)p Ft(\()p Fp(\013)1374 4067 y Fs(0)1418 4052 y Fp(;)17 b(\022)s Ft(\))27 b Fp(;)44 b(\031)t(=)p Ft(2)21 b Fo(\000)h Fp(")28 b(<)h(\022)h Fo(\024)e Fp(\031)t(=)p Ft(2)p Fo(g)43 b Fp(;)542 4202 y(V)600 4217 y Fn(")643 4202 y Ft(\()p Fp(\013)746 4217 y Fs(0)790 4202 y Fp(;)17 b Fo(\000)p Fp(\031)t(=)p Ft(2\))26 b(=)j Fo(f)p Fp(z)j Fo(2)c Fq(H)1537 4161 y Fs(+)1637 4202 y Fp(;)44 b Fo(j)p Fp(z)27 b Fo(\000)22 b Fp(\013)1973 4217 y Fs(0)2018 4202 y Fo(j)27 b Fp(<)h(")g(;)44 b(\031)31 b(>)d Ft(arg)q(\()p Fp(z)f Fo(\000)c Fp(\013)2933 4217 y Fs(0)2977 4202 y Ft(\))28 b Fp(>)g(\031)d Fo(\000)e Fp(")p Fo(g)1133 4351 y([)f(f)p Fp(\013)28 b Fo(2)g Fq(R)33 b Fo(n)22 b Fq(Q)45 b Fp(;)f Ft(0)28 b Fp(<)g(\013)2047 4366 y Fs(0)2113 4351 y Fo(\000)23 b Fp(\013)28 b(<)g(")p Fo(g)1133 4540 y([)1221 4430 y Fk(n)1288 4540 y Ft(\()p Fp(\013;)17 b(\022)s Ft(\))26 b Fo(2)i Fq(Q)40 b Fo(\002)1844 4430 y Fk(h)1891 4540 y Fo(\000)1980 4473 y Fp(\031)p 1980 4518 V 1985 4609 a Ft(2)2053 4540 y Fp(;)17 b Ft(+)2187 4473 y Fp(\031)p 2187 4518 V 2192 4609 a Ft(2)2259 4430 y Fk(i)2350 4540 y Fp(;)44 b Ft(0)28 b Fp(<)g(\013)2669 4555 y Fs(0)2736 4540 y Fo(\000)22 b Fp(\013)28 b(<)g(")3078 4430 y Fk(o)1133 4733 y Fo([)22 b(f)p Ft(\()p Fp(\013)1374 4748 y Fs(0)1418 4733 y Fp(;)17 b(\022)s Ft(\))27 b Fp(;)44 b Fo(\000)p Fp(\031)t(=)p Ft(2)27 b Fo(\024)h Fp(\022)i(<)e Fo(\000)p Fp(\031)t(=)p Ft(2)22 b(+)g Fp(")p Fo(g)45 b Ft(;)169 4993 y(d\))50 b(If)c Fp(\013)489 5008 y Fs(0)583 4993 y Fo(2)51 b Fq(Q)11 b Ft(,)55 b Fo(\000)p Fp(\031)t(=)p Ft(2)49 b Fp(<)i(\022)1316 5008 y Fs(0)1410 4993 y Fp(<)f(\031)t(=)p Ft(2)45 b(a)h(fundamen)m(tal)h(system)f(of)g(neigh)m(b)s(orho)s(o)s(ds) h(of)313 5112 y(\()p Fp(\013)416 5127 y Fs(0)460 5112 y Fp(;)17 b(\022)552 5127 y Fs(0)596 5112 y Ft(\))33 b(is)g(giv)m(en)g(b)m(y)g(\(0)27 b Fp(<)i(")f(<)g(\031)t(=)p Ft(2)21 b Fo(\000)i(j)p Fp(\022)1924 5127 y Fs(0)1968 5112 y Fo(j)p Ft(\))359 5366 y Fp(V)417 5381 y Fn(")460 5366 y Ft(\()p Fp(\013)563 5381 y Fs(0)607 5366 y Fp(;)17 b(\022)699 5381 y Fs(0)743 5366 y Ft(\))27 b(=)h Fo(f)p Fp(z)33 b Fo(2)28 b Fq(H)1208 5325 y Fs(+)1308 5366 y Fp(;)44 b Fo(j)p Fp(z)26 b Fo(\000)d Fp(\013)1644 5381 y Fs(0)1689 5366 y Fo(j)k Fp(<)h(")g(;)44 b(\022)2042 5381 y Fs(0)2109 5366 y Fo(\000)22 b Fp(")28 b(<)h(\031)t(=)p Ft(2)21 b Fo(\000)h Ft(arg)q(\()p Fp(z)27 b Fo(\000)c Fp(\013)3085 5381 y Fs(0)3129 5366 y Ft(\))k Fp(<)i(\022)3348 5381 y Fs(0)3414 5366 y Ft(+)23 b Fp(")p Fo(g)804 5556 y([)892 5445 y Fk(n)959 5556 y Ft(\()p Fp(\013)1062 5571 y Fs(0)1106 5556 y Fp(;)17 b(\022)s Ft(\))26 b Fo(2)i Fq(Q)40 b Fo(\002)1559 5445 y Fk(h)1606 5556 y Fo(\000)1695 5488 y Fp(\031)p 1695 5533 V 1700 5624 a Ft(2)1768 5556 y Fp(;)17 b Ft(+)1902 5488 y Fp(\031)p 1902 5533 V 1907 5624 a Ft(2)1974 5445 y Fk(i)2066 5556 y Fp(;)44 b(\022)2185 5571 y Fs(0)2251 5556 y Fo(\000)23 b Fp(")28 b(<)g(\022)i(<)e(\022)2759 5571 y Fs(0)2826 5556 y Ft(+)23 b Fp(")2972 5445 y Fk(o)3083 5556 y Ft(;)p eop %%Page: 35 35 35 34 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(35)180 383 y(e\))50 b(A)24 b(fundamen)m(tal)g(system)g(of)g(neigh)m (b)s(orho)s(o)s(ds)h(of)f(\()p Fo(1)p Fp(;)17 b(\031)t(=)p Ft(2\))22 b(\(resp.)42 b(\()p Fo(1)p Fp(;)17 b Fo(\000)p Fp(\031)t(=)p Ft(2\))22 b(is)i(giv)m(en)313 503 y(b)m(y)33 b(\(0)27 b Fp(<)h(")h(<)f(\031)t Ft(\))831 745 y Fp(V)889 760 y Fn(")932 745 y Ft(\()p Fo(1)p Fp(;)17 b(\031)t(=)p Ft(2\))25 b(=)j Fo(f)p Fp(z)k Fo(2)d Fq(H)1740 703 y Fs(+)1840 745 y Fp(;)44 b Fo(j)p Fp(z)t Fo(j)28 b Fp(>)g(")2197 703 y Fm(\000)p Fs(1)2332 745 y Fp(;)44 b(\031)31 b(>)d Ft(arg)18 b Fp(z)32 b(>)c(\031)e Fo(\000)c Fp(")p Fo(g)1336 905 y([)g(f)p Fp(\013)28 b Fo(2)g Fq(R)33 b Fo(n)22 b Fq(Q)45 b Fp(;)f(\013)28 b(<)g Fo(\000)p Fp(")2323 864 y Fm(\000)p Fs(1)2431 905 y Fo(g)1336 1095 y([)1424 984 y Fk(n)1491 1095 y Ft(\()p Fp(\013;)17 b(\022)s Ft(\))26 b Fo(2)i Fq(Q)39 b Fo(\002)2047 984 y Fk(h)2094 1095 y Fo(\000)2183 1027 y Fp(\031)p 2183 1072 61 4 v 2188 1163 a Ft(2)2256 1095 y Fp(;)17 b Ft(+)2390 1027 y Fp(\031)p 2390 1072 V 2395 1163 a Ft(2)2462 984 y Fk(i)2553 1095 y Fp(;)44 b(\013)28 b(<)g Fo(\000)p Fp(")2945 1053 y Fm(\000)p Fs(1)3053 984 y Fk(o)1336 1287 y Fo([)22 b(f)p Ft(\()p Fo(1)p Fp(;)17 b(\022)s Ft(\))26 b Fp(;)43 b(\031)t(=)p Ft(2)27 b Fo(\025)h Fp(\022)j(>)d(\031)t(=)p Ft(2)21 b Fo(\000)h Fp(")p Fo(g)45 b Fp(;)754 1448 y(V)812 1463 y Fn(")855 1448 y Ft(\()p Fo(1)p Fp(;)17 b Fo(\000)p Fp(\031)t(=)p Ft(2\))25 b(=)j Fo(f)p Fp(z)k Fo(2)d Fq(H)1740 1407 y Fs(+)1840 1448 y Fp(;)44 b Fo(j)p Fp(z)t Fo(j)28 b Fp(>)g(")2197 1407 y Fm(\000)p Fs(1)2332 1448 y Fp(;)44 b Ft(0)27 b Fp(<)h Ft(arg)18 b Fp(z)32 b(<)d(")p Fo(g)1336 1609 y([)22 b(f)p Fp(\013)28 b Fo(2)g Fq(R)33 b Fo(n)22 b Fq(Q)45 b Fp(;)f(\013)28 b(>)g(")2246 1568 y Fm(\000)p Fs(1)2353 1609 y Fo(g)1336 1798 y([)1424 1688 y Fk(n)1491 1798 y Ft(\()p Fp(\013;)17 b(\022)s Ft(\))26 b Fo(2)i Fq(Q)39 b Fo(\002)2047 1688 y Fk(h)2094 1798 y Fo(\000)2183 1731 y Fp(\031)p 2183 1775 V 2188 1867 a Ft(2)2256 1798 y Fp(;)17 b Ft(+)2390 1731 y Fp(\031)p 2390 1775 V 2395 1867 a Ft(2)2462 1688 y Fk(i)2553 1798 y Fp(;)44 b(\013)28 b(>)g(")2868 1757 y Fm(\000)p Fs(1)2975 1688 y Fk(o)1336 1991 y Fo([)22 b(f)p Ft(\()p Fo(1)p Fp(;)17 b(\022)s Ft(\))26 b Fp(;)43 b Fo(\000)p Fp(\031)t(=)p Ft(2)28 b Fo(\024)g Fp(\022)i(<)e Fo(\000)p Fp(\031)t(=)p Ft(2)22 b(+)g Fp(")p Fo(g)45 b Ft(;)186 2232 y(f)8 b(\))50 b(If)44 b Fo(\000)p Fp(\031)t(=)p Ft(2)g Fp(<)i(\022)875 2247 y Fs(0)964 2232 y Fp(<)f(\031)t(=)p Ft(2)e(a)g(fundamen)m(tal)h(system) f(of)h(neigh)m(b)s(orho)s(o)s(ds)g(of)g(\()p Fo(1)p Fp(;)17 b(\022)3464 2247 y Fs(0)3507 2232 y Ft(\))44 b(is)313 2352 y(giv)m(en)33 b(b)m(y)g(\(0)27 b Fp(<)i(")f(<)g(\031)t(=)p Ft(2)21 b Fo(\000)i(j)p Fp(\022)1469 2367 y Fs(0)1513 2352 y Fo(j)p Ft(\))571 2594 y Fp(V)629 2609 y Fn(")671 2594 y Ft(\()p Fo(1)p Fp(;)17 b(\022)902 2609 y Fs(0)946 2594 y Ft(\))27 b(=)h Fo(f)p Fp(z)k Fo(2)c Fq(H)1411 2552 y Fs(+)1511 2594 y Fp(;)44 b Fo(j)p Fp(z)t Fo(j)27 b Fp(>)i(")1868 2552 y Fm(\000)p Fs(1)2003 2594 y Fp(;)44 b(\022)2122 2609 y Fs(0)2188 2594 y Fo(\000)23 b Fp(")28 b(<)g Ft(arg)18 b Fp(z)27 b Fo(\000)22 b Fp(\031)t(=)p Ft(2)27 b Fp(<)h(\022)3136 2609 y Fs(0)3203 2594 y Ft(+)22 b Fp(")p Fo(g)1006 2743 y([)h(f)p Ft(\()p Fo(1)p Fp(;)17 b(\022)s Ft(\))25 b Fp(;)44 b Fo(j)p Fp(\022)25 b Fo(\000)d Fp(\022)1761 2758 y Fs(0)1806 2743 y Fo(j)27 b Fp(<)h(")p Fo(g)45 b Fp(:)313 3009 y Fr(5.2.2)51 b Ft(One)h(can)g(c)m(hec)m(k)g (that)f(the)g(axioms)e(for)i(a)g(system)g(of)g(neigh)m(b)s(orho)s(o)s (ds)h(are)113 3128 y(v)m(eri\014ed.)83 b(The)46 b(follo)m(wing)f(is)h (the)g(only)f(non)i(trivial)d(prop)s(ert)m(y)i(:)69 b(if)45 b Fp(\020)56 b Fo(2)3082 3103 y Ft(^)3068 3128 y Fq(H)3140 3092 y Fs(+)3257 3128 y Ft(and)47 b Fp(V)68 b Ft(is)113 3248 y(a)41 b(neigh)m(b)s(orho)s(o)s(d)h(of)g Fp(\020)48 b Ft(then)42 b(there)g(exists)f(a)g(neigh)m(b)s(orho)s(o)s(d)h Fp(W)55 b Ft(of)41 b Fp(\020)49 b Ft(suc)m(h)42 b(that)f Fp(V)64 b Ft(is)41 b(a)113 3367 y(neigh)m(b)s(orho)s(o)s(d)d(of)f(eac)m (h)h(p)s(oin)m(t)f(of)g Fp(W)14 b Ft(.)55 b(This)37 b(m)m(ust)g(b)s(e)g (c)m(hec)m(k)m(ed)i(directly)e(for)g(eac)m(h)h(of)f(the)113 3487 y(ab)s(o)m(v)m(e)d(listed)f(cases.)313 3637 y Fr(5.2.3)38 b Ft(It)f(is)h(clear)g(that)f(the)h(top)s(ology)e(of)1988 3612 y(^)1974 3637 y Fq(H)2046 3601 y Fs(+)2156 3637 y Ft(induces)j(on)f Fq(H)2729 3601 y Fs(+)2839 3637 y Ft(the)g(usual)g(top)s(ology)113 3757 y(and)c(that)f Fq(H)594 3721 y Fs(+)700 3757 y Ft(is)g(an)g(op)s(en)h(dense)h(subset)f (of)1884 3731 y(^)1870 3757 y Fq(H)1941 3721 y Fs(+)2013 3757 y Ft(.)313 3907 y Fr(5.2.4)f Ft(The)h(space)1065 3881 y(^)1051 3907 y Fq(H)1123 3871 y Fs(+)1228 3907 y Ft(is)f(compact)f(and)i(Hausdor\013.)44 b(More)33 b(precisely)h (there)f(exists)g(a)113 4027 y(homeomorphism)j(of)p 984 3946 72 4 v 38 w Fq(D)63 b Ft(on)m(to)1337 4001 y(^)1323 4027 y Fq(H)1394 3991 y Fs(+)1504 4027 y Ft(whose)39 b(restriction)e(to)h Fq(D)63 b Ft(is)37 b(a)h(homeomorphism)e(on)m(to) 113 4146 y Fq(H)185 4110 y Fs(+)257 4146 y Ft(.)65 b(One)41 b(can)g(therefore)g(giv)m(e)f(to)1537 4121 y(^)1523 4146 y Fq(H)1594 4110 y Fs(+)1707 4146 y Ft(the)g(structure)i(of)e(a)g(top)s (ological)f(manifold)g(with)113 4266 y(b)s(oundary)-8 b(,)33 b(the)h(b)s(oundary)g(is)f Fp(@)1374 4240 y Ft(^)1361 4266 y Fq(H)1432 4230 y Fs(+)1531 4266 y Ft(=)1650 4240 y(^)1637 4266 y Fq(H)1708 4230 y Fs(+)1802 4266 y Fo(n)22 b Fq(H)1946 4230 y Fs(+)2051 4266 y Ft(and)34 b(is)f(homeomorphic)f(to) h Fq(S)3198 4230 y Fs(1)3237 4266 y Ft(.)313 4416 y Fr(5.2.5)51 b Ft(The)g(action)f(of)g(PSL)q(\(2)p Fp(;)17 b Fq(Z)p Ft(\))46 b(on)51 b Fq(H)1968 4380 y Fs(+)2091 4416 y Ft(b)m(y)g(homographies)f(has)h(a)f Fl(c)-5 b(ontinuous)113 4546 y Ft(extension)34 b(to)f(an)g(action)g(on)1265 4520 y(^)1251 4546 y Fq(H)1323 4510 y Fs(+)1428 4546 y Ft(:)44 b(just)34 b(de\014ne)583 4700 y Fk(\022)673 4781 y Fp(a)104 b(b)677 4900 y(c)h(d)893 4700 y Fk(\023)989 4841 y Fo(\001)22 b Fp(\013)27 b Ft(=)1248 4773 y Fp(a\013)21 b Ft(+)i Fp(b)p 1248 4818 282 4 v 1248 4909 a(c\013)f Ft(+)h Fp(d)1569 4841 y(;)71 b Fo(8)p Fp(\013)28 b Fo(2)g Fq(R)33 b Fo(n)22 b Fq(Q)46 b Fp(;)411 4979 y Fk(\022)501 5060 y Fp(a)104 b(b)506 5179 y(c)h(d)722 4979 y Fk(\023)818 5120 y Fo(\001)21 b Ft(\()p Fp(\013;)c(\022)s Ft(\))26 b(=)1236 4979 y Fk(\022)1321 5052 y Fp(a\013)c Ft(+)g Fp(b)p 1321 5097 V 1321 5188 a(c\013)h Ft(+)f Fp(d)1614 5120 y(;)17 b(\022)1709 4979 y Fk(\023)1826 5120 y Fp(;)71 b Fo(8)p Fp(\013)28 b Fo(2)p 2166 5039 78 4 v 28 w Fq(Q)46 b Fp(;)e Fo(8)p Fp(\022)30 b Fo(2)2570 5009 y Fk(h)2617 5120 y Fo(\000)2706 5052 y Fp(\031)p 2706 5097 61 4 v 2711 5188 a Ft(2)2779 5120 y Fp(;)17 b Ft(+)2913 5052 y Fp(\031)p 2913 5097 V 2918 5188 a Ft(2)2985 5009 y Fk(i)3076 5120 y Fp(:)3402 4980 y Ft(\(5)p Fp(:)p Ft(11\))113 5490 y(Let)34 b Fp(I)41 b Ft(denote)34 b(a)f(compact)f(non)i(trivial)d(in)m(terv)-6 b(al)33 b(in)h Fq(R)44 b Ft(and)34 b Fp(g)c Ft(=)2628 5349 y Fk(\022)2718 5430 y Fp(a)104 b(b)2723 5549 y(c)h(d)2939 5349 y Fk(\023)3040 5490 y Fo(2)28 b Ft(PGL)17 b(\(2)p Fp(;)g Fq(Z)p Ft(\).)p eop %%Page: 36 36 36 35 bop 113 144 a Ft(36)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y Fr(Lemma)37 b(5.4)90 b Fe(Assume)32 b(that)g Fo(\000)p Fp(d=c)39 b(=)-61 b Fo(2)28 b Fp(I)39 b Fe(\(so)32 b(that)g Fp(g)23 b Fo(\001)c Fp(I)35 b Ft(=)29 b Fp(I)2494 347 y Fm(0)2553 383 y Fe(is)j(also)g(a)g(compact)f(in)m (terv)-6 b(al)113 503 y(of)34 b Fq(R)5 b Fe(\).)49 b(If)34 b Fp(')28 b Fo(2)g(O)778 466 y Fs(1)823 503 y Ft(\()p 862 422 72 4 v Fq(C)48 b Fo(n)22 b Fp(I)8 b Ft(\))32 b Fe(has)i(the)f(follo)m(wing)g(prop)s(erties)h(:)158 622 y(\(i\))49 b Fp(')33 b Fe(is)g(real)20 b(;)130 752 y(\(ii\))49 b(the)33 b(harmonic)g(function)h Fo(<)p Fp(e)17 b(')33 b Fe(on)h Fq(H)1755 716 y Fs(+)1860 752 y Fe(has)g(a)f(con)m (tin)m(uous)i(extension)e(to)3196 726 y Ft(^)3182 752 y Fq(H)3254 716 y Fs(+)3326 752 y Fe(.)313 917 y(Then)h(the)f(function) i Fp(L)1203 932 y Fn(g)1249 917 y Fp(')28 b Fo(2)g(O)1518 881 y Fs(1)1562 917 y Ft(\()p 1601 836 V Fq(C)48 b Fo(n)22 b Fp(I)1819 881 y Fm(0)1847 917 y Ft(\))33 b Fe(also)g(has)g(these)h(t) m(w)m(o)g(prop)s(erties.)113 1059 y Fl(Pr)-5 b(o)g(of.)34 b Ft(It)f(is)g(enough)i(to)d(distinguish)i(three)g(cases)g(:)136 1178 y(\(a\))49 b(if)33 b Fp(\013)g Ft(is)g(real)g(and)h(irrational)e (one)i(has)441 1412 y Fo(<)p Fp(e)16 b(L)643 1427 y Fn(g)690 1412 y Fp(')p Ft(\()p Fp(\013)p Ft(\))27 b(=)h(\()p Fp(a)22 b Fo(\000)g Fp(c\013)p Ft(\))1405 1271 y Fk(\024)1458 1412 y Fo(<)p Fp(e)16 b(')1674 1271 y Fk(\022)1760 1345 y Fp(d\013)22 b Fo(\000)h Fp(b)p 1760 1389 282 4 v 1760 1480 a(a)e Fo(\000)i Fp(c\013)2053 1271 y Fk(\023)2149 1412 y Fo(\000)f Fp(')2330 1271 y Fk(\022)2403 1412 y Fo(\000)2492 1345 y Fp(d)p 2492 1389 52 4 v 2496 1480 a(c)2557 1271 y Fk(\023\025)2705 1412 y Fo(\000)g Fp(")2850 1427 y Fn(g)2897 1412 y Fp(c)2940 1371 y Fm(\000)p Fs(1)3047 1412 y Fp(')3112 1371 y Fm(0)3157 1271 y Fk(\022)3230 1412 y Fo(\000)3319 1345 y Fp(d)p 3319 1389 V 3323 1480 a(c)3384 1271 y Fk(\023)3501 1412 y Fp(:)313 1646 y Ft(Note)32 b(that)h(from)f(the)h(assumption)g Fo(\000)p Fp(d=c)39 b(=)-61 b Fo(2)28 b Fp(I)40 b Ft(follo)m(ws)33 b(that)g Fp(')p Ft(\()p Fo(\000)p Fp(d=c)p Ft(\))f(and)i Fp(')3329 1609 y Fm(0)3357 1646 y Ft(\()p Fo(\000)p Fp(d=c)p Ft(\))313 1765 y(are)f(b)s(oth)g(real.)130 1885 y(\(b\))50 b(If)33 b Fp(\013)28 b Ft(=)g Fo(1)33 b Ft(and)h Fp(\022)c Fo(2)1107 1804 y Fk(\002)1149 1885 y Fo(\000)1238 1845 y Fn(\031)p 1238 1862 50 4 v 1243 1919 a Fs(2)1299 1885 y Fp(;)17 b Ft(+)1433 1845 y Fn(\031)p 1433 1862 V 1438 1919 a Fs(2)1494 1804 y Fk(\003)1569 1885 y Ft(is)33 b(arbitrary)f(then)i Fo(<)p Fp(e)17 b(L)2523 1900 y Fn(g)2569 1885 y Fp(')p Ft(\()p Fo(1)p Fp(;)g(\022)s Ft(\))26 b(=)i(0.)141 2014 y(\(c\))50 b(Finally)-8 b(,)31 b(if)i Fp(\013)h Ft(is)f(a)g(rational)f (n)m(um)m(b)s(er)i(and)f Fp(\022)e Fo(2)2142 1934 y Fk(\002)2183 2014 y Fo(\000)2272 1975 y Fn(\031)p 2273 1991 V 2278 2049 a Fs(2)2334 2014 y Fp(;)17 b Ft(+)2468 1975 y Fn(\031)p 2468 1991 V 2473 2049 a Fs(2)2529 1934 y Fk(\003)2603 2014 y Ft(is)33 b(arbitrary)g(one)g(has)313 2258 y Fo(<)p Fp(e)17 b(L)516 2273 y Fn(g)562 2258 y Fp(')p Ft(\()p Fp(\013;)g(\022)s Ft(\))26 b(=)i(\()p Fp(a)16 b Fo(\000)g Fp(c\013)p Ft(\))1359 2117 y Fk(\024)1411 2258 y Fo(<)p Fp(e)h(')1628 2117 y Fk(\022)1714 2190 y Fp(d\013)22 b Fo(\000)g Fp(b)p 1713 2235 282 4 v 1713 2326 a(a)g Fo(\000)h Fp(c\013)2007 2258 y(;)17 b(")2098 2273 y Fn(g)2144 2258 y Fp(\022)2194 2117 y Fk(\023)2289 2258 y Fo(\000)22 b Fp(')2470 2117 y Fk(\022)2544 2258 y Fo(\000)2633 2190 y Fp(d)p 2633 2235 52 4 v 2637 2326 a(c)2697 2117 y Fk(\023\025)2839 2258 y Fo(\000)16 b Fp(")2978 2273 y Fn(g)3025 2258 y Fp(c)3068 2217 y Fm(\000)p Fs(1)3175 2258 y Fp(')3240 2217 y Fm(0)3285 2117 y Fk(\022)3358 2258 y Fo(\000)3447 2190 y Fp(d)p 3447 2235 V 3451 2326 a(c)3511 2117 y Fk(\023)3629 2258 y Fp(:)3579 2484 y Fd(\003)313 2648 y Fr(5.2.6)28 b Ft(If)h Fp(I)36 b Ft(is)28 b(a)g(non)g(trivial)e(compact)i(in)m(terv) -6 b(al)28 b(of)g Fq(R)5 b Ft(,)35 b(w)m(e)29 b(denote)2865 2623 y(^)2846 2648 y Fo(C)2898 2663 y Fg(R)2954 2648 y Ft(\()p 2993 2568 72 4 v Fq(C)38 b Fo(n)12 b Fp(I)c Ft(\))27 b(the)i(space)113 2768 y(of)j(holomorphic)e(functions)j Fp(')28 b Fo(2)g(O)1490 2732 y Fs(1)1535 2768 y Ft(\()p 1574 2687 V Fq(C)44 b Fo(n)18 b Fp(I)8 b Ft(\))31 b(whic)m(h)h(are)f Fl(r)-5 b(e)g(al)31 b Ft(and)h(whose)g(real)f(part)h(on)f Fq(H)3585 2732 y Fs(+)113 2898 y Ft(extends)j(to)f(a)g(con)m(tin)m (uous)i(function)f(on)1722 2872 y(^)1709 2898 y Fq(H)1780 2862 y Fs(+)1852 2898 y Ft(.)44 b(This)34 b(is)f(a)g(Banac)m(h)g(space) i(with)e(the)g(norm)1148 3066 y Fo(k)p Fp(')p Fo(k)1327 3080 y Fs(^)1313 3098 y Fm(C)1393 3066 y Ft(:=)27 b(sup)q Fo(fj)17 b(<)p Fp(e)f(')p Ft(\()p Fp(z)t Ft(\))p Fo(j)28 b Fp(;)44 b(z)32 b Fo(2)d Fq(H)2469 3025 y Fs(+)2541 3066 y Fo(g)1393 3236 y Ft(=)f(sup)q Fo(fj)17 b(<)p Fp(e)f(')p Ft(\()p Fp(z)t Ft(\))p Fo(j)27 b Fp(;)44 b(z)33 b Fo(2)2384 3210 y Ft(^)2370 3236 y Fq(H)2441 3194 y Fs(+)2513 3236 y Fo(g)1393 3405 y Ft(=)28 b(sup)q Fo(fj)17 b(<)p Fp(e)f(')p Ft(\()p Fp(z)t Ft(\))p Fo(j)27 b Fp(;)44 b(z)33 b Fo(2)28 b Fp(@)2442 3379 y Ft(^)2429 3405 y Fq(H)2500 3364 y Fs(+)2572 3405 y Fo(g)3402 3236 y Ft(\(5)p Fp(:)p Ft(12\))113 3579 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Fp(;)17 b Ft(1)p Fp(=)p Ft(\(1)j(+)j Fp(\015)2494 5574 y Fs(0)2538 5559 y Ft(\)]\).)p eop %%Page: 37 37 37 36 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(37)113 383 y Fr(Prop)s(osition)56 b(5.5)i Fe(The)49 b(function)g Fp(T)14 b(')49 b Fe(b)s(elongs)g(to)2293 358 y Ft(^)2275 383 y Fo(C)2327 398 y Fg(R)2383 383 y Ft(\()p 2422 303 72 4 v Fq(C)57 b Fo(n)33 b Ft([0)p Fp(;)17 b Ft(1)p Fp(=)p Ft(\(1)30 b(+)j Fp(\015)3112 398 y Fs(0)3156 383 y Ft(\)]\))47 b Fe(and)i(the)113 503 y(op)s(erator)36 b Fp(T)50 b Fe(from)883 477 y Ft(^)865 503 y Fo(C)917 518 y Fg(R)972 503 y Ft(\()p 1011 422 V Fq(C)g Fo(n)24 b Fp(I)8 b Ft(\))35 b Fe(in)m(to)h(this)g(space)h(is)f(b)s(ounded.)54 b(More)37 b(precisely)-8 b(,)37 b(for)f Fp(k)f Fo(\025)e Ft(0)p Fe(,)113 622 y Ft(2)163 586 y Fn(k)240 622 y Fo(\024)28 b Fp(j)33 b(<)c(j)572 586 y Fm(0)627 622 y Fo(\024)f Ft(2)782 586 y Fn(k)r Fs(+1)965 622 y Fe(one)34 b(has)988 812 y Fo(k)1130 717 y Fk(X)1055 931 y Fn(j)t Fm(\024)p Fn(m)c Ft(1)113 3781 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41 40 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(41)113 383 y Fr(Theorem)38 b(5.10)91 b Fe(F)-8 b(or)33 b Fp(k)e Fo(\025)d Ft(0)p Fe(,)k Fp(m)1458 398 y Fs(1)1504 383 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)1813 398 y Fn(k)1889 383 y Fo(\025)28 b Ft(1)p Fe(,)33 b Fp(z)2151 398 y Fs(0)2224 383 y Fo(2)28 b Fp(H)8 b Ft(\()p Fp(m)2535 398 y Fs(1)2580 383 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)2889 398 y Fn(k)2937 383 y Ft(\))33 b Fe(one)h(has)329 600 y Fo(=)p Fp(m)17 b Fo(B)s Ft(\()p Fp(z)658 615 y Fs(0)704 600 y Ft(\))27 b(=)h Fp(B)951 615 y Fn(f)8 b(inite)1176 600 y Ft(\()p Fp(p)1265 615 y Fn(k)1314 600 y Fp(=q)1408 615 y Fn(k)1457 600 y Ft(\))22 b(+)h(\()p Fp(p)1707 615 y Fn(k)r Fm(\000)p Fs(1)1880 600 y Fo(\000)f Fp(q)2023 615 y Fn(k)r Fm(\000)p Fs(1)2191 600 y Fo(<)p Fp(e)17 b(z)2372 615 y Fs(0)2417 600 y Ft(\))g Fo(=)p Fp(m)g(')2714 615 y Fs(1)2758 600 y Ft(\()p Fp(z)2843 615 y Fn(k)2915 600 y Ft(+)22 b(1\))g(+)27 b(^)-54 b Fp(r)r Ft(\()p Fp(z)3357 615 y Fs(0)3402 600 y Ft(\))113 817 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2479 y Fp(')2001 2494 y Fs(1)2046 2479 y Ft(\()p 2085 2424 51 4 v Fp(z)5 b Ft(\))62 b(=)p 2377 2390 360 4 v 63 w Fp(T)2449 2450 y Fn(k)2498 2479 y Fp(')2563 2494 y Fs(1)2608 2479 y Ft(\()p Fp(z)t Ft(\),)e(hence)c Fo(=)p Fp(m)17 b(T)3370 2443 y Fn(k)3419 2479 y Fp(')3484 2494 y Fs(1)3528 2479 y Ft(\()p Fp(z)t Ft(\))113 2599 y(v)-6 b(anishes)35 b(on)e Fq(R)44 b Ft(outside)34 b(of)1388 2609 y Fk(8)1388 2698 y(<)1388 2878 y(:)1493 2693 y Ft([1)p Fp(=)p Ft(2)p Fp(;)17 b Ft(2])98 b(for)33 b Fp(k)e Ft(=)d(0)f Fp(;)1493 2813 y Ft([0)p Fp(;)17 b Ft(2)p Fp(=)p Ft(3])98 b(for)33 b Fp(k)e Ft(=)d(1)f Fp(;)1493 2932 y Ft([0)p Fp(;)17 b Ft(1])198 b(for)33 b Fp(k)e(>)d Ft(1)f Fp(:)113 3106 y Ft(As)34 b(w)m(e)f(ha)m(v)m(e,)h(on)f(the)h(other)f(hand,)1417 3323 y Fo(j)p Fp(')1510 3282 y Fm(0)1538 3323 y Ft(\()p Fp(z)t Ft(\))p Fo(j)28 b(\024)g Fp(C)1898 3338 y Fn(K)1975 3323 y Fo(j)p Fp(z)t Fo(j)2081 3282 y Fm(\000)p Fs(2)2188 3323 y Fo(k)p Fp(')p Fo(k)1049 b Ft(\(5)p Fp(:)p Ft(22\))113 3541 y(for)34 b(all)e Fp(')c Fo(2)g Fp(E)6 b Ft(\()p Fp(I)i Ft(\),)31 b Fp(K)k Fo(\032)p 1085 3460 72 4 v 28 w Fq(C)48 b Fo(n)22 b Fp(I)41 b Ft(compact,)32 b Fp(z)g Fo(2)d Fp(K)7 b Ft(,)32 b(w)m(e)i(obtain)f(for)g Fp(z)g Fo(2)28 b Ft(\001)401 3759 y Fo(j)17 b(=)p Fp(m)f(')686 3774 y Fs(1)731 3759 y Ft(\()p Fp(z)26 b Ft(+)d Fp(n)p Ft(\))p Fo(j)k(\024)h Fp(cn)1304 3718 y Fm(\000)p Fs(2)1411 3759 y Fo(j)17 b(=)p Fp(m)f(z)t Fo(jk)p Fp(')1824 3774 y Fs(1)1869 3759 y Fo(k)28 b Fp(;)71 b Ft(if)17 b Fp(n)28 b Fo(6)p Ft(=)g(0)p Fp(;)17 b Ft(1)p Fp(;)g Ft(2)25 b Fp(;)44 b(n)28 b Fo(2)g Fq(Z)d Fp(;)329 3920 y Fo(j)17 b(=)p Fp(m)f(T)e(')686 3935 y Fs(1)731 3920 y Ft(\()p Fp(z)26 b Ft(+)d Fp(n)p Ft(\))p Fo(j)k(\024)h Fp(cn)1304 3879 y Fm(\000)p Fs(2)1411 3920 y Fo(j)17 b(=)p Fp(m)f(z)t Fo(jk)p Fp(T)e(')1896 3935 y Fs(1)1941 3920 y Fo(k)28 b Fp(;)71 b Ft(if)17 b Fp(n)28 b Fo(6)p Ft(=)g Fo(\000)p Ft(1)p Fp(;)17 b Ft(0)27 b Fp(;)44 b(n)27 b Fo(2)h Fq(Z)d Fp(;)280 4084 y Fo(j)17 b(=)p Fp(m)f(T)572 4043 y Fn(k)621 4084 y Fp(')686 4099 y Fs(1)731 4084 y Ft(\()p Fp(z)26 b Ft(+)d Fp(n)p Ft(\))p Fo(j)k(\024)h Fp(cn)1304 4043 y Fm(\000)p Fs(2)1411 4084 y Fo(j)17 b(=)p Fp(m)f(z)t Fo(jk)p Fp(T)1831 4043 y Fn(k)1881 4084 y Fp(')1946 4099 y Fs(1)1990 4084 y Fo(k)28 b Fp(;)71 b Ft(if)17 b Fp(k)31 b(>)d Ft(1)g Fp(;)43 b(n)28 b Fo(6)p Ft(=)g Fo(\000)p Ft(1)p Fp(;)17 b Ft(0)p Fp(;)g Ft(1)26 b Fp(;)44 b(n)28 b Fo(2)g Fq(Z)d Fp(:)3402 4204 y Ft(\(5)p Fp(:)p Ft(23\))113 4323 y(Since,)32 b(b)m(y)e(Prop)s(osition)f(4.11,)g(w)m(e)h(ha)m(v)m(e) h Fo(k)p Fp(T)1794 4287 y Fn(k)1843 4323 y Fp(')1908 4338 y Fs(1)1953 4323 y Fo(k)c(\024)h Fp(C)7 b(k)s(G)2346 4287 y Fm(\000)p Fn(k)2458 4323 y Fo(k)p Fp(')2573 4338 y Fs(1)2618 4323 y Fo(k)p Ft(,)30 b(with)g Fp(G)e Ft(=)g(\()3200 4241 y Fo(p)p 3282 4241 50 4 v 3282 4323 a Ft(5)16 b(+)g(1\))p Fp(=)p Ft(2,)113 4443 y(w)m(e)34 b(obtain)113 4617 y Fr(Lemma)39 b(5.11)58 b Fe(F)-8 b(or)32 b Fp(z)h Fo(2)28 b Ft(\001)22 b Fo(n)g Ft([0)p Fp(;)17 b Ft(1])p Fe(,)31 b(if)i(w)m(e)h(write)419 4834 y Fo(=)p Fp(m)17 b Fo(B)t Ft(\()p Fp(z)t Ft(\))28 b(=)g Fo(=)p Fp(m)17 b(')1166 4849 y Fs(1)1211 4834 y Ft(\()p Fp(z)t Ft(\))22 b(+)h Fo(=)p Fp(m)17 b(')1702 4849 y Fs(1)1747 4834 y Ft(\()p Fp(z)26 b Ft(+)d(1\))f(+)g Fo(=)p Fp(m)17 b(')2409 4849 y Fs(1)2454 4834 y Ft(\()p Fp(z)27 b Ft(+)22 b(2\))686 4984 y(+)h Fo(=)p Fp(m)17 b(T)d(')1099 4999 y Fs(1)1144 4984 y Ft(\()p Fp(z)26 b Fo(\000)d Ft(1\))f(+)g Fo(=)p Fp(m)17 b(T)d(')1878 4999 y Fs(1)1923 4984 y Ft(\()p Fp(z)t Ft(\))424 5153 y(+)519 5058 y Fk(X)518 5273 y Fn(k)r(>)p Fs(1)681 5072 y Fk(\002)722 5153 y Fo(=)p Fp(m)j(T)970 5112 y Fn(k)1019 5153 y Fp(')1084 5168 y Fs(1)1129 5153 y Ft(\()p Fp(z)27 b Fo(\000)22 b Ft(1\))g(+)h Fo(=)p Fp(m)17 b(T)1799 5112 y Fn(k)1848 5153 y Fp(')1913 5168 y Fs(1)1957 5153 y Ft(\()p Fp(z)t Ft(\))23 b(+)f Fo(=)p Fp(m)17 b(T)2455 5112 y Fn(k)2505 5153 y Fp(')2570 5168 y Fs(1)2614 5153 y Ft(\()p Fp(z)27 b Ft(+)c(1\))2915 5072 y Fk(\003)2978 5153 y Ft(+)f Fp(r)3122 5168 y Fs(0)3167 5153 y Ft(\()p Fp(z)t Ft(\))28 b Fp(;)51 b Ft(\(5)p Fp(:)p Ft(24\))113 5465 y Fe(then)34 b(w)m(e)g(ha)m(v)m(e)1361 5584 y Fo(j)p Fp(r)1434 5599 y Fs(0)1478 5584 y Ft(\()p Fp(z)t Ft(\))p Fo(j)28 b(\024)g Fp(C)7 b Fo(j)17 b(=)p Fp(m)g(z)t Fo(jk)p Fp(')2259 5599 y Fs(1)2304 5584 y Fo(k)27 b Fp(:)993 b Ft(\(5)p Fp(:)p Ft(25\))p eop %%Page: 42 42 42 41 bop 113 144 a Ft(42)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)313 383 y Fr(5.3.4)33 b Ft(One)i(has)1193 503 y Fo(j)p Fp(')1286 462 y Fm(0)1286 527 y Fs(1)1331 503 y Ft(\()p Fp(z)t Ft(\))p Fo(j)28 b(\024)g Fp(C)c Ft(log\(1)e(+)h Fo(j)p Fp(z)j Fo(\000)d Ft(1)p Fp(=)p Ft(2)p Fo(j)2432 462 y Fm(\000)p Fs(1)2538 503 y Ft(\))113 685 y(in)34 b(the)f(neigh)m(b)s(orho)s(o)s(d)h(of)f(1)p Fp(=)p Ft(2)g(and)1243 907 y Fo(j)p Fp(')1336 866 y Fm(0)1336 932 y Fs(1)1381 907 y Ft(\()p Fp(z)t Ft(\))p Fo(j)27 b(\024)i Fp(C)23 b Ft(log)q(\(1)f(+)g Fo(j)p Fp(z)27 b Fo(\000)c Ft(2)p Fo(j)2382 866 y Fm(\000)p Fs(1)2488 907 y Ft(\))113 1130 y(in)34 b(the)f(neigh)m(b)s(orho)s(o)s(d)h(of)f (2.)44 b(Therefore,)34 b(for)g Fp(z)e Fo(2)c Ft(\001)23 b Fo(n)f(f)p Ft(0)p Fo(g)32 b Ft(w)m(e)i(ha)m(v)m(e)970 1353 y Fo(j)17 b(=)p Fp(m)f(')1255 1368 y Fs(1)1300 1353 y Ft(\()p Fp(z)27 b Ft(+)22 b(2\))p Fo(j)27 b(\024)h Fp(C)7 b Fo(j)17 b(=)p Fp(m)g(z)t Fo(j)g Ft(log)q(\(1)k(+)i Fo(j)p Fp(z)t Fo(j)2599 1312 y Fm(\000)p Fs(1)2706 1353 y Ft(\))k Fp(;)602 b Ft(\(5)p Fp(:)p Ft(26\))113 1576 y(and,)34 b(for)f Fp(z)g Fo(2)28 b Ft(\001)22 b Fo(n)g Fp(D)s Ft(\(1\))920 1798 y Fo(j)17 b(=)p Fp(m)f(')1205 1813 y Fs(1)1250 1798 y Ft(\()p Fp(z)t Ft(\))p Fo(j)28 b(\024)g Fp(C)7 b Fo(j)17 b(=)p Fp(m)g(z)t Fo(j)g Ft(log\(1)22 b(+)h Fo(j)p Fp(z)j Fo(\000)d Ft(1)p Fp(=)p Ft(2)p Fo(j)2650 1757 y Fm(\000)p Fs(1)2756 1798 y Ft(\))k Fp(:)552 b Ft(\(5)p Fp(:)p Ft(27\))113 2021 y(Next)33 b(w)m(e)h(ha)m(v)m(e)113 2197 y Fr(Lemma)39 b(5.12)58 b Fe(F)-8 b(or)32 b Fp(k)f Fo(\025)d Ft(1)33 b Fe(and)h Fp(z)e Fo(2)c Ft(\001)23 b Fo(n)f(f)p Ft(1)p Fo(g)32 b Fe(w)m(e)i(ha)m(v)m(e)607 2419 y Fo(j)17 b(=)p Fp(m)g(T)900 2378 y Fn(k)949 2419 y Fp(')1014 2434 y Fs(1)1059 2419 y Ft(\()p Fp(z)26 b Fo(\000)d Ft(1\))p Fo(j)k(\024)h Fp(C)7 b Fo(j)17 b(=)p Fp(m)g(z)t Fo(j)g Ft(log\(1)22 b(+)h Fo(j)p Fp(z)j Fo(\000)d Ft(1)p Fo(j)2530 2378 y Fm(\000)p Fs(1)2636 2419 y Ft(\))p Fo(k)p Fp(T)2797 2378 y Fn(k)r Fm(\000)p Fs(1)2948 2419 y Fp(')3013 2434 y Fs(1)3058 2419 y Fo(k)k Fp(:)113 2670 y Fl(Pr)-5 b(o)g(of.)34 b Ft(W)-8 b(e)34 b(ha)m(v)m(e)351 3007 y Fo(j)17 b(=)p Fp(m)g(T)644 2966 y Fn(k)693 3007 y Fp(')758 3022 y Fs(1)802 3007 y Ft(\()p Fp(z)t Ft(\))p Fo(j)28 b Ft(=)1091 2803 y Fk(\014)1091 2863 y(\014)1091 2922 y(\014)1091 2982 y(\014)1091 3042 y(\014)1091 3102 y(\014)1124 3007 y Fo(=)p Fp(m)1300 2807 y Fk(2)1300 2986 y(4)1367 3007 y Fp(z)1449 2912 y Fk(X)1434 3126 y Fn(m)p Fm(\025)p Fs(1)1623 2867 y Fk(\022)1697 3007 y Fp(T)1769 2966 y Fn(k)r Fm(\000)p Fs(1)1920 3007 y Fp(')1985 3022 y Fs(1)2046 2867 y Fk(\022)2132 2940 y Ft(1)p 2132 2984 51 4 v 2132 3075 a Fp(z)2216 3007 y Fo(\000)23 b Fp(m)2403 2867 y Fk(\023)2499 3007 y Fo(\000)g Fp(T)2671 2966 y Fn(k)r Fm(\000)p Fs(1)2822 3007 y Fp(')2887 3022 y Fs(1)2932 3007 y Ft(\()p Fo(\000)p Fp(m)p Ft(\))3174 2867 y Fk(\023)3247 2807 y(3)3247 2986 y(5)3314 2803 y(\014)3314 2863 y(\014)3314 2922 y(\014)3314 2982 y(\014)3314 3042 y(\014)3314 3102 y(\014)3391 3007 y Fp(:)113 3344 y Ft(W)-8 b(e)34 b(distinguish)g(t)m(w)m(o)f(cases)h(:)136 3463 y(\(a\))49 b(If)33 b Fo(j)17 b(=)p Fp(m)g(z)683 3427 y Fm(\000)p Fs(1)790 3463 y Fo(j)28 b Ft(=)g Fo(j)p Fp(z)t Fo(j)1057 3427 y Fm(\000)p Fs(2)1164 3463 y Fo(j)17 b(=)p Fp(m)f(z)t Fo(j)28 b(\025)g Ft(1,)33 b(w)m(e)h(c)m(ho)s(ose)g Fp(m)2256 3478 y Fs(0)2329 3463 y Fo(\025)28 b Ft(1)33 b(suc)m(h)h(that)733 3601 y Fk(\014)733 3661 y(\014)733 3721 y(\014)733 3781 y(\014)778 3678 y Ft(1)p 778 3723 V 778 3814 a Fp(z)863 3746 y Ft(+)22 b Fp(m)1049 3761 y Fs(0)1094 3601 y Fk(\014)1094 3661 y(\014)1094 3721 y(\014)1094 3781 y(\014)1155 3746 y Fo(\024)28 b Fp(C)7 b Fo(j)p Fp(z)t Fo(j)1444 3705 y Fm(\000)p Fs(2)1552 3746 y Fo(j)17 b(=)p Fp(m)f(z)t Fo(j)56 b Ft(and)28 b Fp(C)2172 3705 y Fm(\000)p Fs(1)2279 3746 y Fo(j)p Fp(z)t Fo(j)2385 3705 y Fm(\000)p Fs(1)2520 3746 y Fo(\024)g Fp(m)2712 3761 y Fs(0)2785 3746 y Fo(\024)g Fp(C)7 b Fo(j)p Fp(z)t Fo(j)3074 3705 y Fm(\000)p Fs(1)3209 3746 y Fp(:)313 4023 y Ft(One)34 b(has)g(then)603 4143 y Fk(\014)603 4203 y(\014)603 4263 y(\014)603 4323 y(\014)636 4288 y Fp(T)708 4247 y Fn(k)r Fm(\000)p Fs(1)859 4288 y Fp(')924 4303 y Fs(1)985 4147 y Fk(\022)1071 4221 y Ft(1)p 1071 4265 V 1071 4356 a Fp(z)1156 4288 y Fo(\000)22 b Fp(m)1342 4147 y Fk(\023)1438 4288 y Fo(\000)h Fp(T)1610 4247 y Fn(k)r Fm(\000)p Fs(1)1761 4288 y Fp(')1826 4303 y Fs(1)1871 4288 y Ft(\()p Fo(\000)p Fp(m)g Fo(\000)f Fp(m)2283 4303 y Fs(0)2328 4288 y Ft(\))2367 4143 y Fk(\014)2367 4203 y(\014)2367 4263 y(\014)2367 4323 y(\014)2428 4288 y Fo(\024)28 b Fp(C)2623 4221 y Fo(j)17 b(=)p Fp(m)g(z)t Fo(jk)p Fp(T)3044 4184 y Fn(k)r Fm(\000)p Fs(1)3195 4221 y Fp(')3260 4236 y Fs(1)3305 4221 y Fo(k)p 2623 4265 732 4 v 2682 4356 a(j)p Fp(z)t Fo(j)2788 4327 y Fs(2)2833 4356 y Ft(\()p Fp(m)22 b Ft(+)h Fp(m)3168 4371 y Fs(0)3213 4356 y Ft(\))3252 4327 y Fs(2)313 4565 y Ft(from)32 b(whic)m(h)i(it)f (follo)m(ws)g(that)319 4678 y Fk(\014)319 4738 y(\014)319 4798 y(\014)319 4857 y(\014)319 4917 y(\014)319 4977 y(\014)352 4682 y(2)352 4861 y(4)419 4882 y Fp(z)500 4788 y Fk(X)486 5001 y Fn(m)p Fm(\025)p Fs(1)675 4742 y Fk(\022)749 4882 y Fp(T)821 4841 y Fn(k)r Fm(\000)p Fs(1)972 4882 y Fp(')1037 4897 y Fs(1)1098 4742 y Fk(\022)1184 4815 y Ft(1)p 1183 4859 51 4 v 1183 4951 a Fp(z)1268 4882 y Fo(\000)23 b Fp(m)1455 4742 y Fk(\023)1551 4882 y Fo(\000)f Fp(T)1722 4841 y Fn(k)r Fm(\000)p Fs(1)1873 4882 y Fp(')1938 4897 y Fs(1)1983 4882 y Ft(\()p Fo(\000)p Fp(m)h Fo(\000)g Fp(m)2396 4897 y Fs(0)2441 4882 y Ft(\))2480 4742 y Fk(\023)2553 4682 y(3)2553 4861 y(5)2619 4678 y(\014)2619 4738 y(\014)2619 4798 y(\014)2619 4857 y(\014)2619 4917 y(\014)2619 4977 y(\014)2680 4882 y Fo(\024)28 b Fp(C)7 b Fo(j)17 b(=)p Fp(m)g(z)t Fo(jk)p Fp(T)3284 4841 y Fn(k)r Fm(\000)p Fs(1)3435 4882 y Fp(')3500 4897 y Fs(1)3545 4882 y Fo(k)28 b Fp(:)313 5219 y Ft(On)34 b(the)f(other)h (hand)657 5498 y Fo(j)17 b(=)p Fp(m)f(z)961 5372 y Fn(m)1032 5382 y Fi(0)944 5403 y Fk(X)996 5614 y Fs(1)1105 5498 y Fp(T)1177 5457 y Fn(k)r Fm(\000)p Fs(1)1328 5498 y Fp(')1393 5513 y Fs(1)1438 5498 y Ft(\()p Fo(\000)p Fp(m)p Ft(\))p Fo(j)27 b(\024)i Fp(C)7 b Fo(j)17 b(=)p Fp(m)f(z)t Fo(j)h Ft(log)q(\(1)22 b(+)g Fo(j)p Fp(z)t Fo(j)2679 5457 y Fm(\000)p Fs(1)2786 5498 y Ft(\))p Fo(k)p Fp(T)2947 5457 y Fn(k)r Fm(\000)p Fs(1)3098 5498 y Fp(')3163 5513 y Fs(1)3208 5498 y Fo(k)27 b Fp(;)p eop %%Page: 43 43 43 42 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(43)313 383 y(from)32 b(whic)m(h)i(the)g(lemma)d(follo)m(ws)i(in)g (this)g(case.)130 503 y(\(b\))50 b(If)33 b Fo(j)17 b(=)p Fp(m)g(z)683 466 y Fm(\000)p Fs(1)790 503 y Fo(j)28 b(\024)g Ft(1)33 b(w)m(e)g(c)m(ho)s(ose)h Fp(m)1583 518 y Fs(0)1656 503 y Fo(\025)28 b Ft(1)34 b(suc)m(h)g(that)f Fo(\000)p Fp(m)2449 518 y Fs(0)2522 503 y Fo(\024)c(<)p Fp(e)17 b Ft(1)p Fp(=z)32 b Fo(\024)c(\000)p Fp(m)3210 518 y Fs(0)3277 503 y Ft(+)23 b(1)33 b(th)m(us)646 622 y Fk(\014)646 682 y(\014)646 741 y(\014)646 801 y(\014)679 766 y Fo(=)p Fp(m)17 b(T)927 725 y Fn(k)r Fm(\000)p Fs(1)1078 766 y Fp(')1143 781 y Fs(1)1204 626 y Fk(\022)1290 699 y Ft(1)p 1290 743 51 4 v 1290 835 a Fp(z)1375 766 y Fo(\000)22 b Fp(m)1561 626 y Fk(\023)1635 622 y(\014)1635 682 y(\014)1635 741 y(\014)1635 801 y(\014)1696 766 y Fo(\024)28 b Fp(C)7 b Fo(j)17 b(=)p Fp(m)g(z)t Fo(jj)p Fp(z)t Fo(j)2284 725 y Fm(\000)p Fs(2)2391 766 y Ft(\()p Fp(m)22 b Ft(+)h Fp(m)2726 781 y Fs(0)2771 766 y Ft(\))2810 725 y Fm(\000)p Fs(2)2917 766 y Fo(k)p Fp(T)3039 725 y Fn(k)r Fm(\000)p Fs(1)3190 766 y Fp(')3255 781 y Fs(1)3300 766 y Fo(k)k Fp(;)353 901 y Fk(\014)353 961 y(\014)353 1020 y(\014)353 1080 y(\014)387 1045 y Fo(<)p Fp(e)522 905 y Fk(\024)574 1045 y Fp(T)646 1004 y Fn(k)r Fm(\000)p Fs(1)797 1045 y Fp(')862 1060 y Fs(1)924 905 y Fk(\022)1009 978 y Ft(1)p 1009 1022 V 1009 1114 a Fp(z)1094 1045 y Fo(\000)22 b Fp(m)1280 905 y Fk(\023)1376 1045 y Fo(\000)h Fp(T)1548 1004 y Fn(k)r Fm(\000)p Fs(1)1699 1045 y Fp(')1764 1060 y Fs(1)1809 1045 y Ft(\()p Fo(\000)p Fp(m)g Fo(\000)f Fp(m)2221 1060 y Fs(0)2266 1045 y Ft(\))2305 905 y Fk(\025)2358 901 y(\014)2358 961 y(\014)2358 1020 y(\014)2358 1080 y(\014)2419 1045 y Fo(\024)28 b Fp(C)7 b Ft(\()p Fp(m)23 b Ft(+)f Fp(m)2937 1060 y Fs(0)2982 1045 y Ft(\))3021 1004 y Fm(\000)p Fs(2)3128 1045 y Fo(k)p Fp(T)3250 1004 y Fn(k)r Fm(\000)p Fs(1)3401 1045 y Fp(')3466 1060 y Fs(1)3511 1045 y Fo(k)27 b Fp(:)313 1303 y Ft(W)-8 b(e)33 b(th)m(us)h(obtain)1017 1602 y Fo(j)17 b(<)p Fp(e)f(z)t Fo(j)1291 1398 y Fk(\014)1291 1457 y(\014)1291 1517 y(\014)1291 1577 y(\014)1291 1637 y(\014)1291 1696 y(\014)1324 1602 y Fo(=)p Fp(m)1515 1507 y Fk(X)1500 1721 y Fn(m)p Fm(\025)p Fs(1)1690 1602 y Fp(T)1762 1561 y Fn(k)r Fm(\000)p Fs(1)1913 1602 y Fp(')1978 1617 y Fs(1)2039 1461 y Fk(\022)2125 1534 y Ft(1)p 2124 1579 V 2124 1670 a Fp(z)2209 1602 y Fo(\000)23 b Fp(m)2396 1461 y Fk(\023)2470 1398 y(\014)2470 1457 y(\014)2470 1517 y(\014)2470 1577 y(\014)2470 1637 y(\014)2470 1696 y(\014)2531 1602 y Fo(\024)28 b Fp(C)7 b Fo(j)17 b(=)p Fp(m)f(z)t Fo(j)q(k)p Fp(T)3135 1561 y Fn(k)r Fm(\000)p Fs(1)3286 1602 y Fp(')3351 1617 y Fs(1)3396 1602 y Fo(k)27 b Fp(;)469 1796 y Fk(\014)469 1856 y(\014)469 1916 y(\014)469 1975 y(\014)469 2035 y(\014)469 2095 y(\014)502 2000 y Fo(<)p Fp(e)651 1906 y Fk(X)637 2119 y Fn(m)p Fm(\025)p Fs(1)826 1860 y Fk(\024)879 2000 y Fp(T)951 1959 y Fn(k)r Fm(\000)p Fs(1)1102 2000 y Fp(')1167 2015 y Fs(1)1228 1860 y Fk(\022)1314 1933 y Ft(1)p 1313 1977 V 1313 2069 a Fp(z)1398 2000 y Fo(\000)c Fp(m)1585 1860 y Fk(\023)1681 2000 y Fo(\000)f Fp(T)1852 1959 y Fn(k)r Fm(\000)p Fs(1)2004 2000 y Fp(')2069 2015 y Fs(1)2113 2000 y Ft(\()p Fo(\000)p Fp(m)h Fo(\000)g Fp(m)2526 2015 y Fs(0)2571 2000 y Ft(\))2610 1860 y Fk(\025)2662 1796 y(\014)2662 1856 y(\014)2662 1916 y(\014)2662 1975 y(\014)2662 2035 y(\014)2662 2095 y(\014)2723 2000 y Fo(\024)28 b Fp(C)7 b Fo(j)p Fp(z)t Fo(j)q(k)p Fp(T)3135 1959 y Fn(k)r Fm(\000)p Fs(1)3286 2000 y Fp(')3351 2015 y Fs(1)3396 2000 y Fo(k)27 b Fp(;)1386 2360 y Fo(j)1448 2235 y Fn(m)1519 2245 y Fi(0)1431 2266 y Fk(X)1483 2477 y Fs(1)1591 2360 y Fp(T)1663 2319 y Fn(k)r Fm(\000)p Fs(1)1814 2360 y Fp(')1879 2375 y Fs(1)1924 2360 y Ft(\()p Fo(\000)p Fp(m)p Ft(\))p Fo(j)h(\024)g Fp(C)c Ft(log)q(\(1)d(+)i Fo(j)p Fp(z)t Fo(j)2867 2319 y Fm(\000)p Fs(1)2974 2360 y Ft(\))p Fo(k)p Fp(T)3135 2319 y Fn(k)r Fm(\000)p Fs(1)3286 2360 y Fp(')3351 2375 y Fs(1)3396 2360 y Fo(k)k Fp(;)313 2650 y Ft(whic)m(h)34 b(giv)m(e)e(the)i(desired)h(result.)1994 b Fd(\003)113 2847 y Fr(Lemma)39 b(5.13)185 3019 y Ft(1.)50 b Fe(If)33 b Fp(z)g Fo(2)28 b Fp(D)s Ft(\(1\))21 b Fo(n)h Fp(D)s Ft(\(1)p Fp(;)17 b Ft(1\))32 b Fe(w)m(e)i(ha)m(v)m(e)984 3223 y Fo(j)17 b(=)p Fp(m)f(T)e(')1341 3238 y Fs(1)1386 3223 y Ft(\()p Fp(z)t Ft(\))p Fo(j)27 b(\024)i Fp(C)7 b Fo(j)17 b(=)p Fp(m)f(z)t Fo(j)h Ft(log)q(\(1)22 b(+)g Fo(j)p Fp(z)27 b Fo(\000)22 b Ft(2)p Fp(=)p Ft(3)p Fo(j)2785 3182 y Fm(\000)p Fs(1)2892 3223 y Ft(\))27 b Fp(:)185 3427 y Ft(2.)50 b Fe(F)-8 b(or)32 b Fp(k)f(>)d Ft(1)p Fe(,)33 b Fp(z)f Fo(2)c Ft(\001)23 b Fo(n)f(f)p Ft(0)p Fo(g)32 b Fe(w)m(e)i(ha)m(v)m(e)793 3630 y Fo(j)17 b(=)p Fp(m)f(T)1085 3589 y Fn(k)1134 3630 y Fp(')1199 3645 y Fs(1)1244 3630 y Ft(\()p Fp(z)27 b Ft(+)22 b(1\))p Fo(j)27 b(\024)h Fp(C)7 b Fo(j)17 b(=)p Fp(m)g(z)t Fo(j)g Ft(log)q(\(1)k(+)i Fo(j)p Fp(z)t Fo(j)2543 3589 y Fm(\000)p Fs(1)2650 3630 y Ft(\))p Fo(k)p Fp(T)2811 3589 y Fn(k)r Fm(\000)p Fs(2)2962 3630 y Fp(')3027 3645 y Fs(1)3071 3630 y Fo(k)28 b Fp(:)113 3860 y Fl(Pr)-5 b(o)g(of.)42 b Ft(In)f(the)h(\014rst)f(case,)j(in)d(the)g(domain)f(considered,)45 b(w)m(e)c(ha)m(v)m(e)h(1)p Fp(=z)32 b Fo(\000)c Ft(1)40 b Fo(2)h Ft(\001)28 b Fo(n)f Fp(D)s Ft(\(1\))113 3980 y(hence,)35 b(b)m(y)e(\(5.27\),)782 4099 y Fk(\014)782 4159 y(\014)782 4218 y(\014)782 4278 y(\014)815 4243 y Fo(=)p Fp(m)17 b(')1056 4258 y Fs(1)1118 4103 y Fk(\022)1204 4176 y Ft(1)p 1203 4220 V 1203 4312 a Fp(z)1288 4243 y Fo(\000)23 b Ft(1)1438 4103 y Fk(\023)1511 4099 y(\014)1511 4159 y(\014)1511 4218 y(\014)1511 4278 y(\014)1572 4243 y Fo(\024)28 b Fp(C)7 b Fo(j)17 b(=)p Fp(m)g(z)t Fo(j)g Ft(log\(1)22 b(+)h Fo(j)p Fp(z)j Fo(\000)d Ft(2)p Fp(=)p Ft(3)p Fo(j)2788 4202 y Fm(\000)p Fs(1)2894 4243 y Ft(\))k Fp(:)113 4501 y Ft(On)34 b(the)g(other)f(hand)h(w)m(e)g(ha)m(v)m(e)1030 4596 y Fk(\014)1030 4655 y(\014)1030 4715 y(\014)1030 4775 y(\014)1063 4740 y Fo(<)p Fp(e)1198 4600 y Fk(\024)1251 4740 y Fp(')1316 4755 y Fs(1)1377 4600 y Fk(\022)1463 4673 y Ft(1)p 1463 4717 V 1463 4809 a Fp(z)1547 4740 y Fo(\000)23 b Ft(1)1697 4600 y Fk(\023)1792 4740 y Fo(\000)g Fp(')1957 4755 y Fs(1)2002 4740 y Ft(\()p Fo(\000)p Ft(1\))2207 4600 y Fk(\025)2259 4596 y(\014)2259 4655 y(\014)2259 4715 y(\014)2259 4775 y(\014)2320 4740 y Fo(\024)28 b Ft(2)p Fo(k)p Fp(')2590 4755 y Fs(1)2635 4740 y Fo(k)f Fp(;)113 4998 y Ft(and)34 b(for)g Fp(m)28 b Fo(\025)g Ft(2)852 5106 y Fk(\014)852 5166 y(\014)852 5226 y(\014)852 5285 y(\014)885 5251 y Fp(')950 5266 y Fs(1)1011 5110 y Fk(\022)1097 5183 y Ft(1)p 1097 5228 V 1097 5319 a Fp(z)1181 5251 y Fo(\000)23 b Fp(m)1368 5110 y Fk(\023)1464 5251 y Fo(\000)g Fp(')1629 5266 y Fs(1)1674 5251 y Ft(\()p Fo(\000)p Fp(m)p Ft(\))1916 5106 y Fk(\014)1916 5166 y(\014)1916 5226 y(\014)1916 5285 y(\014)1977 5251 y Fo(\024)28 b Fp(C)7 b(m)2247 5209 y Fm(\000)p Fs(2)2355 5251 y Fo(k)p Fp(')2470 5266 y Fs(1)2515 5251 y Fo(k)27 b Fp(;)1150 5385 y Fk(\014)1150 5445 y(\014)1150 5505 y(\014)1150 5564 y(\014)1183 5529 y Fo(=)p Fp(m)17 b(')1424 5544 y Fs(1)1485 5389 y Fk(\022)1571 5462 y Ft(1)p 1571 5507 V 1571 5598 a Fp(z)1656 5529 y Fo(\000)22 b Fp(m)1842 5389 y Fk(\023)1916 5385 y(\014)1916 5445 y(\014)1916 5505 y(\014)1916 5564 y(\014)1977 5529 y Fo(\024)28 b Fp(C)7 b(m)2247 5488 y Fm(\000)p Fs(2)2355 5529 y Fo(j)17 b(=)p Fp(m)f(z)t Fo(jk)p Fp(')2768 5544 y Fs(1)2813 5529 y Fo(k)28 b Fp(;)p eop %%Page: 44 44 44 43 bop 113 144 a Ft(44)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y(from)33 b(whic)m(h)h(the)g(\014rst)f(inequalit)m (y)f(of)i(the)f(Lemma)f(follo)m(ws.)313 520 y(In)h(the)h(second)g(case) g(w)m(e)g(use)g(Lemma)e(5.12)g(to)h(get)504 719 y Fk(\014)504 779 y(\014)504 839 y(\014)504 899 y(\014)537 864 y Fo(=)p Fp(m)18 b(T)786 823 y Fn(k)r Fm(\000)p Fs(1)937 864 y Fp(')1002 879 y Fs(1)1063 723 y Fk(\022)1235 796 y Ft(1)p 1148 841 223 4 v 1148 932 a Fp(z)27 b Ft(+)c(1)1405 864 y Fo(\000)f Ft(1)1554 723 y Fk(\023)1627 719 y(\014)1627 779 y(\014)1627 839 y(\014)1627 899 y(\014)1688 864 y Fo(\024)29 b Fp(C)7 b Fo(j)17 b(=)p Fp(m)f(z)t Fo(j)h Ft(log)q(\(1)22 b(+)g Fo(j)p Fp(z)t Fo(j)2632 823 y Fm(\000)p Fs(1)2739 864 y Ft(\))p Fo(k)p Fp(T)2900 823 y Fn(k)r Fm(\000)p Fs(2)3051 864 y Fp(')3116 879 y Fs(1)3161 864 y Fo(k)27 b Fp(;)113 1208 y Ft(and)34 b(deduce)h(the)f(second)g (inequalit)m(y)f(as)g(ab)s(o)m(v)m(e.)1585 b Fd(\003)313 1438 y Fr(5.3.5)32 b Ft(F)-8 b(or)32 b Fp(z)g Fo(2)d Ft(\001,)j Fp(k)e Fo(\025)e Ft(1,)k(w)m(e)h(set)f(\(as)g(in)h(Section)f (4.2\))f Fp(")2538 1453 y Fn(k)2616 1438 y Ft(=)d Fp(")2767 1453 y Fn(k)2816 1438 y Ft(\()p Fp(z)t Ft(\))g(=)g(0)k(if)g Fp(z)37 b Ft(b)s(elongs)113 1557 y(to)c(some)g Fp(D)s Ft(\()p Fp(m)696 1572 y Fs(1)741 1557 y Fp(;)17 b(:)g(:)g(:)d(;)j(m) 1050 1572 y Fn(k)1099 1557 y Ft(\))33 b(with)g Fp(m)1485 1572 y Fn(k)1562 1557 y Ft(=)28 b(1,)33 b Fp(")1824 1572 y Fn(k)1901 1557 y Ft(=)28 b(1)33 b(otherwise.)313 1694 y(Starting)k(from)g(Lemma)g(5.11,)g(w)m(e)i(use)g(\(5.26\))d(to)i(deal) g(with)f Fo(=)p Fp(m)17 b(')2984 1709 y Fs(1)3029 1694 y Ft(\()p Fp(z)30 b Ft(+)c(2\),)38 b(\(5.27\))113 1813 y(to)j(deal)g(with)g Fo(=)p Fp(m)17 b(')937 1828 y Fs(1)982 1813 y Ft(\()p Fp(z)t Ft(\))42 b(when)g Fp(")f Ft(=)h(1,)h(Lemma)c (5.12)i(to)f(deal)i(with)f Fo(=)p Fp(m)17 b(T)3160 1777 y Fn(k)3209 1813 y Fp(')3274 1828 y Fs(1)3319 1813 y Ft(\()p Fp(z)31 b Fo(\000)d Ft(1\),)113 1933 y(Lemma)h(5.13)f(to)i (deal)g(with)f Fo(=)p Fp(m)17 b(T)1469 1897 y Fn(k)1518 1933 y Fp(')1583 1948 y Fs(1)1628 1933 y Ft(\()p Fp(z)j Ft(+)15 b(1\),)30 b(and)h(also)e(with)h Fo(=)p Fp(m)17 b(T)d(')2896 1948 y Fs(1)2941 1933 y Ft(\()p Fp(z)t Ft(\))30 b(when)g Fp(")3401 1948 y Fs(1)3474 1933 y Ft(=)e(0,)113 2052 y Fp(")159 2067 y Fs(2)232 2052 y Ft(=)g(1.)44 b(This)34 b(giv)m(es)113 2283 y Fr(Lemma)39 b(5.14)91 b Fe(F)-8 b(or)33 b Fp(z)f Fo(2)c Ft(\001)22 b Fo(n)g Ft([0)p Fp(;)17 b Ft(1])32 b Fe(w)m(e)h(ha)m(v)m(e)252 2572 y Fo(=)p Fp(m)17 b Fo(B)s Ft(\()p Fp(z)t Ft(\))28 b(=)g(\(1)22 b Fo(\000)h Fp(")1014 2587 y Fs(1)1059 2572 y Ft(\))17 b Fo(=)p Fp(m)f(')1355 2587 y Fs(1)1400 2572 y Ft(\()p Fp(z)t Ft(\))22 b(+)h Fo(=)p Fp(m)17 b(')1891 2587 y Fs(1)1936 2572 y Ft(\()p Fp(z)26 b Ft(+)d(1\))f(+)g([1)g Fo(\000)h Fp(")2603 2587 y Fs(2)2648 2572 y Ft(\(1)e Fo(\000)i Fp(")2904 2587 y Fs(1)2949 2572 y Ft(\)])17 b Fo(=)p Fp(m)f(T)e(')3345 2587 y Fs(1)3390 2572 y Ft(\()p Fp(z)t Ft(\))647 2752 y(+)747 2657 y Fk(X)746 2872 y Fn(k)r(>)p Fs(1)909 2752 y Fo(=)p Fp(m)j(T)1157 2711 y Fn(k)1206 2752 y Fp(')1271 2767 y Fs(1)1316 2752 y Ft(\()p Fp(z)t Ft(\))22 b(+)h Fp(r)1611 2767 y Fs(0)1655 2752 y Ft(\()p Fp(z)t Ft(\))g(+)f Fp(r)1950 2767 y Fs(1)1995 2752 y Ft(\()p Fp(z)t Ft(\))113 3136 y Fe(and)961 3306 y Fo(j)p Fp(r)1034 3321 y Fs(1)1078 3306 y Ft(\()p Fp(z)t Ft(\))p Fo(j)28 b(\024)g Fp(C)7 b Fo(j)17 b(=)p Fp(m)g(z)t Fo(j)g Ft(log\(1)22 b(+)h Fo(j)17 b(=)p Fp(m)f(z)t Fo(j)2398 3265 y Fm(\000)p Fs(1)2505 3306 y Ft(\))p Fo(k)p Fp(')2659 3321 y Fs(1)2704 3306 y Fo(k)27 b Fp(:)313 3610 y Ft(W)-8 b(e)33 b(next)g(recall)g(that,)g(b)m(y)g(Prop)s(osition)g(4.1,)f(w)m(e) h(ha)m(v)m(e,)h(when)g Fp(z)e Fo(2)c Fp(H)8 b Ft(,)33 b Fp(k)e Fo(\025)d Ft(1)806 3900 y Fo(j)17 b(=)p Fp(m)f(T)1098 3859 y Fn(k)1147 3900 y Fp(')1212 3915 y Fs(1)1257 3900 y Ft(\()p Fp(z)t Ft(\))p Fo(j)28 b(\024)g Fp(C)7 b Fo(j)p Fp(z)t Fo(j)17 b Ft(log\(1)22 b(+)h Fo(j)p Fp(z)t Fo(j)2192 3859 y Fm(\000)p Fs(1)2299 3900 y Ft(\))17 b(sup)2360 3984 y Fn(D)2425 3994 y Ff(1)2521 3900 y Fo(j)p Fp(T)2621 3859 y Fn(k)r Fm(\000)p Fs(1)2771 3900 y Fp(')2836 3915 y Fs(1)2881 3900 y Fo(j)28 b Fp(:)437 b Ft(\(5)p Fp(:)p Ft(28\))313 4332 y Fr(5.3.6)28 b Ft(In)h(the)f(follo)m(wing)g(t)m(w)m (o)g(steps,)i(w)m(e)e(obtain)g(for)h Fo(=)p Fp(m)17 b(T)2577 4296 y Fn(k)2626 4332 y Fp(')2691 4347 y Fs(1)2736 4332 y Ft(\()p Fp(z)t Ft(\))28 b(an)g(appro)m(ximation)113 4452 y(similar)55 b(to)h(Prop)s(osition)f(4.4,)61 b(\014rst)c(for)f (large)g Fp(k)j Ft(\(Prop)s(osition)c(5.15\),)61 b(and)56 b(then)h(for)113 4571 y(small)42 b Fp(k)k Ft(\(Prop)s(osition)c (\(5.16\).)72 b(Here)43 b(w)m(e)h(assume)f(that)f Fp(k)47 b Fo(\025)e Ft(1,)g Fp(m)2835 4586 y Fs(1)2880 4571 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)3189 4586 y Fn(k)3282 4571 y Fo(\025)44 b Ft(1)f(and)113 4691 y Fp(z)159 4706 y Fs(0)232 4691 y Fo(2)28 b Fp(D)s Ft(\()p Fp(m)537 4706 y Fs(1)582 4691 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)891 4706 y Fn(k)940 4691 y Ft(\))33 b(and)h(let)f Fp(l)c(>)f Ft(0)33 b(and)h Fp(')28 b Ft(=)g Fp(T)2061 4655 y Fn(l)2092 4691 y Fp(')2157 4706 y Fs(1)2202 4691 y Ft(.)113 4921 y Fr(Prop)s(osition) 46 b(5.15)97 b Fe(F)-8 b(or)39 b Fp(k)i Fo(\025)e Ft(1)p Fe(,)h Fp(m)1639 4936 y Fs(1)1684 4921 y Fp(;)17 b(:)g(:)g(:)d(;)j(m) 1993 4936 y Fn(k)2080 4921 y Fo(\025)39 b Ft(1)p Fe(,)h Fp(z)2360 4936 y Fs(0)2443 4921 y Fo(2)e Fp(D)s Ft(\()p Fp(m)2758 4936 y Fs(1)2804 4921 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)3113 4936 y Fn(k)3162 4921 y Ft(\))p Fe(,)40 b Fp(l)g(>)e Ft(0)h Fe(w)m(e)113 5040 y(ha)m(v)m(e)448 5310 y Fo(j)17 b(=)p Fp(mT)724 5269 y Fn(k)r Fs(+)p Fn(l)859 5310 y Fp(')924 5325 y Fs(1)969 5310 y Ft(\()p Fp(z)1054 5325 y Fs(0)1099 5310 y Ft(\))22 b Fo(\000)h Ft([)p Fo(=)p Fp(m)16 b(T)1535 5269 y Fn(l)1566 5310 y Fp(')1631 5325 y Fs(1)1676 5310 y Ft(\()p Fp(z)1761 5325 y Fn(k)1810 5310 y Ft(\)]\()p Fp(p)1966 5325 y Fn(k)r Fm(\000)p Fs(1)2138 5310 y Fo(\000)23 b Fp(q)2282 5325 y Fn(k)r Fm(\000)p Fs(1)2450 5310 y Fo(<)p Fp(e)17 b(z)2631 5325 y Fs(0)2676 5310 y Ft(\))p Fo(j)679 5529 y(\024)28 b Fp(C)7 b(q)910 5487 y Fm(\000)p Fs(1)906 5560 y Fn(k)1018 5529 y Fo(j)17 b(=)p Fp(m)f(z)1284 5544 y Fn(k)1333 5529 y Fo(j)h Ft(log\(1)22 b(+)h Fo(j)17 b(=)p Fp(m)f(z)1983 5544 y Fn(k)2032 5529 y Fo(j)2060 5488 y Fm(\000)p Fs(1)2167 5529 y Ft(\))22 b Fo(\002)2328 5389 y Fk(\032)2419 5475 y Fo(k)p Fp(T)2541 5438 y Fn(l)p Fm(\000)p Fs(2)2673 5475 y Fp(')2738 5490 y Fs(1)2783 5475 y Fo(k)100 b Fe(if)33 b Fp(l)c(>)f Ft(1)p Fe(,)2419 5594 y Fo(k)p Fp(')2534 5609 y Fs(1)2579 5594 y Fo(k)304 b Fe(if)33 b Fp(l)c Ft(=)f(1)p Fe(.)p eop %%Page: 45 45 45 44 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(45)113 383 y Fl(Pr)-5 b(o)g(of.)34 b Ft(As)g(w)m(e)f(did)h(in)f(the)h (pro)s(of)f(of)h(Prop)s(osition)e(4.4)g(w)m(e)i(write)274 721 y Fp(T)346 679 y Fn(k)395 721 y Fp(')p Ft(\()p Fp(z)545 736 y Fs(0)590 721 y Ft(\))27 b(=)h(\()p Fp(p)850 736 y Fn(k)r Fm(\000)p Fs(1)1023 721 y Fo(\000)23 b Fp(q)1167 736 y Fn(k)r Fm(\000)p Fs(1)1318 721 y Fp(z)1364 736 y Fs(0)1409 721 y Ft(\))p Fp(')p Ft(\()p Fp(z)1598 736 y Fn(k)1647 721 y Ft(\))f(+)1858 596 y Fn(k)1808 626 y Fk(X)1811 838 y Fn(j)t Fs(=1)1952 721 y Ft(\()p Fp(p)2041 736 y Fn(j)t Fm(\000)p Fs(1)2207 721 y Fo(\000)g Fp(q)2350 736 y Fn(j)t Fm(\000)p Fs(1)2495 721 y Fp(z)2541 736 y Fs(0)2586 721 y Ft(\))p Fp(T)2697 679 y Fn(k)r Fm(\000)p Fn(j)2845 721 y Fp(')p Ft(\()p Fp(z)2995 736 y Fn(j)3059 721 y Fo(\000)h Ft(1\))650 1084 y(+)800 960 y Fn(k)750 990 y Fk(X)753 1202 y Fn(j)t Fs(=1)894 1084 y Ft(\()p Fp(p)983 1099 y Fn(j)t Fm(\000)p Fs(1)1149 1084 y Fo(\000)g Fp(q)1293 1099 y Fn(j)t Fm(\000)p Fs(1)1437 1084 y Fp(z)1483 1099 y Fs(0)1528 1084 y Ft(\))p Fp(")1613 1099 y Fn(j)1655 1084 y Fp(T)1727 1043 y Fn(k)r Fm(\000)p Fn(j)1876 1084 y Fp(')p Ft(\()p Fp(z)2026 1099 y Fn(j)2090 1084 y Ft(+)g(1\))650 1448 y(+)800 1324 y Fn(k)750 1354 y Fk(X)753 1566 y Fn(j)t Fs(=1)894 1448 y Ft(\()p Fp(p)983 1463 y Fn(j)t Fm(\000)p Fs(2)1149 1448 y Fo(\000)g Fp(q)1293 1463 y Fn(j)t Fm(\000)p Fs(2)1437 1448 y Fp(z)1483 1463 y Fs(0)1528 1448 y Ft(\))p Fp(R)1644 1407 y Fs(\()p Fn(m)1746 1417 y Fj(j)1782 1407 y Fs(\))1818 1448 y Ft(\()p Fp(T)1929 1407 y Fn(k)r Fm(\000)p Fn(j)2077 1448 y Fp(')p Ft(\)\()p Fp(z)2266 1463 y Fn(j)t Fm(\000)p Fs(1)2410 1448 y Ft(\))656 1733 y(=)28 b(\()p Fp(p)850 1748 y Fn(k)r Fm(\000)p Fs(1)1023 1733 y Fo(\000)23 b Fp(q)1167 1748 y Fn(k)r Fm(\000)p Fs(1)1318 1733 y Fp(z)1364 1748 y Fs(0)1409 1733 y Ft(\))p Fp(')p Ft(\()p Fp(z)1598 1748 y Fn(k)1647 1733 y Ft(\))f(+)1829 1708 y(^)1808 1733 y Fp(R)1885 1681 y Fs([)p Fn(k)r Fs(])1884 1760 y(1)1978 1733 y Ft(\()p Fp(')p Ft(\)\()p Fp(z)2206 1748 y Fs(0)2251 1733 y Ft(\))g(+)2433 1708 y(^)2411 1733 y Fp(R)2488 1681 y Fs([)p Fn(k)r Fs(])2487 1760 y(2)2582 1733 y Ft(\()p Fp(')p Ft(\)\()p Fp(z)2810 1748 y Fs(0)2854 1733 y Ft(\))g(+)3036 1708 y(^)3015 1733 y Fp(R)3092 1681 y Fs([)p Fn(k)r Fs(])3091 1760 y(0)3185 1733 y Ft(\()p Fp(')p Ft(\)\()p Fp(z)3413 1748 y Fs(0)3458 1733 y Ft(\))3402 1853 y(\(5)p Fp(:)p Ft(29\))113 1972 y(where)335 2212 y Fp(R)412 2171 y Fs(\()p Fn(m)514 2181 y Fj(j)550 2171 y Fs(\))586 2212 y Ft(\()p Fp( )t Ft(\)\()p Fp(z)818 2227 y Fn(j)t Fm(\000)p Fs(1)961 2212 y Ft(\))28 b(=)1147 2117 y Fk(X)1133 2331 y Fn(m)p Fm(\025)p Fs(1)1322 2212 y Fp( )1391 2171 y Fm(0)1419 2212 y Ft(\()p Fo(\000)p Fp(m)p Ft(\))22 b(+)h Fp(z)1829 2227 y Fn(j)t Fm(\000)p Fs(1)2225 2117 y Fk(X)1990 2336 y Fn(m)p Fm(\025)p Fs(1)13 b Fn(;)g Fm(j)p Fn(m)p Fm(\000)p Fn(m)2441 2346 y Fj(j)2478 2336 y Fm(j\024)p Fs(1)2620 2212 y Fp( )t Ft(\()p Fo(\000)p Fp(m)p Ft(\))1022 2510 y Fo(\000)23 b Fp(z)1168 2525 y Fn(j)t Fm(\000)p Fs(1)1564 2415 y Fk(X)1329 2634 y Fn(m)p Fm(\025)p Fs(1)13 b Fn(;)g Fm(j)p Fn(m)p Fm(\000)p Fn(m)1780 2644 y Fj(j)1817 2634 y Fm(j)p Fn(>)p Fs(1)1943 2510 y Ft([)p Fp( )t Ft(\()p Fp(z)2125 2525 y Fn(j)2188 2510 y Ft(+)22 b Fp(m)2374 2525 y Fn(j)2439 2510 y Fo(\000)h Fp(m)p Ft(\))f Fo(\000)h Fp( )t Ft(\()p Fo(\000)p Fp(m)p Ft(\)])k Fp(:)3402 2410 y Ft(\(5)p Fp(:)p Ft(30\))113 2868 y(W)-8 b(e)34 b(will)e(use)i(rep)s(eatedly)g(the)f(follo)m(wing)g (inequalities)989 3112 y Fo(j)17 b(=)p Fp(m)p Ft(\()p Fp(w)1303 3127 y Fs(1)1348 3112 y Fp(w)1419 3127 y Fs(2)1464 3112 y Ft(\))p Fo(j)27 b(\024)h(j)p Fp(w)1762 3127 y Fs(1)1807 3112 y Fo(jj)17 b(=)p Fp(m)f(w)2126 3127 y Fs(2)2171 3112 y Fo(j)22 b Ft(+)g Fo(j)p Fp(w)2419 3127 y Fs(2)2464 3112 y Fo(jj)17 b(=)p Fp(m)f(w)2783 3127 y Fs(1)2828 3112 y Fo(j)27 b Fp(;)491 b Ft(\(5)p Fp(:)p Ft(31\))880 3262 y Fo(j)p Fp(p)958 3277 y Fn(j)t Fm(\000)p Fs(1)1124 3262 y Fo(\000)22 b Fp(q)1267 3277 y Fn(j)t Fm(\000)p Fs(1)1412 3262 y Fp(z)1458 3277 y Fs(0)1503 3262 y Fo(j)27 b(\024)h Fp(cq)1754 3219 y Fm(\000)p Fs(1)1750 3290 y Fn(j)1889 3262 y Fp(;)1485 b Ft(\(5)p Fp(:)p Ft(32\))626 3431 y Fo(j)17 b(=)p Fp(m)p Ft(\()p Fp(p)919 3446 y Fn(j)t Fm(\000)p Fs(1)1085 3431 y Fo(\000)22 b Fp(q)1228 3446 y Fn(j)t Fm(\000)p Fs(1)1373 3431 y Fp(z)1419 3446 y Fs(0)1464 3431 y Ft(\))p Fo(j)27 b Ft(=)h Fp(q)1707 3446 y Fn(j)t Fm(\000)p Fs(1)1852 3431 y Fo(j)17 b(=)p Fp(m)f(z)2118 3446 y Fs(0)2163 3431 y Fo(j)27 b(\024)h Fp(C)7 b(q)2445 3446 y Fn(j)t Fm(\000)p Fs(1)2590 3431 y Fp(q)2638 3388 y Fm(\000)p Fs(2)2634 3461 y Fn(k)2745 3431 y Fo(j)17 b(=)p Fp(m)g(z)3012 3446 y Fn(k)3061 3431 y Fo(j)27 b Fp(:)258 b Ft(\(5)p Fp(:)p Ft(33\))113 3675 y(By)33 b(Prop)s(osition)f (4.1,)g(as)i Fp(l)29 b(>)f Ft(0,)33 b(for)g(1)28 b Fo(\024)g Fp(j)33 b Fo(\024)28 b Fp(k)37 b Ft(w)m(e)c(ha)m(v)m(e)1034 3920 y Fo(j)p Fp(T)1134 3879 y Fn(k)r Fm(\000)p Fn(j)1282 3920 y Fp(')p Ft(\()p Fp(z)1432 3935 y Fn(j)1496 3920 y Fo(\000)23 b Ft(1\))p Fo(j)k(\024)h Fp(C)c Ft(sup)1946 4004 y Fn(D)2011 4014 y Ff(1)2107 3920 y Fo(j)p Fp(T)2207 3879 y Fn(k)r Fm(\000)p Fn(j)t Fs(+)p Fn(l)p Fm(\000)p Fs(1)2543 3920 y Fp(')2608 3935 y Fs(1)2653 3920 y Fo(j)j Fp(;)666 b Ft(\(5)p Fp(:)p Ft(34\))113 4228 y(and,)34 b(b)m(y)f(Lemma)f(5.12,)303 4473 y Fo(j)17 b(=)p Fp(m)g(T)596 4432 y Fn(k)r Fm(\000)p Fn(j)744 4473 y Fp(')p Ft(\()p Fp(z)894 4488 y Fn(j)958 4473 y Fo(\000)23 b Ft(1\))p Fo(j)k(\024)h Fp(C)7 b Fo(j)17 b(=)p Fp(m)g(z)1652 4488 y Fn(j)1694 4473 y Fo(j)g Ft(log\(1)22 b(+)g Fo(j)p Fp(z)2151 4488 y Fn(j)2215 4473 y Fo(\000)h Ft(1)p Fo(j)2393 4432 y Fm(\000)p Fs(1)2499 4473 y Ft(\))p Fo(k)p Fp(T)2660 4432 y Fn(k)r Fm(\000)p Fn(j)t Fs(+)p Fn(l)p Fm(\000)p Fs(1)2997 4473 y Fp(')3062 4488 y Fs(1)3107 4473 y Fo(k)k Fp(:)190 b Ft(\(5)p Fp(:)p Ft(35\))113 4718 y(W)-8 b(e)34 b(observ)m(e)g(that)f(for)g Fp(j)h(<)28 b(k)d Fo(\000)e Ft(1)561 4962 y(log)q(\(1)e(+)i Fo(j)p Fp(z)974 4977 y Fn(j)1038 4962 y Fo(\000)g Ft(1)p Fo(j)1216 4921 y Fm(\000)p Fs(1)1322 4962 y Ft(\))28 b Fo(\024)g Fp(C)c Ft(log\(1)e(+)h Fo(j)p Fp(z)2002 4977 y Fn(j)t Fs(+1)2145 4962 y Fo(j)2173 4921 y Fm(\000)p Fs(1)2279 4962 y Ft(\))28 b Fo(\024)g Fp(C)c Ft(log\(1)e(+)h Fp(m)2972 4977 y Fn(j)t Fs(+2)3115 4962 y Ft(\))k Fp(;)113 5207 y Ft(and)34 b(one)g(also)f(has) 443 5428 y(log)q(\(1)22 b(+)g Fo(j)p Fp(z)856 5443 y Fn(k)r Fm(\000)p Fs(1)1030 5428 y Fo(\000)g Ft(1)p Fo(j)1207 5387 y Fm(\000)p Fs(1)1314 5428 y Ft(\))27 b Fo(\024)h Fp(C)c Ft(log)q(\(1)e(+)g Fo(j)p Fp(z)1993 5443 y Fn(k)2042 5428 y Fo(j)2070 5387 y Fm(\000)p Fs(1)2177 5428 y Ft(\))27 b Fo(\024)h Fp(C)c Ft(log)q(\(1)e(+)g Fo(j)17 b(=)p Fp(m)g(z)3049 5443 y Fn(k)3098 5428 y Fo(j)3126 5387 y Fm(\000)p Fs(1)3233 5428 y Ft(\))27 b Fp(;)545 5589 y Ft(log)q(\(1)22 b(+)g Fo(j)p Fp(z)958 5604 y Fn(k)1030 5589 y Fo(\000)g Ft(1)p Fo(j)1207 5548 y Fm(\000)p Fs(1)1314 5589 y Ft(\))27 b Fo(\024)h Ft(log)q(\(1)22 b(+)g Fo(j)17 b(=)p Fp(m)g(z)2091 5604 y Fn(k)2140 5589 y Fo(j)2168 5548 y Fm(\000)p Fs(1)2274 5589 y Ft(\))28 b Fp(:)p eop %%Page: 46 46 46 45 bop 113 144 a Ft(46)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y(Th)m(us,)34 b(from)f(\(5.31\){\(5.35\))d(w)m(e)k (get)508 666 y Fo(j)17 b(=)p Fp(m)750 641 y Ft(^)729 666 y Fp(R)806 615 y Fs([)p Fn(k)r Fs(])805 693 y(1)899 666 y Ft(\()p Fp(')p Ft(\)\()p Fp(z)1127 681 y Fs(0)1172 666 y Ft(\))p Fo(j)27 b(\024)h Fp(C)7 b(q)1497 624 y Fm(\000)p Fs(1)1493 696 y Fn(k)1604 666 y Fo(j)17 b(=)p Fp(m)g(z)1871 681 y Fn(k)1920 666 y Fo(j)g Ft(log\(1)22 b(+)g Fo(j)17 b(=)p Fp(m)g(z)2570 681 y Fn(k)2619 666 y Fo(j)2647 625 y Fm(\000)p Fs(1)2753 666 y Ft(\))p Fo(k)p Fp(T)2914 625 y Fn(l)p Fm(\000)p Fs(1)3047 666 y Fp(')3112 681 y Fs(1)3157 666 y Fo(k)27 b Fp(:)113 930 y Ft(Similarly)-8 b(,)30 b(b)m(y)k(Lemma)d(5.13,)h(w)m(e)i(ha)m(v)m(e,)f(for)h Fp(j)f(<)c(k)s Ft(,)j Fp(l)e Fo(\025)e Ft(1)33 b(or)g Fp(j)g Fo(\024)c Fp(k)s Ft(,)j Fp(l)e(>)e Ft(1)389 1194 y Fo(j)17 b(=)p Fp(m)f(T)681 1153 y Fn(k)r Fm(\000)p Fn(j)830 1194 y Fp(')p Ft(\()p Fp(z)980 1209 y Fn(j)1044 1194 y Ft(+)23 b(1\))p Fo(j)k(\024)h Fp(C)7 b Fo(j)17 b(=)p Fp(m)f(z)1737 1209 y Fn(j)1780 1194 y Fo(j)h Ft(log\(1)k(+)i Fo(j)p Fp(z)2237 1209 y Fn(j)2279 1194 y Fo(j)2307 1153 y Fm(\000)p Fs(1)2414 1194 y Ft(\))p Fo(k)p Fp(T)2575 1153 y Fn(k)r Fm(\000)p Fn(j)t Fs(+)p Fn(l)p Fm(\000)p Fs(2)2911 1194 y Fp(')2976 1209 y Fs(1)3021 1194 y Fo(k)28 b Fp(;)275 b Ft(\(5)p Fp(:)p Ft(36\))113 1458 y(and,)34 b(b)m(y)f(Prop)s(osition)f(4.2,)1034 1722 y Fo(j)p Fp(T)1134 1681 y Fn(k)r Fm(\000)p Fn(j)1282 1722 y Fp(')p Ft(\()p Fp(z)1432 1737 y Fn(j)1496 1722 y Ft(+)23 b(1\))p Fo(j)k(\024)h Fp(C)c Ft(sup)1946 1805 y Fn(D)2011 1815 y Ff(1)2107 1722 y Fo(j)p Fp(T)2207 1681 y Fn(k)r Fm(\000)p Fn(j)t Fs(+)p Fn(l)p Fm(\000)p Fs(2)2543 1722 y Fp(')2608 1737 y Fs(1)2653 1722 y Fo(j)j Fp(:)666 b Ft(\(5)p Fp(:)p Ft(37\))113 2064 y(On)34 b(the)g(other)f(hand,)h(when)g Fp(j)g Ft(=)28 b Fp(k)s Ft(,)33 b Fp(l)c Ft(=)f(1)33 b(\(i.e.)44 b Fp(T)2074 2028 y Fn(k)r Fm(\000)p Fn(j)2222 2064 y Fp(')28 b Ft(=)g Fp(T)14 b(')2557 2079 y Fs(1)2602 2064 y Ft(\))33 b(one)g(has)1052 2328 y Fo(j)17 b(=)p Fp(m)f(T)e(')1409 2343 y Fs(1)1454 2328 y Ft(\()p Fp(z)1539 2343 y Fn(k)1611 2328 y Ft(+)22 b(1\))p Fo(j)27 b(\024)h Fp(C)7 b Fo(j)17 b(=)p Fp(m)g(z)2304 2343 y Fn(k)2353 2328 y Fo(jk)p Fp(T)d(')2568 2343 y Fs(1)2613 2328 y Fo(k)27 b Fp(;)1245 2477 y Fo(j)p Fp(T)14 b(')1410 2492 y Fs(1)1454 2477 y Ft(\()p Fp(z)1539 2492 y Fn(k)1611 2477 y Ft(+)22 b(1\))p Fo(j)27 b(\024)h Fp(C)7 b Fo(k)p Fp(T)14 b(')2224 2492 y Fs(1)2269 2477 y Fo(k)28 b Fp(;)113 2741 y Ft(\(note)i(that)f(as)g Fp(T)14 b(')837 2756 y Fs(1)910 2741 y Fo(2)28 b Fp(E)6 b Ft(\([0)p Fp(;)17 b Ft(2)p Fp(=)p Ft(3]\),)26 b Fp(z)1563 2756 y Fn(k)1627 2741 y Ft(+)14 b(1)30 b(is)g(w)m(ell)f(separated)h(from)f(the)h(b)s (oundary\).)42 b(This)113 2861 y(giv)m(es)163 3179 y Fo(j)17 b(=)p Fp(m)405 3154 y Ft(^)384 3179 y Fp(R)461 3127 y Fs([)p Fn(k)r Fs(])460 3205 y(2)554 3179 y Ft(\()p Fp(')p Ft(\)\()p Fp(z)782 3194 y Fs(0)826 3179 y Ft(\))p Fo(j)28 b(\024)g Fp(C)7 b(q)1152 3136 y Fm(\000)p Fs(1)1148 3209 y Fn(k)1259 3179 y Fo(j)17 b(=)p Fp(m)f(z)1525 3194 y Fn(k)1575 3179 y Fo(j)h Ft(log\(1)j(+)i Fo(j)p Fp(z)2030 3194 y Fn(k)2079 3179 y Fo(j)2107 3138 y Fm(\000)p Fs(1)2213 3179 y Ft(\))f Fo(\002)2371 3038 y Fk(\032)2463 3124 y Fo(k)p Fp(T)2585 3088 y Fn(l)p Fm(\000)p Fs(2)2717 3124 y Fp(')2782 3139 y Fs(1)2827 3124 y Fo(k)100 b Ft(\()p Fp(l)29 b(>)f Ft(1\))2463 3244 y Fo(k)p Fp(')2578 3259 y Fs(1)2622 3244 y Fo(k)305 b Ft(\()p Fp(l)29 b Ft(=)f(1\))3325 3179 y Fp(:)49 b Ft(\(5)p Fp(:)p Ft(38\))113 3497 y(Finally)32 b(one)i(has)531 3761 y Fo(j)p Fp(R)636 3720 y Fs(\()p Fn(m)738 3730 y Fj(j)774 3720 y Fs(\))810 3761 y Ft(\()p Fp( )t Ft(\)\()p Fp(z)1042 3776 y Fn(j)t Fm(\000)p Fs(1)1185 3761 y Ft(\))p Fo(j)27 b(\024)i Fp(C)7 b Fo(k)p Fp( )t Fo(k)27 b Fp(;)1715 b Ft(\(5)p Fp(:)p Ft(39\))338 3923 y Fo(j)17 b(=)p Fp(m)g(R)636 3881 y Fs(\()p Fn(m)738 3891 y Fj(j)774 3881 y Fs(\))810 3923 y Ft(\()p Fp( )t Ft(\)\()p Fp(z)1042 3938 y Fn(j)t Fm(\000)p Fs(1)1185 3923 y Ft(\))p Fo(j)27 b(\024)i Fp(C)7 b Fo(k)p Fp( )t Fo(k)p Ft([)p Fo(j)17 b(=)p Fp(m)e(z)1925 3938 y Fn(j)t Fm(\000)p Fs(1)2069 3923 y Fo(j)i Ft(log\(1)22 b(+)h Fp(m)2540 3938 y Fn(j)2582 3923 y Ft(\))f(+)g Fo(<)p Fp(e)17 b(z)2923 3938 y Fn(j)t Fm(\000)p Fs(1)3068 3923 y Fo(j)g(=)p Fp(m)f(z)3334 3938 y Fn(j)3376 3923 y Fo(j)p Ft(])1279 4072 y Fo(\024)29 b Fp(cm)1515 4029 y Fm(\000)p Fs(1)1515 4100 y Fn(j)1622 4072 y Fo(j)17 b(=)p Fp(m)f(z)1888 4087 y Fn(j)1931 4072 y Fo(jk)p Fp( )t Fo(k)26 b Fp(;)1220 b Ft(\(5)p Fp(:)p Ft(40\))113 4336 y(giving)999 4494 y Fo(j)17 b(=)p Fp(m)g(R)1297 4442 y Fs([)p Fn(k)r Fs(])1296 4521 y(0)1390 4494 y Ft(\()p Fp(')p Ft(\)\()p Fp(z)1618 4509 y Fs(0)1663 4494 y Ft(\))p Fo(j)27 b(\024)h Fp(cq)1953 4451 y Fm(\000)p Fs(1)1949 4524 y Fn(k)2060 4494 y Fo(j)17 b(=)p Fp(m)f(z)2326 4509 y Fn(k)2376 4494 y Fo(jk)p Fp(T)2526 4453 y Fn(l)2556 4494 y Fp(')2621 4509 y Fs(1)2666 4494 y Fo(k)27 b Fp(:)631 b Ft(\(5)p Fp(:)p Ft(41\))113 4708 y(T)-8 b(o)33 b(conclude)i(our)e(pro)s(of)g(w)m(e)h(only)f(need)h(to)f (observ)m(e)h(that)693 4972 y Fo(j)17 b(=)p Fp(m)o Ft(\()p Fp(p)985 4987 y Fn(k)r Fm(\000)p Fs(1)1158 4972 y Fo(\000)23 b Fp(q)1302 4987 y Fn(k)r Fm(\000)p Fs(1)1453 4972 y Fp(z)1499 4987 y Fs(0)1544 4972 y Ft(\))17 b Fo(<)p Fp(e)g(')p Ft(\()p Fp(z)1885 4987 y Fn(k)1934 4972 y Ft(\))p Fo(j)27 b(\024)h Fp(C)7 b(q)2259 4929 y Fm(\000)p Fs(1)2255 5002 y Fn(k)2367 4972 y Fo(j)17 b(=)p Fp(m)f(z)2633 4987 y Fn(k)2682 4972 y Fo(jk)p Fp(T)2832 4931 y Fn(l)2862 4972 y Fp(')2927 4987 y Fs(1)2972 4972 y Fo(k)28 b Fp(;)324 b Ft(\(5)p Fp(:)p Ft(42\))113 5236 y(to)33 b(get)g(the)h(desired)g (result.)2396 b Fd(\003)313 5440 y Fr(5.3.7)48 b Ft(F)-8 b(or)47 b Fp(k)56 b Fo(\025)c Ft(1,)f Fp(m)1242 5455 y Fs(1)1287 5440 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)1596 5455 y Fn(k)1697 5440 y Fo(\025)53 b Ft(1,)e Fp(z)2002 5455 y Fs(0)2099 5440 y Fo(2)h Fp(D)s Ft(\()p Fp(m)2428 5455 y Fs(1)2474 5440 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)2783 5455 y Fn(k)2832 5440 y Ft(\))47 b(w)m(e)h(no)m(w)h(consider)113 5559 y Fo(=)p Fp(m)17 b(T)361 5523 y Fn(k)410 5559 y Fp(')475 5574 y Fs(1)520 5559 y Ft(\()p Fp(z)605 5574 y Fs(0)650 5559 y Ft(\))33 b(\(i.e.)43 b(the)34 b(case)g Fp(l)29 b Ft(=)f(0)33 b(left)h(out)f(from)f(Prop)s(osition)h(5.15\))f (:)p eop %%Page: 47 47 47 46 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(47)113 383 y Fr(Prop)s(osition)39 b(5.16)58 b Fe(F)-8 b(or)33 b Fp(k)d Fo(\025)e Ft(1)p Fe(,)33 b Fp(m)1558 398 y Fs(1)1603 383 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)1912 398 y Fn(k)1989 383 y Fo(\025)28 b Ft(1)p Fe(,)33 b Fp(z)2251 398 y Fs(0)2323 383 y Fo(2)28 b Fp(D)s Ft(\()p Fp(m)2628 398 y Fs(1)2674 383 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)2983 398 y Fn(k)3031 383 y Ft(\))33 b Fe(w)m(e)h(ha)m(v)m(e)169 570 y Fo(j)17 b(=)p Fp(mT)445 529 y Fn(k)494 570 y Fp(')559 585 y Fs(1)604 570 y Ft(\()p Fp(z)689 585 y Fs(0)734 570 y Ft(\))22 b Fo(\000)g Ft(\()p Fp(p)983 585 y Fn(k)r Fm(\000)p Fs(1)1156 570 y Fo(\000)h Fp(q)1300 585 y Fn(k)r Fm(\000)p Fs(1)1468 570 y Fo(<)p Fp(e)17 b(z)1649 585 y Fs(0)1694 570 y Ft(\)[\(1)k Fo(\000)i Fp(")2017 585 y Fn(k)r Fs(+1)2167 570 y Ft(\))17 b Fo(=)p Fp(m)f(')2463 585 y Fs(1)2508 570 y Ft(\()p Fp(z)2593 585 y Fn(k)2642 570 y Ft(\))22 b(+)h Fp(")2849 585 y Fn(k)2915 570 y Fo(=)p Fp(m)17 b(')3156 585 y Fs(1)3201 570 y Ft(\()p Fp(z)3286 585 y Fn(k)3357 570 y Ft(+)23 b(1\)])p Fo(j)401 732 y(\024)28 b Fp(C)7 b(q)632 690 y Fm(\000)p Fs(1)628 762 y Fn(k)739 732 y Fo(j)17 b(=)p Fp(m)g(z)1006 747 y Fn(k)1055 732 y Fo(j)g Ft(log\(1)22 b(+)g Fo(j)17 b(=)p Fp(m)g(z)1705 747 y Fn(k)1754 732 y Fo(j)1782 691 y Fm(\000)p Fs(1)1889 732 y Ft(\))p Fo(k)p Fp(')2043 747 y Fs(1)2087 732 y Fo(k)27 b Fp(:)113 938 y Fl(Pr)-5 b(o)g(of.)34 b Ft(W)-8 b(e)34 b(no)m(w)g(write)e(\(with)h Fp(")1357 902 y Fm(0)1357 966 y Fn(k)1434 938 y Ft(=)28 b(0)33 b(if)h Fp(m)1801 953 y Fn(k)1878 938 y Fo(\024)28 b Ft(2,)33 b Fp(")2140 902 y Fm(0)2140 966 y Fn(k)2217 938 y Ft(=)28 b(1)33 b(otherwise\))297 1128 y Fp(T)369 1087 y Fn(k)418 1128 y Fp(')483 1143 y Fs(1)527 1128 y Ft(\()p Fp(z)612 1143 y Fs(0)657 1128 y Ft(\))28 b(=)g(\()p Fp(p)918 1143 y Fn(k)r Fm(\000)p Fs(1)1091 1128 y Fo(\000)22 b Fp(q)1234 1143 y Fn(k)r Fm(\000)p Fs(1)1386 1128 y Fp(z)1432 1143 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1749 y Fp(z)1553 1764 y Fs(0)1598 1749 y Ft(\))p Fp(")1683 1764 y Fn(j)1725 1749 y Fp(T)1797 1708 y Fn(k)r Fm(\000)p Fn(j)1945 1749 y Fp(')2010 1764 y Fs(1)2055 1749 y Ft(\()p Fp(z)2140 1764 y Fn(j)2204 1749 y Ft(+)h(1\))718 2113 y(+)868 1989 y Fn(k)818 2019 y Fk(X)821 2231 y Fn(j)t Fs(=1)962 2113 y Ft(\()p Fp(p)1051 2128 y Fn(j)t Fm(\000)p Fs(2)1217 2113 y Fo(\000)f Fp(q)1360 2128 y Fn(j)t Fm(\000)p Fs(2)1505 2113 y Fp(z)1551 2128 y Fs(0)1596 2113 y Ft(\))1656 2088 y(~)1635 2113 y Fp(R)1712 2072 y Fs(\()p Fn(m)1814 2082 y Fj(j)1850 2072 y Fs(\))1886 2113 y Ft(\()p Fp(T)1997 2072 y Fn(k)r Fm(\000)p Fn(j)2145 2113 y Fp(')2210 2128 y Fs(1)2255 2113 y Ft(\)\()p Fp(z)2379 2128 y Fn(j)t Fm(\000)p Fs(1)2523 2113 y Ft(\))3402 1671 y(\(5)p Fp(:)p Ft(43\))113 2418 y(where)423 2393 y(~)401 2418 y Fp(R)478 2382 y Fs(\()p Fn(m)580 2392 y Fj(j)617 2382 y Fs(\))653 2418 y Ft(\()p Fp(T)764 2382 y Fn(k)r Fm(\000)p Fn(j)912 2418 y Fp(')977 2433 y Fs(1)1022 2418 y Ft(\))27 b(=)i Fp(R)1271 2382 y Fs(\()p 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Fp(R)812 3601 y Fs([)p Fn(k)r Fs(])811 3679 y(1)905 3653 y Ft(\()p Fp(')1009 3668 y Fs(1)1054 3653 y Ft(\))g(is)g(dealt)g(with)g(as)g(in)g (\(5.31\){\(5.35\))e(to)h(get)461 3856 y Fo(j)17 b(=)p Fp(m)f(R)758 3804 y Fs([)p Fn(k)r Fs(])757 3882 y(1)852 3856 y Ft(\()p Fp(')956 3871 y Fs(1)1001 3856 y Ft(\)\()p Fp(z)1125 3871 y Fs(0)1169 3856 y Ft(\))p Fo(j)27 b(\024)h Fp(C)7 b(q)1494 3813 y Fm(\000)p Fs(1)1490 3886 y Fn(k)1602 3856 y Fo(j)17 b(=)p Fp(m)f(z)1868 3871 y Fn(k)1917 3856 y Fo(j)h Ft(log\(1)22 b(+)h Fo(j)17 b(=)p Fp(m)f(z)2567 3871 y Fn(k)2616 3856 y Fo(j)2644 3815 y Fm(\000)p Fs(1)2751 3856 y Ft(\))p Fo(k)p Fp(')2905 3871 y Fs(1)2949 3856 y Fo(k)28 b Fp(:)347 b Ft(\(5)p Fp(:)p Ft(45\))113 4062 y(The)34 b(middle)f(sum)g Fp(R)933 4010 y Fs([)p Fn(k)r Fs(])932 4089 y(2)1026 4062 y Ft(\()p Fp(')1130 4077 y Fs(1)1175 4062 y Ft(\))g(satis\014es)h(the)f(same)g(estimate,)f(pro)m (v)m(ed)i(as)f(in)g(\(5.36\){\(5.38\).)313 4182 y(By)f(\(5.26\))g(w)m (e)h(ha)m(v)m(e)163 4364 y Fo(j)17 b(=)p Fp(m)p Ft([\()p Fp(p)484 4379 y Fn(k)r Fm(\000)p Fs(1)645 4364 y Fo(\000)11 b Fp(q)777 4379 y Fn(k)r Fm(\000)p Fs(1)929 4364 y Fp(z)975 4379 y Fs(0)1020 4364 y Ft(\))p Fp(')1124 4379 y Fs(1)1168 4364 y Ft(\()p Fp(z)1253 4379 y Fn(k)1313 4364 y Ft(+)g(2\)])p Fo(j)27 b(\024)h Fp(q)1726 4321 y Fm(\000)p Fs(1)1722 4394 y Fn(k)1833 4364 y Ft([)p Fp(C)7 b Fo(j)17 b(=)p Fp(m)f(z)2205 4379 y Fn(k)2255 4364 y Fo(j)h Ft(log\(1)11 b(+)g Fo(j)p Fp(z)2690 4379 y Fn(k)2738 4364 y Fo(j)2766 4322 y Fm(\000)p Fs(1)2872 4364 y Ft(\))g(+)g(1])p Fo(k)p Fp(')3203 4379 y Fs(1)3247 4364 y Fo(k)28 b Fp(:)49 b Ft(\(5)p Fp(:)p Ft(46\))113 4545 y(Similarly)-8 b(,)30 b(if)k Fp(z)694 4560 y Fn(k)782 4545 y Fp(=)-61 b Fo(2)28 b Fp(D)s Ft(\(1\))33 b(\(i.e.)43 b(if)33 b Fp(")1458 4560 y Fn(k)r Fs(+1)1636 4545 y Ft(=)28 b(1\))33 b(w)m(e)h(ha)m(v)m(e) 181 4727 y Fo(j)17 b(=)p Fp(m)p Ft([\()p Fp(p)502 4742 y Fn(k)r Fm(\000)p Fs(1)674 4727 y Fo(\000)23 b Fp(q)818 4742 y Fn(k)r Fm(\000)p Fs(1)969 4727 y Fp(z)1015 4742 y Fs(0)1060 4727 y Ft(\))p Fp(')1164 4742 y Fs(1)1209 4727 y Ft(\()p Fp(z)1294 4742 y Fn(k)1343 4727 y Ft(\)])p Fo(j)k(\024)h Fp(C)7 b(q)1696 4685 y Fm(\000)p Fs(1)1692 4757 y Fn(k)1803 4727 y Fo(j)17 b(=)p Fp(m)g(z)2070 4742 y Fn(k)2119 4727 y Fo(j)g Ft(log\(1)22 b(+)g Fo(j)p Fp(z)2576 4742 y Fn(k)2647 4727 y Fo(\000)h Ft(1)p Fp(=)p Ft(2)p Fo(j)2925 4686 y Fm(\000)p Fs(1)3031 4727 y Ft(\))p Fo(k)p Fp(')3185 4742 y Fs(1)3230 4727 y Fo(k)k Fp(;)67 b Ft(\(5)p Fp(:)p Ft(47\))113 4909 y(according)34 b(to)f(\(5.27\).)42 b(Finally)-8 b(,)32 b(for)h Fp(")28 b Ft(=)g(0)p Fp(;)17 b Ft(1)638 5091 y Fo(j)g(=)p Fp(m)o Ft(\()p Fp(p)930 5106 y Fn(k)r Fm(\000)p Fs(1)1103 5091 y Fo(\000)23 b Fp(q)1247 5106 y Fn(k)r Fm(\000)p Fs(1)1398 5091 y Fp(z)1444 5106 y Fs(0)1489 5091 y Ft(\))17 b Fo(<)p Fp(e)g(')1745 5106 y Fs(1)1789 5091 y Ft(\()p Fp(z)1874 5106 y Fn(k)1946 5091 y Ft(+)22 b Fp(")p Ft(\))p Fo(j)28 b(\024)g Fp(C)7 b(q)2417 5049 y Fm(\000)p Fs(1)2413 5121 y Fn(k)2524 5091 y Fo(j)17 b(=)p Fp(m)g(z)2791 5106 y Fn(k)2840 5091 y Fo(jk)p Fp(')2983 5106 y Fs(1)3027 5091 y Fo(k)28 b Fp(;)269 b Ft(\(5)p Fp(:)p Ft(48\))113 5273 y(and)34 b(from)f(\(5.43\){\(5.48\))d(w)m(e)k(get)f(our)g(result.)1682 b Fd(\003)313 5440 y Fr(5.3.8)34 b Ft(Starting)g(from)g(Lemma)e(5.14,)h (w)m(e)i(no)m(w)g(mak)m(e)e(use)i(of)f(Prop)s(ositions)g(5.15)f(and)113 5559 y(5.16)g(to)f(obtain)h(a)g(simpler)g(appro)m(ximation)e(for)j Fo(=)p Fp(m)17 b Fo(B)s Ft(\()p Fp(z)t Ft(\).)p eop %%Page: 48 48 48 47 bop 113 144 a Ft(48)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y Fr(Prop)s(osition)39 b(5.17)58 b Fe(F)-8 b(or)33 b Fp(k)d Fo(\025)e Ft(0)p Fe(,)33 b Fp(m)1558 398 y Fs(1)1603 383 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)1912 398 y Fn(k)1989 383 y Fo(\025)28 b Ft(1)p Fe(,)33 b Fp(z)2251 398 y Fs(0)2323 383 y Fo(2)28 b Fp(H)8 b Ft(\()p Fp(m)2634 398 y Fs(1)2679 383 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)2988 398 y Fn(k)3037 383 y Ft(\))33 b Fe(w)m(e)h(ha)m(v)m(e)533 649 y Fo(=)p Fp(m)17 b Fo(B)s Ft(\()p Fp(z)862 664 y Fs(0)907 649 y Ft(\))28 b(=)1129 525 y Fn(k)1079 555 y Fk(X)1087 769 y Fn(l)p Fs(=0)1239 649 y Fo(=)p Fp(m)17 b(')1480 664 y Fs(1)1525 649 y Ft(\()p Fp(z)1610 664 y Fn(l)1663 649 y Ft(+)23 b(1\)\()p Fp(p)1941 664 y Fn(l)p Fm(\000)p Fs(1)2095 649 y Fo(\000)f Fp(q)2238 664 y Fn(l)p Fm(\000)p Fs(1)2388 649 y Fo(<)p Fp(e)17 b(z)2569 664 y Fs(0)2614 649 y Ft(\))p Fp(")2699 664 y Fn(l)968 1002 y Ft(+)1068 877 y Fn(k)r Fm(\000)p Fs(1)1069 907 y Fk(X)1077 1122 y Fn(l)p Fs(=0)1230 1002 y Fo(=)p Fp(m)g(')1471 1017 y Fs(1)1516 1002 y Ft(\()p Fp(z)1601 1017 y Fn(l)1632 1002 y Ft(\)\()p Fp(p)1760 1017 y Fn(l)p Fm(\000)p Fs(1)1914 1002 y Fo(\000)23 b Fp(q)2058 1017 y Fn(l)p Fm(\000)p Fs(1)2207 1002 y Fo(<)p Fp(e)17 b(z)2388 1017 y Fs(0)2433 1002 y Ft(\)\(1)22 b Fo(\000)g Fp(")2728 1017 y Fn(l)p Fs(+1)2860 1002 y Ft(\))g(+)h Fp(r)s Ft(\()p Fp(z)3154 1017 y Fs(0)3198 1002 y Ft(\))113 1259 y Fe(with)33 b Fp(")386 1274 y Fs(0)459 1259 y Ft(=)28 b(1)p Fe(,)33 b(and)1137 1379 y Fo(j)p Fp(r)s Ft(\()p 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Ft(\))p Fo(j)27 b(\024)h Fp(C)7 b Fo(j)p Fp(z)1775 2193 y Fs(0)1820 2178 y Fo(j)17 b Ft(log\(1)22 b(+)h Fo(j)p Fp(z)2278 2193 y Fs(0)2322 2178 y Fo(j)2350 2137 y Fm(\000)p Fs(1)2457 2178 y Ft(\))k Fp(:)851 b Ft(\(5)p Fp(:)p Ft(50\))113 2336 y(As)34 b(w)m(e)f(also)g(ha)m(v)m(e,)g(for)h Fp(z)1063 2351 y Fs(0)1136 2336 y Fo(2)28 b Fp(H)8 b Ft(,)1319 2504 y Fo(j)17 b(=)p Fp(m)f(')1604 2519 y Fs(1)1649 2504 y Ft(\()p Fp(z)1734 2519 y Fs(0)1779 2504 y Ft(\))p Fo(j)27 b(\024)h Fp(C)7 b Fo(j)17 b(=)p Fp(m)g(z)2323 2519 y Fs(0)2368 2504 y Fo(j)27 b Fp(;)113 2672 y Ft(w)m(e)34 b(obtain)759 2791 y Fo(j)17 b(=)p Fp(m)f Fo(B)s Ft(\()p Fp(z)t Ft(\))23 b Fo(\000)f(=)p Fp(m)17 b(')1538 2806 y Fs(1)1583 2791 y Ft(\()p Fp(z)1668 2806 y Fs(0)1735 2791 y Ft(+)23 b(1\))p Fo(j)k(\024)h Fp(C)7 b Fo(j)p Fp(z)2236 2806 y Fs(0)2281 2791 y Fo(j)17 b Ft(log\(1)k(+)i Fo(j)p Fp(z)2738 2806 y Fs(0)2783 2791 y Fo(j)2811 2750 y Fm(\000)p Fs(1)2917 2791 y Ft(\))28 b Fp(:)390 b Ft(\(5)p Fp(:)p Ft(51\))113 2949 y(Next)33 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b(T)1711 3369 y Fn(l)1742 3405 y Fp(')1807 3420 y Fs(1)1851 3405 y Ft(\()p Fp(z)1936 3420 y Fs(0)1981 3405 y Ft(\),)44 b Fp(l)g(>)e(k)j Ft(and)e(Prop)s(osition)e(5.16)g(to)h (deal)113 3524 y(with)33 b Fo(=)p Fp(m)17 b(T)588 3488 y Fn(l)619 3524 y Fp(')684 3539 y Fs(1)729 3524 y Ft(\()p Fp(z)814 3539 y Fs(0)859 3524 y Ft(\),)32 b(0)c Fp(<)g(l)h Fo(\024)f Fp(k)s Ft(.)44 b(W)-8 b(e)34 b(ha)m(v)m(e)699 3712 y Fo(j)744 3618 y Fk(X)750 3832 y Fn(l>k)904 3712 y Fo(=)p Fp(m)17 b(T)1152 3671 y Fn(l)1183 3712 y Fp(')1248 3727 y Fs(1)1293 3712 y Ft(\()p Fp(z)1378 3727 y Fs(0)1422 3712 y Ft(\))22 b Fo(\000)h Ft(\()p Fp(p)1672 3727 y Fn(k)r Fm(\000)p Fs(1)1845 3712 y Fo(\000)g Fp(q)1989 3727 y Fn(k)r Fm(\000)p Fs(1)2140 3712 y Fp(z)2186 3727 y Fs(0)2231 3712 y Ft(\))2287 3618 y Fk(X)2295 3832 y Fn(l>)p Fs(0)2447 3712 y Fo(=)p Fp(m)17 b(T)2695 3671 y Fn(l)2726 3712 y Fp(')2791 3727 y Fs(1)2835 3712 y Ft(\()p Fp(z)2920 3727 y Fn(k)2970 3712 y Ft(\))p Fo(j)1489 3975 y(\024)28 b Fp(C)7 b(q)1720 3932 y 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b(w)m(e)h(ha)m(v)m(e)g(in)f(Prop)s(osition)f(5.16)g(and) i(Lemma)e(5.14)399 4762 y Fp(q)447 4719 y Fm(\000)p Fs(1)443 4792 y Fn(l)554 4762 y Fo(j)17 b(=)p Fp(m)f(z)820 4777 y Fn(l)851 4762 y Fo(j)h Ft(log\(1)22 b(+)g Fo(j)17 b(=)p Fp(m)g(z)1501 4777 y Fn(l)1532 4762 y Fo(j)1560 4721 y Fm(\000)p Fs(1)1666 4762 y Ft(\))28 b Fo(\024)g Fp(C)7 b(q)1960 4777 y Fn(l)1991 4762 y Fp(q)2039 4719 y Fm(\000)p Fs(2)2035 4792 y Fn(k)2146 4762 y Fo(j)17 b(=)p Fp(m)g(z)2413 4777 y Fn(k)2462 4762 y Fo(j)g Ft(log\(1)22 b(+)g Fo(j)17 b(=)p Fp(m)g(z)3112 4777 y Fn(l)3142 4762 y Fo(j)3170 4721 y Fm(\000)p Fs(1)3277 4762 y Ft(\))27 b Fp(;)113 4930 y Ft(and)432 5047 y Fn(k)382 5077 y Fk(X)434 5288 y Fs(0)542 5171 y Fp(q)586 5186 y Fn(l)617 5171 y Fp(q)665 5129 y Fm(\000)p Fs(1)661 5201 y Fn(k)789 5171 y Ft(log)q(\(1)21 b(+)i Fo(j)17 b(=)p Fp(m)f(z)1394 5186 y Fn(l)1425 5171 y Fo(j)1453 5130 y Fm(\000)p Fs(1)1560 5171 y Ft(\))27 b Fo(\024)h Fp(C)1876 5047 y Fn(k)1826 5077 y Fk(X)1878 5288 y Fs(0)1987 5171 y Fp(q)2031 5186 y Fn(l)2062 5171 y Fp(q)2110 5129 y Fm(\000)p Fs(1)2106 5201 y Fn(k)2233 5171 y Ft(log)q(\(1)22 b(+)g Fp(C)7 b(q)2698 5129 y Fm(\000)p Fs(2)2694 5201 y Fn(l)2806 5171 y Fp(q)2854 5130 y Fs(2)2850 5196 y Fn(k)2899 5171 y Fo(j)17 b(=)p Fp(m)g(z)3166 5186 y Fn(k)3215 5171 y Fo(j)3243 5130 y Fm(\000)p Fs(1)3350 5171 y Ft(\))1626 5424 y Fo(\024)28 b Fp(C)c Ft(log)q(\(1)e(+)g Fo(j)17 b(=)p Fp(m)g(z)2432 5439 y Fn(k)2481 5424 y Fo(j)2509 5383 y Fm(\000)p Fs(1)2616 5424 y Ft(\))27 b Fp(:)3579 5584 y Fd(\003)p eop %%Page: 49 49 49 48 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(49)313 383 y Fr(5.3.9)33 b Ft(W)-8 b(e)34 b(can)g(\014nally)e (complete)h(the)g Fe(pro)s(of)h(of)f(Theorem)g(5.10)p Ft(.)313 503 y(Let)g Fp(k)e Fo(\025)e Ft(0,)k Fp(m)878 518 y Fs(1)923 503 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)1232 518 y Fn(k)1309 503 y Fo(\025)29 b Ft(1.)44 b(As)34 b(w)m(e)g(already)f (kno)m(w,)g(the)h(domain)e Fp(H)8 b Ft(\()p Fp(m)3215 518 y Fs(1)3260 503 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)3569 518 y Fn(k)3618 503 y Ft(\))113 622 y(meets)32 b Fq(R)43 b Ft(in)33 b(a)f(unique)g(p)s(oin)m(t)g(whic)m(h)h(is)g Fp(p)1712 637 y Fn(k)1761 622 y Fp(=q)1855 637 y Fn(k)1904 622 y Ft(.)44 b(Let)32 b(us)h(denote)g Fp(x)2657 637 y Fs(0)2729 622 y Ft(=)28 b Fp(p)2884 637 y Fn(k)2933 622 y Fp(=q)3027 637 y Fn(k)3109 622 y Ft(and)33 b(consider)113 742 y(its)g(con)m(tin)m(ued)i(fraction)996 967 y Fp(x)1053 924 y Fm(\000)p Fs(1)1053 994 y Fn(i)1187 967 y Ft(=)28 b Fp(m)1379 982 y Fn(i)p Fs(+1)1536 967 y Ft(+)23 b Fp(x)1693 982 y Fn(i)p Fs(+1)1854 967 y Fp(;)72 b Ft(0)27 b Fo(\024)h Fp(i)g(<)h(k)h(;)72 b(x)2543 982 y Fn(k)2620 967 y Ft(=)28 b(0)113 1191 y(\(of)34 b(course)g Fp(x)628 1206 y Fn(i)694 1191 y Ft(is)f(the)h(p)s(oin)m(t)f(of)h(in)m(tersection)f(of)h Fp(H)8 b Ft(\()p Fp(m)2205 1206 y Fn(i)p Fs(+1)2339 1191 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)2648 1206 y Fn(k)2697 1191 y Ft(\))33 b(with)g Fq(R)5 b Ft(\).)313 1311 y(Let)42 b Fp(z)546 1326 y Fs(0)635 1311 y Fo(2)h Fp(H)8 b Ft(\()p Fp(m)961 1326 y Fs(1)1006 1311 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)1315 1326 y Fn(k)1364 1311 y Ft(\),)45 b Fo(=)p Fp(m)17 b(z)1698 1326 y Fs(0)1786 1311 y Fp(>)44 b Ft(0.)71 b(W)-8 b(e)43 b(will)f(then)h(ha)m(v)m(e)g(\()p Fo(\000)p Ft(1\))3115 1275 y Fn(l)3162 1311 y Fo(=)p Fp(m)17 b(z)3384 1326 y Fn(l)3458 1311 y Fp(>)44 b Ft(0,)113 1431 y(0)28 b Fo(\024)g Fp(l)h Fo(\024)g Fp(k)s Ft(.)44 b(On)34 b(the)f(other)h(hand) g(one)f(has,)h(for)f(0)28 b Fo(\024)g Fp(l)h(<)f(k)919 1697 y(")965 1712 y Fn(l)1013 1697 y Fo(=)p Fp(m)17 b(')1254 1712 y Fs(1)1299 1697 y Ft(\()p Fp(x)1395 1712 y Fn(l)1447 1697 y Ft(+)23 b(1)f(+)h(\()p Fo(\000)p Ft(1\))1924 1656 y Fn(l)1954 1697 y Fp(i)p Ft(0\))k(=)i(\()p Fo(\000)p Ft(1\))2415 1656 y Fn(l)2445 1697 y Fp(")2491 1712 y Fn(l)2539 1697 y Ft(log)2715 1629 y(1)p 2696 1674 88 4 v 2696 1765 a Fp(x)2753 1780 y Fn(l)2823 1697 y Fp(;)551 b Ft(\(5)p Fp(:)p Ft(54\))113 1965 y(and)34 b(for)g(0)27 b Fo(\024)h Fp(l)i(<)e(k)d Fo(\000)e Ft(1)461 2225 y(\(1)f Fo(\000)h Fp(")718 2240 y Fn(l)p Fs(+1)850 2225 y Ft(\))17 b Fo(=)p Fp(m)f(')1146 2240 y Fs(1)1191 2225 y Ft(\()p Fp(x)1287 2240 y Fn(l)1339 2225 y Ft(+)23 b(\()p Fo(\000)p Ft(1\))1644 2184 y Fn(l)1674 2225 y Fp(i)p Ft(0\))28 b(=)g(\(1)22 b Fo(\000)g Fp(")2186 2240 y Fn(l)p Fs(+1)2318 2225 y Ft(\)\()p Fo(\000)p Ft(1\))2562 2184 y Fn(l)2592 2225 y Fp(x)2649 2240 y Fn(l)2696 2225 y Ft(log)2923 2158 y(1)p 2854 2202 189 4 v 2854 2293 a Fp(x)2911 2308 y Fn(l)p Fs(+1)3402 2225 y Ft(\(5)p Fp(:)p Ft(55\))113 2516 y(since)420 2475 y Fn(x)465 2486 y Fj(l)p 369 2493 177 4 v 369 2550 a Fs(1)p Fm(\000)p Fn(x)516 2561 y Fj(l)585 2516 y Ft(=)762 2477 y Fs(1)p 702 2493 160 4 v 702 2550 a Fn(x)747 2561 y Fj(l)p Fi(+1)907 2516 y Ft(when)34 b(1)22 b Fo(\000)g Fp(")1384 2531 y Fn(l)p Fs(+1)1544 2516 y Fo(6)p Ft(=)28 b(0.)44 b(\(5.54\))32 b(and)h(\(5.55\))f(imply)g (that)680 2743 y Fn(k)r Fm(\000)p Fs(1)681 2773 y Fk(X)733 2984 y Fs(0)843 2867 y Fo(=)p Fp(m)17 b(')1084 2882 y Fs(1)1128 2867 y Ft(\()p Fp(x)1224 2882 y Fn(l)1277 2867 y Ft(+)23 b(1)e(+)i(\()p Fo(\000)p Ft(1\))1753 2826 y Fn(l)1783 2867 y Fp(i)p Ft(0\)\()p Fp(p)1995 2882 y Fn(l)p Fm(\000)p Fs(1)2150 2867 y Fo(\000)g Fp(q)2294 2882 y Fn(l)p Fm(\000)p Fs(1)2427 2867 y Fp(x)2484 2882 y Fs(0)2528 2867 y Ft(\))p Fp(")2613 2882 y Fn(l)848 3217 y Ft(+)948 3092 y Fn(k)r Fm(\000)p Fs(2)949 3122 y Fk(X)1001 3333 y Fs(0)1110 3217 y Fo(=)p Fp(m)17 b(')1351 3232 y Fs(1)1396 3217 y Ft(\()p Fp(x)1492 3232 y Fn(l)1545 3217 y Ft(+)22 b(\()p Fo(\000)p Ft(1\))1849 3175 y Fn(l)1880 3217 y Fp(i)p Ft(0\)\()p Fp(p)2092 3232 y Fn(l)p Fm(\000)p Fs(1)2246 3217 y Fo(\000)h Fp(q)2390 3232 y Fn(l)p Fm(\000)p Fs(1)2523 3217 y Fp(x)2580 3232 y Fs(0)2624 3217 y Ft(\)\(1)f Fo(\000)g Fp(")2919 3232 y Fn(l)p Fs(+1)3051 3217 y Ft(\))854 3566 y(=)959 3441 y Fn(k)r Fm(\000)p Fs(1)960 3471 y Fk(X)1012 3682 y Fs(0)1121 3566 y Fp(\014)1177 3581 y Fn(l)p Fm(\000)p Fs(1)1310 3566 y Ft(\()p Fp(x)1406 3581 y Fs(0)1451 3566 y Ft(\))17 b(log)1682 3498 y(1)p 1663 3543 88 4 v 1663 3634 a Fp(x)1720 3649 y Fn(l)1790 3566 y Ft(:=)28 b Fp(B)1999 3581 y Fn(f)8 b(inite)2224 3566 y Ft(\()p Fp(p)2313 3581 y Fn(k)2362 3566 y Fp(=q)2456 3581 y Fn(k)2505 3566 y Ft(\))28 b Fp(:)3402 3213 y Ft(\(5)p Fp(:)p Ft(56\))113 3876 y(When)57 b(w)m(e)e(set)g(this)g(expression)h(in)f(Theorem)g (5.10,)k(and)c(compare)f(the)i(result)f(with)113 3996 y(expression)46 b(for)e Fo(=)p Fp(m)17 b(B)5 b Ft(\()p Fp(z)1110 4011 y Fs(0)1155 3996 y Ft(\))44 b(giv)m(en)g(b)m(y)g(Prop)s (osion)g(5.17,)i(w)m(e)f(see)g(that)f(w)m(e)h(ha)m(v)m(e)g(to)f(deal) 113 4116 y(with)33 b(the)h(follo)m(wing)f(expressions.)45 b(F)-8 b(or)32 b(0)c Fo(\024)g Fp(l)i Fo(\024)e Fp(k)d Fo(\000)d Ft(1,)33 b(and)h Fp(")2544 4131 y Fn(l)2603 4116 y Ft(=)28 b(1)33 b(:)113 4340 y Fr(A)200 4355 y Fn(l)258 4340 y Ft(=)28 b Fo(=)p Fp(m)17 b(')604 4355 y Fs(1)649 4340 y Ft(\()p Fp(z)734 4355 y Fn(l)785 4340 y Ft(+)k(1\)\()p Fp(p)1061 4355 y Fn(l)p Fm(\000)p Fs(1)1213 4340 y Fo(\000)g Fp(q)1355 4355 y Fn(l)p Fm(\000)p Fs(1)1505 4340 y Fo(<)p Fp(e)c(z)1686 4355 y Fs(0)1731 4340 y Ft(\))j Fo(\000)h(=)p Fp(m)c(')2129 4355 y Fs(1)2174 4340 y Ft(\()p Fp(x)2270 4355 y Fn(l)2320 4340 y Ft(+)k(1)f(+)h(\()p Fo(\000)p Ft(1\))2791 4299 y Fn(l)2821 4340 y Fp(i)p Ft(0\)\()p Fp(p)3033 4355 y Fn(l)p Fm(\000)p Fs(1)3186 4340 y Fo(\000)g Fp(q)3328 4355 y Fn(l)p Fm(\000)p Fs(1)3461 4340 y Fp(x)3518 4355 y Fs(0)3563 4340 y Ft(\))27 b Fp(:)113 4565 y Ft(F)-8 b(or)33 b(0)28 b Fo(\024)g Fp(l)h Fo(\024)f Fp(k)d Fo(\000)e Ft(2,)33 b(and)h Fp(")1167 4580 y Fn(l)p Fs(+1)1326 4565 y Ft(=)28 b(0)33 b(:)277 4790 y Fr(B)358 4805 y Fn(l)416 4790 y Ft(=)c Fo(=)p Fp(m)17 b(')763 4805 y Fs(1)808 4790 y Ft(\()p Fp(z)893 4805 y Fn(l)923 4790 y Ft(\)\()p Fp(p)1051 4805 y Fn(l)p Fm(\000)p Fs(1)1205 4790 y Fo(\000)23 b Fp(q)1349 4805 y Fn(l)p Fm(\000)p Fs(1)1499 4790 y Fo(<)p Fp(e)17 b(z)1680 4805 y Fs(0)1725 4790 y Ft(\))22 b Fo(\000)g(=)p Fp(m)17 b(')2126 4805 y Fs(1)2171 4790 y Ft(\()p Fp(x)2267 4805 y Fn(l)2320 4790 y Ft(+)22 b(\()p Fo(\000)p Ft(1\))2624 4749 y Fn(l)2655 4790 y Fp(i)p Ft(0\)\()p Fp(p)2867 4805 y Fn(l)p Fm(\000)p Fs(1)3021 4790 y Fo(\000)h Fp(q)3165 4805 y Fn(l)p Fm(\000)p Fs(1)3298 4790 y Fp(x)3355 4805 y Fs(0)3399 4790 y Ft(\))28 b Fp(:)113 5015 y Ft(And)34 b(when)g Fp(")638 5030 y Fn(k)715 5015 y Ft(=)28 b(0)33 b(:)239 5240 y Fr(C)322 5255 y Fn(l)380 5240 y Ft(=)c Fo(=)p Fp(m)17 b(')727 5255 y Fs(1)772 5240 y Ft(\()p Fp(z)857 5255 y Fn(k)928 5240 y Ft(+)22 b(1\)\()p Fp(p)1205 5255 y Fn(k)r Fm(\000)p Fs(1)1378 5240 y Fo(\000)h Fp(q)1522 5255 y Fn(k)r Fm(\000)p Fs(1)1690 5240 y Fo(<)p Fp(e)17 b(z)1871 5255 y Fs(0)1916 5240 y Ft(\))22 b Fo(\000)g(=)p Fp(m)17 b(')2317 5255 y Fs(1)2362 5240 y Ft(\()p Fp(z)2447 5255 y Fn(k)r Fm(\000)p Fs(1)2598 5240 y Ft(\)\()p Fp(p)2726 5255 y Fn(k)r Fm(\000)p Fs(2)2899 5240 y Fo(\000)22 b Fp(q)3042 5255 y Fn(k)r Fm(\000)p Fs(2)3211 5240 y Fo(<)p Fp(e)16 b(z)3391 5255 y Fs(0)3437 5240 y Ft(\))27 b Fp(:)313 5465 y Ft(First)33 b(of)g(all)f(one)i(has)1338 5584 y Fo(j)17 b(<)p Fp(e)f(z)1563 5599 y Fs(0)1631 5584 y Fo(\000)22 b Fp(x)1787 5599 y Fs(0)1832 5584 y Fo(j)27 b(\024)h Fp(C)7 b(q)2118 5542 y Fm(\000)p Fs(2)2114 5614 y Fn(k)2226 5584 y Fo(j)p Fp(z)2300 5599 y Fn(k)2349 5584 y Fo(j)27 b Fp(:)p eop %%Page: 50 50 50 49 bop 113 144 a Ft(50)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y(On)34 b(the)g(other)f(hand,)h(near)g(1)f(one)h (has)1413 612 y Fo(j)p Fp(')1506 571 y Fm(0)1506 637 y Fs(1)1550 612 y Ft(\()p Fp(z)t Ft(\))p Fo(j)28 b(\024)g Fp(C)7 b Fo(j)p Fp(z)27 b Fo(\000)c Ft(1)p Fo(j)2196 571 y Fm(\000)p Fs(1)2330 612 y Fp(:)113 841 y Ft(F)-8 b(or)33 b(0)27 b Fo(\024)i Fp(l)g(<)f(k)36 b Ft(\(resp.)45 b(0)27 b Fo(\024)h Fp(l)i(<)e(k)c Fo(\000)f Ft(1\))32 b(the)h(distances)h(of)g Fp(x)2429 856 y Fn(l)2492 841 y Ft(and)g Fp(z)2732 856 y Fn(l)2795 841 y Ft(from)f(0)g(\(resp.)44 b(1\))33 b(are)113 960 y(comparable.)44 b(Th)m(us,)34 b(for)f(0)28 b Fo(\024)g Fp(l)h(<)f(k)564 1189 y Fo(j)17 b(=)p Fp(m)g(')850 1204 y Fs(1)895 1189 y Ft(\()p Fp(z)980 1204 y Fn(l)1032 1189 y Ft(+)23 b(1\))f Fo(\000)g(=)p Fp(m)17 b(')1583 1204 y Fs(1)1628 1189 y Ft(\()p Fp(x)1724 1204 y Fn(l)1777 1189 y Ft(+)22 b(1)g(+)h(\()p Fo(\000)p Ft(1\))2253 1148 y Fn(l)2283 1189 y Fp(i)p Ft(0\))p Fo(j)k(\024)i Fp(C)7 b(x)2702 1146 y Fm(\000)p Fs(1)2702 1219 y Fn(l)2809 1189 y Fo(j)p Fp(z)2883 1204 y Fn(l)2936 1189 y Fo(\000)22 b Fp(x)3092 1204 y Fn(l)3123 1189 y Fo(j)27 b Fp(;)113 1418 y Ft(and,)34 b(for)f(0)28 b Fo(\024)g Fp(l)h(<)f(k)e Fo(\000)c Ft(1)607 1647 y Fo(j)17 b(=)p Fp(m)g(')893 1662 y Fs(1)937 1647 y Ft(\()p Fp(z)1022 1662 y Fn(l)1053 1647 y Ft(\))22 b Fo(\000)h(=)p Fp(m)17 b(')1455 1662 y Fs(1)1499 1647 y Ft(\()p Fp(x)1595 1662 y Fn(l)1648 1647 y Ft(+)22 b(\()p Fo(\000)p Ft(1\))1952 1605 y Fn(l)1983 1647 y Fp(i)p Ft(0\))p Fo(j)27 b(\024)h Fp(C)7 b Fo(j)p Fp(x)2429 1662 y Fn(l)2482 1647 y Fo(\000)23 b Ft(1)p Fo(j)2660 1605 y Fm(\000)p Fs(1)2766 1647 y Fo(j)p Fp(z)2840 1662 y Fn(l)2893 1647 y Fo(\000)f Fp(x)3049 1662 y Fn(l)3080 1647 y Fo(j)27 b Fp(:)113 1875 y Ft(W)-8 b(e)26 b(ha)m(v)m(e)f(here)h Fo(j)p Fp(z)780 1890 y Fn(l)816 1875 y Fo(\000)5 b Fp(x)955 1890 y Fn(l)986 1875 y Fo(j)27 b(\024)i Fp(C)7 b Fo(j)p Fp(z)1299 1890 y Fn(k)1348 1875 y Fo(j)p Fp(q)1424 1839 y Fs(2)1420 1904 y Fn(l)1468 1875 y Fp(q)1516 1833 y Fm(\000)p Fs(2)1512 1905 y Fn(k)1623 1875 y Ft(,)27 b Fp(x)1735 1833 y Fm(\000)p Fs(1)1735 1905 y Fn(l)1869 1875 y Fo(\024)h Fp(C)7 b(m)2139 1890 y Fn(l)p Fs(+1)2296 1875 y Ft(and)26 b Fo(j)p Fp(x)2567 1890 y Fn(l)2602 1875 y Fo(\000)5 b Ft(1)p Fo(j)2762 1839 y Fm(\000)p Fs(1)2897 1875 y Fo(\024)29 b Fp(cx)3103 1833 y Fm(\000)p Fs(1)3103 1905 y Fn(l)p Fs(+1)3262 1875 y Fo(\024)f Fp(cm)3497 1890 y Fn(l)p Fs(+2)3629 1875 y Ft(.)113 1995 y(W)-8 b(e)34 b(th)m(us)g(get,)f(taking)f(\(5.54\))g(and)i(\(5.55\))d(in)m(to) i(accoun)m(t,)885 2271 y Fo(j)p Fr(A)1000 2286 y Fn(l)1030 2271 y Fo(j)28 b(\024)g Fp(C)7 b Ft(\()p Fp(q)1352 2286 y Fn(l)p Fm(\000)p Fs(1)1485 2271 y Fp(q)1533 2228 y Fm(\000)p Fs(2)1529 2301 y Fn(k)1640 2271 y Fo(j)p Fp(z)1714 2286 y Fn(k)1763 2271 y Fo(j)17 b Ft(log)1983 2203 y(1)p 1964 2248 88 4 v 1964 2339 a Fp(x)2021 2354 y Fn(l)2086 2271 y Ft(+)23 b Fp(x)2243 2228 y Fm(\000)p Fs(1)2243 2301 y Fn(l)2349 2271 y Fo(j)p Fp(z)2423 2286 y Fn(l)2476 2271 y Fo(\000)g Fp(x)2633 2286 y Fn(l)2663 2271 y Fo(j)p Fp(q)2739 2228 y Fm(\000)p Fs(1)2735 2301 y Fn(l)2846 2271 y Ft(\))1086 2481 y Fo(\024)28 b Fp(C)7 b Fo(j)p Fp(z)1343 2496 y Fn(k)1392 2481 y Fo(j)p Fp(q)1468 2439 y Fm(\000)p Fs(2)1464 2511 y Fn(k)1575 2481 y Ft(\()p Fp(q)1658 2496 y Fn(l)p Fm(\000)p Fs(1)1807 2481 y Ft(log)17 b Fp(m)2039 2496 y Fn(l)p Fs(+1)2193 2481 y Ft(+)23 b Fp(q)2337 2496 y Fn(l)2368 2481 y Fp(m)2455 2496 y Fn(l)p Fs(+1)2587 2481 y Ft(\))1086 2644 y Fo(\024)28 b Fp(C)7 b Fo(j)p Fp(z)1343 2659 y Fn(k)1392 2644 y Fo(j)p Fp(q)1468 2601 y Fm(\000)p Fs(2)1464 2674 y Fn(k)1575 2644 y Fp(q)1619 2659 y Fn(l)p Fs(+1)1778 2644 y Fp(;)709 2929 y Fo(j)p Fr(B)818 2944 y Fn(l)848 2929 y Fo(j)28 b(\024)g Fp(C)7 b Ft(\()p Fp(q)1170 2944 y Fn(l)p Fm(\000)p Fs(1)1303 2929 y Fp(q)1351 2887 y Fm(\000)p Fs(2)1347 2959 y Fn(k)1458 2929 y Fo(j)p Fp(z)1532 2944 y Fn(k)1581 2929 y Fo(j)17 b Ft(log)1852 2862 y(1)p 1783 2906 189 4 v 1783 2998 a Fp(x)1840 3013 y Fn(l)p Fs(+1)2005 2929 y Ft(+)23 b Fo(j)p Fp(x)2190 2944 y Fn(l)2242 2929 y Fo(\000)g Ft(1)p Fo(j)2420 2888 y Fm(\000)p Fs(1)2526 2929 y Fo(j)p Fp(z)2600 2944 y Fn(l)2653 2929 y Fo(\000)f Fp(x)2809 2944 y Fn(l)2840 2929 y Fo(j)p Fp(q)2916 2887 y Fm(\000)p Fs(1)2912 2959 y Fn(l)3023 2929 y Ft(\))904 3148 y Fo(\024)28 b Fp(C)7 b Fo(j)p Fp(z)1161 3163 y Fn(k)1210 3148 y Fo(j)p Fp(q)1286 3106 y Fm(\000)p Fs(2)1282 3178 y Fn(k)1393 3148 y Ft(\()p Fp(q)1476 3163 y Fn(l)p Fm(\000)p Fs(1)1625 3148 y Ft(log)18 b Fp(m)1858 3163 y Fn(l)p Fs(+2)2012 3148 y Ft(+)k Fp(q)2155 3163 y Fn(l)2186 3148 y Fp(m)2273 3163 y Fn(l)p Fs(+2)2405 3148 y Ft(\))904 3311 y Fo(\024)28 b Fp(C)7 b Fo(j)p Fp(z)1161 3326 y Fn(k)1210 3311 y Fo(j)p Fp(q)1286 3268 y Fm(\000)p 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Fn(k)r Fm(\000)p Fs(2)808 4963 y Fo(\000)f Fp(q)951 4978 y Fn(k)r Fm(\000)p Fs(2)1120 4963 y Fo(<)p Fp(e)16 b(z)1300 4978 y Fs(0)1345 4963 y Ft(\))h Fo(=)p Fp(m)g(z)1623 4978 y Fn(k)r Fm(\000)p Fs(1)1791 4823 y Fk(\022)1896 4896 y Fp(\031)p 1876 4940 100 4 v 1876 5032 a Ft(12)2010 4963 y(+)2127 4896 y(1)p 2121 4940 61 4 v 2121 5032 a Fp(\031)2210 4963 y Ft(log)h(2)k Fo(\000)g(<)p Fp(e)17 b(')2727 4978 y Fs(1)2772 4963 y Ft(\(1)22 b(+)g Fp(z)3028 4978 y Fn(k)3078 4963 y Ft(\))3117 4823 y Fk(\023)3234 4963 y Fp(:)113 5246 y Ft(But)33 b(w)m(e)h(ha)m(v)m(e)g(the)f(follo)m (wing)g(inequalities)1092 5385 y Fk(\014)1092 5445 y(\014)1092 5505 y(\014)1092 5564 y(\014)1157 5462 y Fp(\031)p 1138 5507 100 4 v 1138 5598 a Ft(12)1271 5529 y(+)1388 5462 y(1)p 1383 5507 61 4 v 1383 5598 a Fp(\031)1472 5529 y Ft(log)17 b(2)22 b Fo(\000)h(<)p Fp(e)17 b(')1989 5544 y Fs(1)2034 5529 y Ft(\(1)k(+)i Fp(z)2290 5544 y Fn(k)2339 5529 y Ft(\))2378 5385 y Fk(\014)2378 5445 y(\014)2378 5505 y(\014)2378 5564 y(\014)2439 5529 y Fo(\024)28 b Fp(C)35 b(;)p eop %%Page: 51 51 51 50 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(51)817 371 y Fo(j)p Ft(\()p Fp(p)934 386 y Fn(k)r Fm(\000)p Fs(2)1107 371 y Fo(\000)23 b Fp(q)1251 386 y Fn(k)r Fm(\000)p Fs(2)1419 371 y Fo(<)p Fp(e)17 b(z)1600 386 y Fs(0)1645 371 y Ft(\))g Fo(=)p Fp(m)f(z)1922 386 y Fn(k)r Fm(\000)p Fs(1)2073 371 y Fo(j)28 b(\024)g Fp(C)7 b(q)2360 328 y Fm(\000)p Fs(1)2356 401 y Fn(k)r Fm(\000)p Fs(1)2508 371 y Fo(j)17 b(=)p Fp(m)f(z)2774 386 y Fn(k)r Fm(\000)p Fs(1)2925 371 y Fo(j)2129 537 y(\024)28 b Fp(C)7 b(q)2360 494 y Fm(\000)p Fs(1)2356 567 y Fn(k)2467 537 y Fo(j)17 b(=)p Fp(m)f(z)2733 552 y Fn(k)2783 537 y Fo(j)27 b Fp(;)208 746 y Fo(j)p Ft(\()p Fp(p)325 761 y Fn(k)r Fm(\000)p Fs(1)498 746 y Fo(\000)22 b Fp(q)641 761 y Fn(k)r Fm(\000)p Fs(1)809 746 y Fo(<)p Fp(e)17 b(z)990 761 y Fs(0)1035 746 y Ft(\))22 b(+)h(\()p Fp(p)1285 761 y Fn(k)r Fm(\000)p Fs(2)1458 746 y Fo(\000)f Fp(q)1601 761 y Fn(k)r Fm(\000)p 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y Fm(\000)p Fs(1)2577 1437 y Ft(\))k Fp(;)113 1631 y Ft(and)34 b(the)g(pro)s(of)f(of)h (Theorem)e(5.10)h(is)g(complete.)1579 b Fd(\003)113 1815 y Fl(R)-5 b(emark)36 b Ft(5.18)65 b(W)-8 b(e)34 b(recall)f(that)g(the)g (term)f(in)m(v)m(olv)m(ed)i(in)f(Theo)s(orem)f(5.10)h(satis\014es)499 2094 y(\()p Fp(p)588 2109 y Fn(k)r Fm(\000)p Fs(1)762 2094 y Fo(\000)22 b Fp(q)905 2109 y Fn(k)r Fm(\000)p Fs(1)1073 2094 y Fo(<)p Fp(e)17 b(z)1254 2109 y Fs(0)1299 2094 y Ft(\))g Fo(=)p Fp(m)g(')1596 2109 y Fs(1)1640 2094 y Ft(\()p Fp(z)1725 2109 y Fn(k)1797 2094 y Ft(+)22 b(1\))28 b(=)2118 1953 y Fk(\022)2191 2094 y Fp(q)2239 2051 y Fm(\000)p Fs(1)2235 2124 y Fn(k)2363 2094 y Ft(log)2570 2026 y(1)p 2520 2071 151 4 v 2520 2162 a Fo(j)p Fp(z)2594 2177 y Fn(k)2643 2162 y Fo(j)2683 1953 y Fk(\023)2772 2094 y Ft(\(1)22 b(+)h(o)16 b(\(1\)\))27 b Fp(;)113 2387 y Ft(as)34 b Fp(z)282 2402 y Fn(k)359 2387 y Fo(!)27 b Ft(0.)313 2539 y Fr(5.3.10)36 b Ft(Here)g(w)m(e)g(consider)h(the)f (imaginary)e(part)h(of)h 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4014 y Fo(=)p Fp(m)k Fo(B)s Ft(\()p Fp(w)25 b Ft(+)e Fp(\013)p Ft(\))28 b(=)g Fp(B)5 b Ft(\()p Fp(\013)p Ft(\))27 b Fp(:)185 4313 y Fe(2.)50 b(Let)33 b Fp(\013)g Fe(b)s(e)h(an)f(irrational)f(diophan)m(tine)i(n)m(um)m(b)s (er)g(and)g Ft(0)27 b Fp(<)i(h)f(<)g Ft(1)p Fp(=)p Ft(2)k Fe(suc)m(h)j(that)1383 4557 y Ft(lim)17 b(inf)1422 4617 y Fn(q)s Fm(!1)1676 4557 y Fo(k)p Fp(q)t(\013)p Fo(k)1888 4572 y Fg(Z)1939 4557 y Fp(q)1987 4516 y Fs(1)p Fn(=h)p Fm(\000)p Fs(1)2249 4557 y Ft(=)28 b(+)p Fo(1)g Fp(;)313 4849 y Fe(where)34 b Fo(k)27 b(k)728 4864 y Fg(Z)813 4849 y Fe(denotes)34 b(the)g(distance)g(from)e(the)i(nearest)g(in)m (teger.)44 b(Then)1411 5093 y Ft(lim)1248 5174 y Fn(w)r Fm(!)p Fs(0)14 b Fn(;)f(w)r Fm(2)1614 5157 y Fs(~)1592 5174 y Fn(W)1667 5185 y Fj(h)1729 5093 y Fo(=)p Fp(m)k Fo(B)s Ft(\()p Fp(w)25 b Ft(+)e Fp(\013)p Ft(\))k(=)h Fp(B)5 b Ft(\()p Fp(\013)p Ft(\))27 b Fp(:)113 5440 y Fl(Pr)-5 b(o)g(of.)33 b Ft(W)-8 b(e)32 b(b)s(egin)f(b)m(y)h(stating)f (a)g(useful)i(Lemma)d(\(whose)h(pro)s(of)h(is)g(an)f(easy)h(adaptation) e(of)113 5559 y(the)k(argumen)m(ts)f(of)h(5.3.9)e(and)h(is)g(left)h(to) f(the)g(reader\).)p eop %%Page: 52 52 52 51 bop 113 144 a Ft(52)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y Fr(Lemma)45 b(5.20)58 b Fe(Let)39 b Fp(k)g Fo(\025)f Ft(1)p Fe(,)h Fp(m)1381 398 y Fs(1)1426 383 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)1735 398 y Fn(k)1821 383 y Fo(\025)37 b Ft(1)p Fe(,)j Fp(p)2103 398 y Fn(k)2152 383 y Fp(=q)2246 398 y Fn(k)2334 383 y Fe(b)s(e)f(the)g(p)s(oin)m(t)f (of)h(in)m(tersection)g(of)113 503 y Fp(H)8 b Ft(\()p Fp(m)330 518 y Fs(1)375 503 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)684 518 y Fn(k)733 503 y Ft(\))33 b Fe(with)g Fq(R)5 b Fe(.)50 b(F)-8 b(or)33 b(all)f Fp(x)c Fo(2)g Fp(D)s Ft(\()p Fp(m)1883 518 y Fs(1)1928 503 y Fp(;)17 b(:)g(:)g(:)d(;)j(m)2237 518 y Fn(k)2286 503 y Ft(\))22 b Fo(\\)g Fq(R)44 b Fe(one)34 b(has)870 837 y Fo(j)p Fp(B)974 852 y Fn(f)8 b(inite)1199 837 y Ft(\()p Fp(p)1288 852 y Fn(k)1337 837 y Fp(=q)1431 852 y Fn(k)1480 837 y Ft(\))22 b Fo(\000)1641 712 y Fn(k)r Fm(\000)p Fs(1)1642 742 y Fk(X)1694 953 y Fs(0)1803 837 y Fp(\014)1859 852 y Fn(l)p Fm(\000)p Fs(1)1992 837 y Ft(\()p Fp(x)p Ft(\))17 b(log)2319 769 y(1)p 2300 814 88 4 v 2300 905 a Fp(x)2357 920 y Fn(l)2400 837 y Fo(j)27 b(\024)h Fp(C)7 b(x)2695 852 y Fn(k)2745 837 y Fp(q)2793 794 y Fm(\000)p Fs(1)2789 867 y Fn(k)113 1164 y Fe(where)34 b Ft(\()p Fp(x)497 1179 y Fn(i)530 1164 y Ft(\))569 1179 y Fn(i)p Fm(\025)p Fs(0)737 1164 y Fe(is)g(the)f(con)m(tin)m(ued)i (fraction)e(of)h Fp(x)p Fe(.)313 1372 y Ft(Assume)f(no)m(w)g(that)g Fp(\013)28 b Fo(2)g Ft(\(0)p Fp(;)17 b Ft(1\))32 b(is)h(irrational)f (with)h(con)m(tin)m(ued)h(fraction)1041 1608 y Fp(\013)28 b Ft(=)g(1)p Fp(=m)1425 1623 y Fs(1)1492 1608 y Ft(+)22 b(1)p Fp(=m)1778 1623 y Fs(2)1845 1608 y Ft(+)h Fp(:)17 b(:)g(:)j Ft(+)j(1)p Fp(=m)2370 1623 y Fn(k)2441 1608 y Ft(+)g Fp(:)17 b(:)g(:)42 b(:)113 1844 y Ft(Let)29 b(\()p Fp(p)376 1859 y Fn(k)425 1844 y Fp(=q)519 1859 y Fn(k)568 1844 y Ft(\))607 1859 y Fn(k)r Fm(\025)p Fs(0)786 1844 y Ft(denote)g(the)f(sequence)i(of)f(the)f(partial)f(fractions.)43 b(Let)29 b Fp(w)i Ft(b)s(e)d(a)g(p)s(oin)m(t)g(close)113 1964 y(to)33 b(0)h(and)f Fp(z)g Ft(=)c Fp(\013)22 b Ft(+)h Fp(w)s Ft(.)44 b(F)-8 b(or)33 b Fo(j)17 b(=)p Fp(m)g(z)t Fo(j)28 b(\024)g Ft(1)p Fp(=)p Ft(2,)33 b Fp(z)38 b Ft(b)s(elongs)c(to) f(a)g(domain)f Fp(V)23 b Ft(\()p Fp(p=q)t Ft(\))33 b(\(de\014ned)i(in) 113 2083 y(Section)f(5.3.1\))e(and)i(w)m(e)f(distinguish)h(t)m(w)m(o)f (cases.)149 2203 y(\(I\))50 b(Here)29 b(w)m(e)g(assume)h(that)e Fp(p=q)k Ft(=)c Fp(p)1569 2218 y Fn(k)1618 2203 y Fp(=q)1712 2218 y Fn(k)1790 2203 y Ft(is)h(one)h(of)f(the)h(partial)e(fractions)h (of)g Fp(\013)p Ft(.)43 b(One)30 b(has)313 2322 y(then)k(b)m(y)f(Prop)s (osition)f(A1.1)1388 2558 y Fo(j)p Fp(\013)22 b Fo(\000)g Fp(p)1651 2573 y Fn(k)1700 2558 y Fp(=q)1794 2573 y Fn(k)1844 2558 y Fo(j)27 b(\025)h Ft(\(2)p Fp(q)2137 2573 y Fn(k)2186 2558 y Fp(q)2230 2573 y Fn(k)r Fs(+1)2381 2558 y Ft(\))2420 2517 y Fm(\000)p Fs(1)2554 2558 y Fp(:)271 2794 y Ft(\(I.1\))49 b(If)34 b Fp(w)c Fo(2)e Fp(W)901 2809 y Fn(H)1010 2794 y Ft(one)34 b(gets)1428 3030 y Fo(j)p Fp(z)27 b Fo(\000)c Fp(p)1679 3045 y Fn(k)1728 3030 y Fp(=q)1822 3045 y Fn(k)1871 3030 y Fo(j)k(\025)i Fp(c)2075 2989 y Fm(\000)p Fs(1)2182 3030 y Ft(\()p Fp(q)2265 3045 y Fn(k)2314 3030 y Fp(q)2358 3045 y Fn(k)r Fs(+1)2508 3030 y Ft(\))2547 2989 y Fm(\000)p Fn(H)2712 3030 y Fp(;)512 3266 y Ft(th)m(us)1542 3385 y Fo(j)p Fp(z)1616 3400 y Fn(k)1665 3385 y Fo(j)1693 3344 y Fm(\000)p Fs(1)1827 3385 y Fo(\024)f Fp(cq)2023 3343 y Fm(\000)p Fs(2)2019 3416 y Fn(k)2130 3385 y Ft(\()p Fp(q)2213 3400 y Fn(k)2263 3385 y Fp(q)2307 3400 y Fn(k)r Fs(+1)2457 3385 y Ft(\))2496 3344 y Fn(H)2599 3385 y Fp(;)512 3573 y Ft(and)986 3648 y(1)p 964 3692 94 4 v 964 3783 a Fp(q)1008 3798 y Fn(k)1086 3715 y Ft(log)17 b Fo(j)p Fp(z)1305 3730 y Fn(k)1354 3715 y Fo(j)1382 3674 y Fm(\000)p Fs(1)1517 3715 y Fo(\024)28 b Fp(q)1670 3672 y Fm(\000)p Fs(1)1666 3745 y Fn(k)1777 3715 y Ft([)p Fp(c)22 b Ft(+)g Fp(H)j Ft(log)17 b Fp(q)2266 3730 y Fn(k)r Fs(+1)2439 3715 y Ft(+)22 b(\()p Fp(H)30 b Fo(\000)23 b Ft(2\))17 b(log)f Fp(q)3084 3730 y Fn(k)3134 3715 y Ft(])27 b Fp(;)512 3950 y Ft(whic)m(h)36 b(is)g(small)e(when)i Fp(\013)g Ft(is)f(a)g(Brjuno)i(n)m(um)m(b)s(er)f(and)g Fp(k)i Ft(is)e(large.)50 b(Lemma)34 b(5.20,)512 4070 y(Remark)e(5.18,)g(and)i(Theorem)f(5.10)f(lead)h(to)g(the)h(desired)g (conclusion.)271 4189 y(\(I.2\))49 b(If)34 b Fp(w)c Fo(2)836 4164 y Ft(~)807 4189 y Fp(W)901 4204 y Fn(h)986 4189 y Ft(w)m(e)k(will)e(ha)m(v)m(e)1383 4425 y Fo(j)p Fp(z)26 b Fo(\000)d Fp(p)1633 4440 y Fn(k)1682 4425 y Fp(=q)1776 4440 y Fn(k)1825 4425 y Fo(j)k(\025)i Ft(exp[)p Fo(\000)p Fp(c)p Ft(\()p Fp(q)2369 4440 y Fn(k)2418 4425 y Fp(q)2462 4440 y Fn(k)r Fs(+1)2613 4425 y Ft(\))2652 4384 y Fn(h)2703 4425 y Ft(])e Fp(;)512 4661 y Ft(th)m(us)1420 4781 y Fo(j)p Fp(z)1494 4796 y Fn(k)1543 4781 y Fo(j)1571 4739 y Fm(\000)p Fs(1)1706 4781 y Fo(\024)h Fp(cq)1902 4738 y Fm(\000)p Fs(2)1898 4811 y Fn(k)2025 4781 y Ft(exp)q([)p Fp(c)p Ft(\()p Fp(q)2332 4796 y Fn(k)2381 4781 y Fp(q)2425 4796 y Fn(k)r Fs(+1)2575 4781 y Ft(\))2614 4739 y Fn(h)2665 4781 y Ft(])g Fp(;)512 4968 y Ft(and)1536 5043 y(1)p 1514 5087 V 1514 5178 a Fp(q)1558 5193 y Fn(k)1636 5110 y Ft(log)17 b Fo(j)p Fp(z)1855 5125 y Fn(k)1904 5110 y Fo(j)1932 5069 y Fm(\000)p Fs(1)2067 5110 y Fo(\024)28 b Fp(cq)2263 5069 y Fn(h)2259 5135 y(k)r Fs(+1)2409 5110 y Fp(q)2457 5067 y Fn(h)p Fm(\000)p Fs(1)2453 5140 y Fn(k)2639 5110 y Fp(;)512 5345 y Ft(whic)m(h)38 b(is)g(small)d(if)j Fp(\013)g Ft(satis\014es)g(the)g(diophan)m(tine)g(condition)f(w)m(e)h (ha)m(v)m(e)g(assumed)512 5465 y(and)48 b Fp(k)i Ft(is)d(large.)86 b(Once)49 b(again)e(the)g(conclusion)h(follo)m(ws)g(from)e(Lemma)g (5.20,)512 5584 y(Remark)32 b(5.18,)g(and)i(Theorem)f(5.10.)p eop %%Page: 53 53 53 52 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(53)111 383 y(\(I)s(I\))49 b(Here)36 b(w)m(e)g(assume)f(that)h Fp(p=q)j Ft(is)d Fl(not)g Ft(one)g(of)g(the)g(partial)e(fractions)j(of) e Fp(\013)p Ft(.)52 b(W)-8 b(e)36 b(denote)313 503 y(\()p Fp(p)402 466 y Fm(0)402 531 y Fn(l)432 503 y Fp(=q)530 466 y Fm(0)526 531 y Fn(l)558 503 y Ft(\))597 518 y Fs(0)p Fm(\024)p Fn(l)p Fm(\024)p Fn(L)885 503 y Ft(the)k(partial)e(fractions) h(of)g Fp(p=q)k Ft(and)d Fp(k)i Ft(the)d(largest)g(in)m(teger)h(suc)m (h)g(that)313 622 y Fp(p)363 586 y Fm(0)363 650 y Fn(k)412 622 y Fp(=q)510 586 y Fm(0)506 650 y Fn(k)583 622 y Ft(=)28 b Fp(p)738 637 y Fn(k)787 622 y Fp(=q)881 637 y Fn(k)930 622 y Ft(.)44 b(Clearly)30 b(one)h(has)h Fp(k)f(<)d(L)j Ft(and)g Fp(p)2230 586 y Fm(0)2230 650 y Fn(L)2290 622 y Fp(=q)2388 586 y Fm(0)2384 650 y Fn(L)2472 622 y Ft(=)d Fp(p=q)t Ft(.)43 b(By)30 b(a)h(classical)g(result)313 742 y(\([HW],)h(Theorem)h(184,)f(p.)44 b(153\))1613 1033 y Fo(j)p Fp(\013)22 b Fo(\000)g Fp(p=q)t Fo(j)28 b(\025)2193 966 y Ft(1)p 2147 1010 143 4 v 2147 1101 a(2)p Fp(q)2245 1072 y Fs(2)2329 1033 y Fp(:)313 1347 y Ft(F)-8 b(or)32 b Fp(w)f Fo(2)716 1322 y Ft(~)687 1347 y Fp(W)781 1362 y Fn(h)866 1347 y Ft(one)j(has)1469 1472 y Fo(j)p Fp(z)26 b Fo(\000)d Fp(p=q)t Fo(j)k(\025)i Ft(exp[)p Fo(\000)p Fp(cq)2326 1431 y Fs(2)p Fn(h)2417 1472 y Ft(])f Fp(;)313 1666 y Ft(th)m(us)1454 1791 y Fo(j)p Fp(z)1528 1806 y Fn(L)1588 1791 y Fo(j)1616 1750 y Fm(\000)p Fs(1)1750 1791 y Fo(\024)g Fp(cq)1946 1750 y Fm(\000)p Fs(2)2070 1791 y Ft(exp[)p Fp(cq)2341 1750 y Fs(2)p Fn(h)2432 1791 y Ft(])g Fp(;)313 1985 y Ft(and)1482 2064 y(1)p 1482 2109 50 4 v 1483 2200 a Fp(q)1560 2132 y Ft(log)17 b Fo(j)p Fp(z)1779 2147 y Fn(L)1839 2132 y Fo(j)1867 2091 y Fm(\000)p Fs(1)2001 2132 y Fo(\024)29 b Fp(q)2155 2091 y Fm(\000)p Fs(1)2262 2132 y Fp(cq)2353 2091 y Fs(2)p Fn(h)2472 2132 y Fp(;)313 2379 y Ft(whic)m(h)d(is)f(small)f(since)i Fp(h)i(<)h Ft(1)p Fp(=)p Ft(2.)40 b(T)-8 b(aking)25 b(in)m(to)g(accoun) m(t)h(Theorem)f(5.10,)h(Lemma)e(5.20,)313 2499 y(and)32 b(Remark)f(5.18)g(w)m(e)i(only)e(need)j(to)d(c)m(hec)m(k)j(that)d Fp(B)2361 2514 y Fn(f)8 b(inite)2586 2499 y Ft(\()p Fp(p)2675 2514 y Fn(k)2724 2499 y Fp(=q)2818 2514 y Fn(k)2867 2499 y Ft(\))32 b(and)g Fp(B)3206 2514 y Fn(f)8 b(inite)3431 2499 y Ft(\()p Fp(p=q)t Ft(\))313 2618 y(are)33 b(close.)313 2740 y(Let)g(us)h(in)m(tro)s(duce)1112 3024 y Fp(\032)27 b Ft(=)60 b(max)1296 3089 y Fn(k)r Fm(\024)p Fn(l)g Ft(0)33 b Fo(9)p Fp(n)745 2117 y Fs(0)823 2102 y Ft(suc)m(h)i(that)e Fo(8)p Ft(\()p Fp(u)1415 2117 y Fn(l)1446 2102 y Ft(\))1485 2117 y Fn(l)p Fm(\025)p Fs(0)1645 2102 y Fo(2)28 b Fp(K)40 b Ft(one)33 b(has)2224 2028 y Fk(P)2329 2132 y Fn(l>n)2465 2142 y Fi(0)2527 2102 y Fo(j)p Fp(u)2612 2117 y Fn(l)2642 2102 y Fo(j)27 b(\024)h Fp(")p Ft(.)113 2475 y Fr(A1.)51 b(App)s(endix)36 b(1)j(:)51 b(real)37 b(con)m(tin)m(ued)f(fractions)113 2704 y Ft(In)54 b(this)f(app)s(endix)g(w)m(e)g(recall)g(some)f(elemen)m (tary)g(facts)i(on)f(standard)g(real)g(con)m(tin)m(ued)113 2824 y(fractions)38 b(\(w)m(e)f(refer)h(to)e([MMY],)h(and)g(references) i(therein,)g(for)e(more)f(general)h(con)m(tin)m(ued)113 2943 y(fractions\).)313 3066 y(W)-8 b(e)33 b(will)f(consider)j(the)e (iteration)g(of)g(the)h(Gauss)g(map)1491 3314 y Fp(A)27 b Ft(:)g(\(0)p Fp(;)17 b Ft(1\))26 b Fo(7!)i Ft([0)p Fp(;)17 b Ft(1])26 b Fp(;)1097 b Ft(\()p Fp(A)p Ft(1)p Fp(:)p Ft(1\))113 3562 y(de\014ned)36 b(b)m(y)1391 3745 y Fp(A)p Ft(\()p Fp(x)p Ft(\))27 b(=)1778 3677 y(1)p 1745 3722 117 4 v 1775 3813 a Fp(x)1896 3745 y Fo(\000)1995 3604 y Fk(\024)2126 3677 y Ft(1)p 2093 3722 V 2123 3813 a Fp(x)2254 3604 y Fk(\025)2351 3745 y Fp(:)998 b Ft(\()p Fp(A)p Ft(1)p Fp(:)p Ft(2\))113 3996 y(Let)1081 4173 y Fp(G)28 b Ft(=)1304 4023 y Fo(p)p 1387 4023 50 4 v 82 x Ft(5)22 b(+)h(1)p 1304 4150 305 4 v 1432 4241 a(2)1648 4173 y Fp(;)44 b(g)31 b Ft(=)d Fp(G)1982 4132 y Fm(\000)p Fs(1)2117 4173 y Ft(=)2234 4023 y Fo(p)p 2317 4023 50 4 v 82 x Ft(5)22 b Fo(\000)h Ft(1)p 2234 4150 305 4 v 2362 4241 a(2)2578 4173 y Fp(:)113 4405 y Ft(T)-8 b(o)36 b(eac)m(h)g Fp(x)c Fo(2)h Fq(R)i Fo(n)23 b Fq(Q)54 b Ft(w)m(e)36 b(asso)s(ciate)g(a)f(con)m(tin)m(ued)j(fraction)e (expansion)g(b)m(y)g(iterating)f Fp(A)h Ft(as)113 4524 y(follo)m(ws.)44 b(Let)1338 4653 y Fp(x)1395 4668 y Fs(0)1467 4653 y Ft(=)28 b Fp(x)22 b Fo(\000)h Ft([)p Fp(x)p Ft(])k Fp(;)116 b(a)2088 4668 y Fs(0)2160 4653 y Ft(=)28 b([)p Fp(x)p Ft(])f Fp(;)944 b Ft(\()p Fp(A)p Ft(1)p Fp(:)p Ft(3\))113 4851 y(then)34 b Fp(x)28 b Ft(=)g Fp(a)583 4866 y Fs(0)650 4851 y Ft(+)22 b Fp(x)806 4866 y Fs(0)851 4851 y Ft(.)44 b(W)-8 b(e)33 b(no)m(w)h(de\014ne)h(inductiv)m(ely)e (for)g(all)g Fp(n)27 b Fo(\025)h Ft(0)1086 5153 y Fp(x)1143 5168 y Fn(n)p Fs(+1)1325 5153 y Ft(=)h Fp(A)p Ft(\()p Fp(x)1602 5168 y Fn(n)1655 5153 y Ft(\))f Fp(;)116 b(a)1919 5168 y Fn(n)p Fs(+1)2101 5153 y Ft(=)2206 5013 y Fk(\024)2301 5086 y Ft(1)p 2271 5130 112 4 v 2271 5222 a Fp(x)2328 5237 y Fn(n)2394 5013 y Fk(\025)2474 5153 y Fo(\025)28 b Ft(1)g Fp(;)692 b Ft(\()p Fp(A)p Ft(1)p Fp(:)p Ft(4\))113 5456 y(th)m(us)1438 5584 y Fp(x)1495 5543 y Fm(\000)p Fs(1)1495 5609 y Fn(n)1630 5584 y Ft(=)28 b Fp(a)1788 5599 y Fn(n)p Fs(+1)1965 5584 y Ft(+)22 b Fp(x)2121 5599 y Fn(n)p Fs(+1)2304 5584 y Fp(:)1045 b Ft(\()p Fp(A)p Ft(1)p Fp(:)p Ft(5\))p eop %%Page: 56 56 56 55 bop 113 144 a Ft(56)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y(Therefore)35 b(w)m(e)f(ha)m(v)m(e)270 609 y Fp(x)27 b Ft(=)h Fp(a)512 624 y Fs(0)579 609 y Ft(+)23 b Fp(x)736 624 y Fs(0)808 609 y Ft(=)28 b Fp(a)966 624 y Fs(0)1032 609 y Ft(+)1280 542 y(1)p 1144 586 321 4 v 1144 678 a Fp(a)1197 693 y Fs(1)1264 678 y Ft(+)22 b Fp(x)1420 693 y Fs(1)1504 609 y Ft(=)29 b Fp(:)17 b(:)g(:)26 b Ft(=)i Fp(a)1912 624 y Fs(0)1978 609 y Ft(+)2597 542 y(1)p 2090 586 1064 4 v 2090 730 a Fp(a)2143 745 y Fs(1)2209 730 y Ft(+)2706 663 y(1)p 2321 707 821 4 v 2321 869 a Fp(a)2374 884 y Fs(2)2440 869 y Ft(+)2545 799 y(.)2584 829 y(.)2623 859 y(.)2678 869 y(+)2935 802 y(1)p 2790 846 340 4 v 2790 937 a Fp(a)2843 952 y Fn(n)2919 937 y Ft(+)23 b Fp(x)3076 952 y Fn(n)3193 609 y Fp(;)156 b Ft(\()p Fp(A)p Ft(1)p Fp(:)p Ft(6\))113 1116 y(and)34 b(w)m(e)g(will)e(write)1371 1236 y Fp(x)c Ft(=)g([)p Fp(a)1642 1251 y Fs(0)1686 1236 y Fp(;)17 b(a)1784 1251 y Fs(1)1827 1236 y Fp(;)g(:)g(:)g(:)d(;)j(a)2102 1251 y Fn(n)2156 1236 y Fp(;)g(:)g(:)g(:)m Ft(])27 b Fp(:)978 b Ft(\()p Fp(A)p Ft(1)p Fp(:)p Ft(7\))113 1409 y(The)34 b(n)m(th-con)m(v)m(ergen)m(t)h(is)f(de\014ned)h(b)m(y)824 1588 y Fp(p)874 1603 y Fn(n)p 824 1632 105 4 v 827 1723 a Fp(q)871 1738 y Fn(n)968 1655 y Ft(=)28 b([)p Fp(a)1154 1670 y Fs(0)1198 1655 y Fp(;)17 b(a)1296 1670 y Fs(1)1340 1655 y Fp(;)g(:)g(:)g(:)d(;)j(a)1615 1670 y Fn(n)1668 1655 y Ft(])27 b(=)i Fp(a)1882 1670 y Fs(0)1948 1655 y Ft(+)2450 1588 y(1)p 2060 1632 831 4 v 2060 1776 a Fp(a)2113 1791 y Fs(1)2179 1776 y Ft(+)2560 1708 y(1)p 2291 1753 588 4 v 2291 1915 a Fp(a)2344 1930 y Fs(2)2410 1915 y Ft(+)2515 1845 y(.)2554 1875 y(.)2593 1905 y(.)2648 1915 y(+)2788 1847 y(1)p 2760 1892 107 4 v 2760 1983 a Fp(a)2813 1998 y Fn(n)2930 1655 y Fp(:)419 b Ft(\()p Fp(A)p Ft(1)p Fp(:)p Ft(8\))113 2162 y(The)34 b(n)m(umerators)f Fp(p)892 2177 y Fn(n)980 2162 y Ft(and)h(denominators)e Fp(q)1838 2177 y Fn(n)1926 2162 y Ft(are)i(recursiv)m(ely)f(determined) h(b)m(y)1128 2366 y Fp(p)1178 2381 y Fm(\000)p Fs(1)1313 2366 y Ft(=)28 b Fp(q)1462 2381 y Fm(\000)p Fs(2)1597 2366 y Ft(=)g(1)56 b Fp(;)99 b(p)1985 2381 y Fm(\000)p Fs(2)2120 2366 y Ft(=)28 b Fp(q)2269 2381 y Fm(\000)p Fs(1)2404 2366 y Ft(=)g(0)55 b Fp(;)735 b Ft(\()p Fp(A)p Ft(1)p Fp(:)p Ft(9\))113 2570 y(and)34 b(for)g(all)e Fp(n)27 b Fo(\025)i Ft(0)902 2775 y Fp(p)952 2790 y Fn(n)1034 2775 y Ft(=)f Fp(a)1192 2790 y Fn(n)1246 2775 y Fp(p)1296 2790 y Fn(n)p Fm(\000)p Fs(1)1474 2775 y Ft(+)23 b Fp(p)1624 2790 y Fn(n)p Fm(\000)p Fs(2)1808 2775 y Fp(;)116 b(q)1996 2790 y Fn(n)2078 2775 y Ft(=)28 b Fp(a)2236 2790 y Fn(n)2290 2775 y Fp(q)2334 2790 y Fn(n)p Fm(\000)p Fs(1)2513 2775 y Ft(+)23 b Fp(q)2657 2790 y Fn(n)p Fm(\000)p Fs(2)2841 2775 y Fp(:)458 b Ft(\()p Fp(A)p Ft(1)p Fp(:)p Ft(10\))113 2979 y(Moreo)m(v)m(er)1330 3199 y Fp(x)27 b Ft(=)1531 3131 y Fp(p)1581 3146 y Fn(n)1658 3131 y Ft(+)c Fp(p)1808 3146 y Fn(n)p Fm(\000)p Fs(1)1964 3131 y Fp(x)2021 3146 y Fn(n)p 1531 3176 544 4 v 1537 3267 a Fp(q)1581 3282 y Fn(n)1658 3267 y Ft(+)g Fp(q)1802 3282 y Fn(n)p Fm(\000)p Fs(1)1958 3267 y Fp(x)2015 3282 y Fn(n)2115 3199 y Fp(;)1184 b Ft(\()p Fp(A)p Ft(1)p Fp(:)p Ft(11\))1275 3456 y Fp(x)1332 3471 y Fn(n)1414 3456 y Ft(=)28 b Fo(\000)1711 3388 y Fp(q)1755 3403 y Fn(n)1810 3388 y Fp(x)22 b Fo(\000)g Fp(p)2038 3403 y Fn(n)p 1609 3433 586 4 v 1609 3524 a Fp(q)1653 3539 y Fn(n)p Fm(\000)p Fs(1)1810 3524 y Fp(x)g Fo(\000)g Fp(p)2038 3539 y Fn(n)p Fm(\000)p Fs(1)2234 3456 y Fp(;)1065 b Ft(\()p Fp(A)p Ft(1)p Fp(:)p Ft(12\))1288 3662 y Fp(q)1332 3677 y Fn(n)1387 3662 y Fp(p)1437 3677 y Fn(n)p Fm(\000)p Fs(1)1615 3662 y Fo(\000)23 b Fp(p)1765 3677 y Fn(n)1819 3662 y Fp(q)1863 3677 y Fn(n)p Fm(\000)p Fs(1)2047 3662 y Ft(=)29 b(\()p Fo(\000)p Ft(1\))2358 3621 y Fn(n)2467 3662 y Fp(:)832 b Ft(\()p Fp(A)p Ft(1)p Fp(:)p Ft(13\))113 3866 y(Let)393 4070 y Fp(\014)449 4085 y Fn(n)531 4070 y Ft(=)29 b(\005)712 4029 y Fn(n)712 4095 y(i)p Fs(=0)845 4070 y Fp(x)902 4085 y Fn(i)963 4070 y Ft(=)f(\()p Fo(\000)p Ft(1\))1273 4029 y Fn(n)1327 4070 y Ft(\()p Fp(q)1410 4085 y Fn(n)1465 4070 y Fp(x)22 b Fo(\000)g Fp(p)1693 4085 y Fn(n)1748 4070 y Ft(\))99 b(for)34 b Fp(n)27 b Fo(\025)h Ft(0)p Fp(;)116 b Ft(and)34 b Fp(\014)2675 4085 y Fm(\000)p Fs(1)2810 4070 y Ft(=)28 b(1)55 b Fp(:)279 b Ft(\()p Fp(A)p Ft(1)p Fp(:)p Ft(14\))113 4275 y(F)-8 b(rom)27 b(the)h(de\014nitions)g(giv)m(en)g(one)g(easily)f (pro)m(v)m(es)h(b)m(y)g(induction)g(the)g(follo)m(wing)f(prop)s (osition)113 4394 y(\(see)34 b([MMY]\))113 4566 y Fr(Prop)s(osition)39 b(A1.1)90 b Fe(F)-8 b(or)33 b(all)f Fp(x)c Fo(2)g Fq(R)33 b Fo(n)22 b Fq(Q)51 b Fe(and)34 b(for)f(all)f Fp(n)c Fo(\025)g Ft(1)33 b Fe(one)h(has)158 4704 y(\(i\))248 b Fo(j)p Fp(q)584 4719 y Fn(n)638 4704 y Fp(x)22 b Fo(\000)h Fp(p)867 4719 y Fn(n)921 4704 y Fo(j)28 b Ft(=)1385 4665 y(1)p 1094 4681 633 4 v 1094 4743 a Fp(q)1138 4758 y Fn(n)p Fs(+1)1316 4743 y Ft(+)22 b Fp(q)1459 4758 y Fn(n)1514 4743 y Fp(x)1571 4758 y Fn(n)p Fs(+1)1738 4704 y Fe(,)33 b(so)g(that)2149 4665 y Ft(1)p 2149 4681 50 4 v 2149 4749 a(2)2238 4704 y Fp(<)28 b(\014)2399 4719 y Fn(n)2454 4704 y Fp(q)2498 4719 y Fn(n)p Fs(+1)2681 4704 y Fp(<)g Ft(1)20 b Fe(;)130 4880 y(\(ii\))49 b Fp(\014)369 4895 y Fn(n)451 4880 y Fo(\024)28 b Fp(g)608 4844 y Fn(n)695 4880 y Fe(and)33 b Fp(q)932 4895 y Fn(n)1015 4880 y Fo(\025)1132 4841 y Ft(1)p 1132 4857 V 1132 4925 a(2)1194 4880 y Fp(G)1272 4844 y Fn(n)p Fm(\000)p Fs(1)1461 4880 y Fe(.)113 5106 y Ft(Note)c(that)g(from)g Fl(\(ii\))f Ft(it)h(follo)m(ws)g(that)1597 5031 y Fk(P)1702 5056 y Fm(1)1702 5136 y Fn(k)r Fs(=0)1881 5062 y(log)15 b Fn(q)2035 5073 y Fj(k)p 1881 5083 197 4 v 1939 5140 a Fn(q)1976 5151 y Fj(k)2119 5106 y Ft(and)2309 5031 y Fk(P)2414 5056 y Fm(1)2414 5136 y Fn(k)r Fs(=0)2613 5067 y(1)p 2593 5083 81 4 v 2593 5140 a Fn(q)2630 5151 y Fj(k)2714 5106 y Ft(are)29 b(alw)m(a)m(ys)g(con)m(v)m(ergen)m(t)113 5226 y(and)34 b(their)f(sum)g(is)g(uniformly)g(b)s(ounded.)313 5345 y(With)50 b(the)h(notations)f(of)h(Section)g(2.1,)j(equation)c (\(A1.5\))f(can)i(b)s(e)g(written)f Fp(x)3468 5360 y Fn(n)3579 5345 y Ft(=)113 5465 y Fp(g)t Ft(\()p Fp(a)257 5480 y Fn(n)p Fs(+1)411 5465 y Ft(\))p Fp(x)507 5480 y Fn(n)p Fs(+1)662 5465 y Ft(,)d(th)m(us)f(w)m(e)f(ha)m(v)m(e)g Fp(x)1424 5480 y Fs(0)1516 5465 y Ft(=)i Fp(g)t Ft(\()p Fp(a)1784 5480 y Fs(1)1828 5465 y Ft(\))p Fp(g)t Ft(\()p Fp(a)2011 5480 y Fs(2)2054 5465 y Ft(\))17 b Fo(\001)g(\001)g(\001)d Fp(g)t Ft(\()p Fp(a)2386 5480 y Fn(n)2439 5465 y Ft(\))p Fp(x)2535 5480 y Fn(n)2588 5465 y Ft(.)79 b(The)45 b(follo)m(wing)g(c)m (harac-)113 5584 y(terization)33 b(of)g(the)h(monoid)e Fo(M)h Ft(de\014ned)i(in)e(Section)h(2.1)f(is)g(therefore)h(relev)-6 b(an)m(t.)p eop %%Page: 57 57 57 56 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(57)113 428 y Fr(Prop)s(osition)41 b(A1.2)57 b Fe(Let)36 b Fp(g)t Ft(\()p Fp(m)p Ft(\))30 b(=)1561 287 y Fk(\022)1651 368 y Ft(0)119 b(1)1651 488 y(1)100 b Fp(m)1905 287 y Fk(\023)1978 428 y Fe(,)36 b(where)g Fp(m)31 b Fo(\025)g Ft(1)p Fe(.)50 b Fo(M)35 b Fe(is)g(the)h Fl(fr)-5 b(e)g(e)36 b(monoid)p Fe(,)113 607 y(with)f(unit,)f(generated)i(b)m(y)f(the)f (elemen)m(ts)h Fp(g)t Ft(\()p Fp(m)p Ft(\))p Fe(,)f Fp(m)c Fo(\025)h Ft(1)j Fe(:)47 b(eac)m(h)35 b(elemen)m(t)g Fp(g)i Fe(of)e Fo(M)f Fe(can)h(b)s(e)113 727 y(written)e(as)1078 846 y Fp(g)d Ft(=)f Fp(g)t Ft(\()p Fp(m)1440 861 y Fs(1)1484 846 y Ft(\))17 b Fo(\001)g(\001)g(\001)d Fp(g)t Ft(\()p Fp(m)1850 861 y Fn(r)1893 846 y Ft(\))28 b Fp(;)71 b(r)31 b Fo(\025)d Ft(0)f Fp(;)44 b(m)2476 861 y Fn(i)2537 846 y Fo(\025)29 b Ft(1)113 1036 y Fe(and)34 b(this)f(decomp)s(osition)g (is)g(unique.)113 1185 y Fl(Pr)-5 b(o)g(of.)36 b Fe(Let)f Fo(M)713 1149 y Fm(\003)793 1185 y Fe(b)s(e)g(the)g(monoid)f(with)h (unit)f(generated)i(b)m(y)f(the)g Fp(g)t Ft(\()p Fp(m)p Ft(\))p Fp(;)17 b(m)29 b Fo(\025)i Ft(1)p Fe(.)48 b(If)35 b Fp(m)c Fo(\025)g Ft(1)113 1305 y Fe(one)j(has)g Fp(g)t Ft(\()p Fp(m)p Ft(\))26 b Fo(2)j(M)j Fe(and)i Fo(M)p Fp(g)t Ft(\()p Fp(m)p Ft(\))26 b Fo(\032)j(M)p Fe(,)j(th)m(us)i Fo(M)2146 1268 y Fm(\003)2219 1305 y Fo(\032)29 b(M)p Fe(.)313 1484 y Ft(Con)m(v)m(ersely)45 b(let)g Fp(g)50 b Fo(2)e(M)p Ft(,)f Fp(g)k Fo(6)p Ft(=)d(id.)79 b(W)-8 b(e)46 b(no)m(w)f(pro)m(v)m(e)h(that)f(there)g(exists)g(a)g(unique)113 1658 y(in)m(teger)39 b Fp(m)d Fo(\025)g Ft(1)i(suc)m(h)h(that)1222 1517 y Fk(\022)1312 1598 y Fp(a)1365 1562 y Fm(0)1496 1598 y Fp(b)1539 1562 y Fm(0)1316 1718 y Fp(c)1359 1681 y Fm(0)1492 1718 y Fp(d)1544 1681 y Fm(0)1588 1517 y Fk(\023)1697 1658 y Ft(=)d Fp(g)1862 1622 y Fm(0)1925 1658 y Ft(=)g Fp(g)t Ft(\()p Fp(g)t Ft(\()p Fp(m)p Ft(\)\))2385 1622 y Fm(\000)p Fs(1)2525 1658 y Fo(2)g(M)f Fp(;)j Ft(whic)m(h)g (leads)h(to)e(the)113 1837 y(conditions)e Fp(b)634 1801 y Fm(0)692 1837 y Ft(=)c Fp(a)f(;)77 b(d)1040 1801 y Fm(0)1098 1837 y Ft(=)31 b Fp(c)f(;)77 b(a)1437 1801 y Fm(0)1495 1837 y Ft(=)31 b Fp(b)23 b Fo(\000)g Fp(ma)31 b(;)77 b(c)2088 1801 y Fm(0)2146 1837 y Ft(=)31 b Fp(d)23 b Fo(\000)h Fp(mc)30 b(:)35 b Ft(W)-8 b(e)35 b(consider)h(separately) 113 1957 y(three)e(cases.)174 2076 y(1\))50 b Fp(a)33 b Ft(=)i(0,)j(th)m(us)g Fp(b)c Ft(=)g Fp(c)g Ft(=)h(1)i(and)h Fp(g)f Ft(=)d Fp(g)t Ft(\()p Fp(d)p Ft(\))i(where)i Fp(d)c Fo(\025)h Ft(1.)55 b(If)37 b(there)h(w)m(ere)g Fp(m)d Fo(\025)f Ft(1)j(suc)m(h)313 2196 y(that)f Fp(g)584 2160 y Fm(0)644 2196 y Ft(=)e Fp(g)t Ft(\()p Fp(g)t Ft(\()p Fp(m)p Ft(\)\))1102 2160 y Fm(\000)p Fs(1)1240 2196 y Fo(2)f(M)p Ft(,)k(and)g Fp(g)1773 2160 y Fm(0)1833 2196 y Fo(6)p Ft(=)d(id,)j(one)g(should)g(ha)m(v)m(e)g Fp(b)2873 2160 y Fm(0)2934 2196 y Ft(=)d(0,)j(th)m(us)g Fp(a)3435 2160 y Fm(0)3496 2196 y Ft(=)d(0)313 2315 y(and)f Fp(a)559 2279 y Fm(0)587 2315 y Fp(d)639 2279 y Fm(0)689 2315 y Fo(\000)23 b Fp(b)832 2279 y Fm(0)859 2315 y Fp(c)902 2279 y Fm(0)958 2315 y Ft(=)28 b(0)33 b(whic)m(h)h(is)f(imp)s(ossible.) 174 2435 y(2\))50 b Fp(a)39 b Ft(=)g(1,)j(th)m(us)f Fp(b;)17 b(c)39 b Fo(\025)g Ft(1)h(and)h Fp(d)e Ft(=)h Fp(bc)27 b Fo(\006)g Ft(1.)65 b(W)-8 b(e)40 b(therefore)i(ha)m(v)m(e)f Fp(b)2896 2399 y Fm(0)2963 2435 y Ft(=)e(1)h(and)h Fp(a)3423 2399 y Fm(0)3490 2435 y Ft(=)f(0)313 2555 y(or)k Fp(a)499 2518 y Fm(0)574 2555 y Ft(=)j(1.)79 b(If)45 b Fp(a)1019 2518 y Fm(0)1094 2555 y Ft(=)i(0)e(then)g Fp(b)i Ft(=)h Fp(m)d Ft(and)g Fp(c)2146 2518 y Fm(0)2221 2555 y Ft(=)j Fp(d)29 b Fo(\000)i Fp(bc)p Ft(,)47 b(whic)m(h)f(is)e(admissible)h(if) 313 2674 y(and)37 b(only)g(if)g Fp(d)d Ft(=)h Fp(bc)25 b Ft(+)g(1.)56 b(If)37 b Fp(a)1529 2638 y Fm(0)1591 2674 y Ft(=)e(1)i(then)h Fp(b)c Ft(=)g Fp(m)26 b Ft(+)f(1)37 b(from)f(whic)m(h)i(it)f(follo)m(ws)g(that)313 2794 y Fp(c)356 2758 y Fm(0)411 2794 y Ft(=)29 b Fp(d)12 b Fo(\000)g Fp(mc)27 b Ft(=)h Fp(bc)12 b Fo(\006)g Ft(1)g Fo(\000)g Fp(mc)27 b Ft(=)h Fp(c)12 b Fo(\006)g Ft(1,)28 b(whic)m(h)h(is)f (admissible)f(if)h(and)h(only)e(if)h Fp(d)g Ft(=)g Fp(bc)12 b Fo(\000)g Ft(1,)313 2913 y(and)33 b(then)h Fp(b;)17 b(c)27 b Fo(\025)h Ft(2.)174 3033 y(3\))50 b Fp(a)36 b(>)h Ft(1.)60 b(Since)40 b Fp(a)973 2997 y Fm(0)1026 3033 y Fo(^)27 b Fp(b)1162 2997 y Fm(0)1226 3033 y Ft(=)37 b(1)i(the)g(relation)f(0)e Fo(\024)i Fp(a)2231 2997 y Fm(0)2295 3033 y Ft(=)f Fp(b)26 b Fo(\000)g Fp(ma)37 b Fo(\024)g Fp(b)2915 2997 y Fm(0)2980 3033 y Ft(=)g Fp(a)h Ft(determines)313 3152 y(uniquely)f Fp(m)f Fo(\025)f Ft(1)j(and)g(one)g(has)g(0)d Fp(<)g(a)1861 3116 y Fm(0)1924 3152 y Fp(<)g(b)2079 3116 y Fm(0)2107 3152 y Ft(.)57 b(But)37 b(one)i(also)e(has)h Fp(b)3011 3116 y Fm(0)3074 3152 y Ft(=)d Fp(a)g Fo(\024)g Fp(c)g Ft(=)h Fp(d)3629 3116 y Fm(0)313 3272 y Ft(and)d Fo(j)p Fp(a)587 3236 y Fm(0)615 3272 y Fp(d)667 3236 y Fm(0)717 3272 y Fo(\000)22 b Fp(b)859 3236 y Fm(0)887 3272 y Fp(c)930 3236 y Fm(0)958 3272 y Fo(j)27 b Ft(=)h(1,)33 b(from)g(whic)m(h)h(one)f(easily)g(gets)g Fp(d)2464 3236 y Fm(0)2520 3272 y Fo(\025)28 b Fp(c)2668 3236 y Fm(0)2724 3272 y Fo(\025)g Fp(a)2882 3236 y Fm(0)2909 3272 y Ft(.)642 b Fd(\003)113 3541 y Fr(A2.)51 b(App)s(endix)36 b(2)j(:)51 b(Hyp)s(erfunctions)113 3690 y(A2.1)31 b Ft(W)-8 b(e)31 b(follo)m(w)g(here)h([H],)d(Chapter)j(9.)43 b(Let)31 b Fp(K)37 b Ft(b)s(e)32 b(a)e(non)i(empt)m(y)e(compact)g(subset)i(of)f Fq(R)5 b Ft(.)113 3810 y(A)36 b Fl(hyp)-5 b(erfunction)37 b(with)h(supp)-5 b(ort)38 b(in)d Fp(K)43 b Ft(is)35 b(a)h(linear)g (functional)g Fp(u)g Ft(on)f(the)i(space)f Fo(O)s Ft(\()p Fp(K)7 b Ft(\))35 b(of)113 3929 y(functions)g(analytic)c(in)i(a)g (neigh)m(b)s(orho)s(o)s(d)g(of)h Fp(K)39 b Ft(suc)m(h)c(that)d(for)h (all)f(neigh)m(b)s(orho)s(o)s(d)i Fp(V)55 b Ft(of)33 b Fp(K)113 4049 y Ft(there)h(is)f(a)g(constan)m(t)h Fp(C)1024 4064 y Fn(V)1121 4049 y Fp(>)29 b Ft(0)k(suc)m(h)h(that)1123 4288 y Fo(j)p Fp(u)p Ft(\()p Fp(')p Ft(\))p Fo(j)27 b(\024)h Fp(C)1582 4303 y Fn(V)1668 4288 y Ft(sup)1711 4372 y Fn(V)1834 4288 y Fo(j)p Fp(')p Fo(j)f Fp(;)100 b Fo(8)p Fp(')28 b Fo(2)g(O)s Ft(\()p Fp(V)22 b Ft(\))28 b Fp(:)113 4586 y Ft(W)-8 b(e)49 b(denote)h(b)m(y)f Fp(A)866 4550 y Fm(0)893 4586 y Ft(\()p Fp(K)7 b Ft(\))48 b(the)h(space)h(of)e(h)m (yp)s(erfunctions)j(with)d(supp)s(ort)i(in)e Fp(K)7 b Ft(.)90 b(It)48 b(is)h(a)113 4706 y(F)-8 b(r)m(\023)-47 b(ec)m(het)36 b(space)f(:)47 b(a)34 b(seminorm)f(is)i(asso)s(ciated)g (to)f(eac)m(h)h(neigh)m(b)s(orho)s(o)s(d)g Fp(V)57 b Ft(of)35 b Fp(K)7 b Ft(.)48 b(One)35 b(has)113 4825 y(the)f(follo)m (wing)f(prop)s(osition.)p eop %%Page: 58 58 58 57 bop 113 144 a Ft(58)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y Fr(Prop)s(osition)31 b(A1.1)84 b Fe(The)27 b(spaces)h Fp(A)1614 347 y Fm(0)1642 383 y Ft(\()p Fp(K)7 b Ft(\))26 b Fe(and)h Fo(O)2107 347 y Fs(1)2152 383 y Ft(\()p 2191 303 72 4 v Fq(C)34 b Fo(n)9 b Fp(K)e Ft(\))26 b Fe(are)h(canonically)f(isomorphic.)113 503 y(T)-8 b(o)33 b(eac)m(h)h Fp(u)28 b Fo(2)g Fp(A)738 466 y Fm(0)766 503 y Ft(\()p Fp(K)7 b Ft(\))32 b Fe(corresp)s(onds)j Fp(')28 b Fo(2)g(O)1784 466 y Fs(1)1829 503 y Ft(\()p 1868 422 V Fq(C)48 b Fo(n)21 b Fp(K)7 b Ft(\))33 b Fe(giv)m(en)g(b)m(y) 1290 732 y Fp(')p Ft(\()p Fp(z)t Ft(\))28 b(=)g Fp(u)p Ft(\()p Fp(c)1755 747 y Fn(z)1801 732 y Ft(\))f Fp(;)44 b Fo(8)p Fp(z)33 b Fo(2)28 b Fq(C)48 b Fo(n)22 b Fp(K)35 b(;)113 962 y Fe(where)56 b Fp(c)466 977 y Fn(z)512 962 y Ft(\()p Fp(x)p Ft(\))63 b(=)867 922 y Fs(1)p 862 939 50 4 v 862 996 a Fn(\031)990 922 y Fs(1)p 935 939 149 4 v 935 996 a Fn(x)p Fm(\000)p Fn(z)1096 962 y Fe(.)108 b(Con)m(v)m(ersely)55 b(to)f(eac)m(h)i Fp(')63 b Fo(2)h(O)2487 925 y Fs(1)2532 962 y Ft(\()p 2571 881 72 4 v Fq(C)e Fo(n)36 b Fp(K)7 b Ft(\))54 b Fe(corresp)s(onds)i(the)113 1081 y(h)m(yp)s(erfunction)1124 1246 y Fp(u)p Ft(\()p Fp( )t Ft(\))27 b(=)1510 1179 y Fp(i)p 1472 1223 111 4 v 1472 1314 a Ft(2)p Fp(\031)1610 1110 y Fk(Z)1666 1337 y Fn(\015)1734 1246 y Fp(')p Ft(\()p Fp(z)t Ft(\))p Fp( )t Ft(\()p Fp(z)t Ft(\))p Fp(dz)32 b(;)44 b Fo(8)p Fp( )31 b Fo(2)d Fp(A)113 1507 y Fe(where)34 b Fp(\015)j Fe(is)32 b(an)m(y)g(piecewise)i Fo(C)1266 1471 y Fs(1)1343 1507 y Fe(path)e(winding)h(around)g Fp(K)39 b Fe(in)32 b(the)h(p)s(ositiv)m(e)f(direction.)43 b(W)-8 b(e)113 1640 y(will)33 b(also)f(use)j(the)e(notation)g Fp(u)p Ft(\()p Fp(x)p Ft(\))27 b(=)1597 1600 y(1)p 1580 1617 85 4 v 1580 1686 a(2)p Fp(i)1676 1640 y Ft([)p Fp(')p Ft(\()p Fp(x)22 b Ft(+)g Fp(i)p Ft(0\))g Fo(\000)h Fp(')p Ft(\()p Fp(x)f Fo(\000)g Fp(i)p Ft(0\)])33 b Fe(for)h(short.)113 1864 y Fr(A2.2)h Ft(Let)g Fq(T)624 1828 y Fs(1)702 1864 y Ft(=)c Fq(R)5 b Fp(=)p Fq(Z)33 b Fo(\032)e Fq(C)19 b Fp(=)p Fq(Z)p Ft(.)52 b(A)34 b Fl(hyp)-5 b(erfunction)37 b(on)d Fq(T)k Ft(is)d(a)f(linear)h(fun)m(tional)g Fp(U)46 b Ft(on)34 b(the)113 1984 y(space)i Fo(O)s Ft(\()p Fq(T)565 1947 y Fs(1)613 1984 y Ft(\))e(of)h(functions)h(analytic)e(in)g(a)h (complex)e(neigh)m(b)s(orho)s(o)s(d)i(of)g Fq(T)3011 1947 y Fs(1)3094 1984 y Ft(suc)m(h)h(that)e(for)113 2103 y(all)f(neigh)m(b)s(orho)s(o)s(d)g Fp(V)56 b Ft(of)33 b Fq(T)k Ft(there)d(exists)f Fp(C)1805 2118 y Fn(V)1902 2103 y Fp(>)28 b Ft(0)33 b(suc)m(h)i(that)1102 2333 y Fo(j)p Fp(U)11 b Ft(\(\010\))p Fo(j)26 b(\024)i Fp(C)1589 2348 y Fn(V)1675 2333 y Ft(sup)1718 2416 y Fn(V)1842 2333 y Fo(j)p Ft(\010)p Fo(j)f Fp(;)99 b Fo(8)p Ft(\010)28 b Fo(2)g(O)s Ft(\()p Fp(V)22 b Ft(\))28 b Fp(:)113 2627 y Ft(W)-8 b(e)27 b(will)f(denote)h Fp(A)849 2591 y Fm(0)877 2627 y Ft(\()p Fq(T)979 2591 y Fs(1)1027 2627 y Ft(\).)41 b(the)27 b(F)-8 b(r)m(\023)-47 b(ec)m(het)28 b(space)f(of)g(h)m(yp)s (erfunctions)i(with)d(supp)s(ort)h(in)g Fq(T)p Ft(.)45 b(F)-8 b(or)113 2747 y Fp(U)39 b Fo(2)28 b Fp(A)389 2711 y Fm(0)417 2747 y Ft(\()p Fq(T)p Ft(\),)35 b(let)781 2721 y(^)763 2747 y Fp(U)11 b Ft(\()p Fp(n)p Ft(\))27 b(:=)h Fp(U)11 b Ft(\()p Fp(e)1304 2762 y Fm(\000)p Fn(n)1420 2747 y Ft(\))32 b(with)g Fp(e)1763 2762 y Fn(n)1818 2747 y Ft(\()p Fp(z)t Ft(\))c(=)g Fp(e)2125 2711 y Fs(2)p Fn(\031)s(inz)2337 2747 y Ft(.)44 b(The)32 b(doubly)h(in\014nite)f (sequence)113 2866 y(\()169 2841 y(^)152 2866 y Fp(U)11 b Ft(\()p Fp(n)p Ft(\)\))408 2881 y Fn(n)p Fm(2)p Fg(Z)595 2866 y Ft(satis\014es)1461 2986 y Fo(j)1506 2961 y Ft(^)1489 2986 y Fp(U)g Ft(\()p Fp(n)p Ft(\))p Fo(j)26 b Fp(<)i(C)1936 3001 y Fn(")1979 2986 y Fp(e)2025 2945 y Fs(2)p Fn(\031)s Fm(j)p Fn(n)p Fm(j)p Fn(")2281 2986 y Fp(:)113 3171 y Ft(for)j(all)f Fp(")e(>)g Ft(0)i(and)h(for)g(all)f Fp(n)d Fo(2)h Fq(Z)g Ft(with)i(a)h(suitably)e(c)m(hosen)j Fp(C)2481 3186 y Fn(")2552 3171 y Fp(>)c Ft(0.)43 b(Con)m(v)m(ersely)31 b(an)m(y)f(suc)m(h)113 3290 y(sequence)35 b(is)e(the)h(F)-8 b(ourier)33 b(expansion)h(of)f(a)g(unique)h(h)m(yp)s(erfunction)h(with) e(supp)s(ort)h(in)f Fq(T)p Ft(.)313 3410 y(Let)25 b Fo(O)562 3425 y Fs(\006)650 3410 y Ft(denote)i(the)e(complex)g(v)m(ector)h (space)g(of)g(holomorphic)f(functions)h(\010)45 b(:)27 b Fq(C)g Fo(n)7 b Fq(R)44 b Fo(!)113 3529 y Fq(C)20 b Ft(,)37 b(1{p)s(erio)s(dic,)29 b(b)s(ounded)j(at)e Fo(\006)p Fp(i)p Fo(1)g Ft(and)g(suc)m(h)i(that)d(\010\()p Fo(\006)p Fp(i)p Fo(1)p Ft(\))f(:=)g(lim)2798 3544 y Fm(=)p Fn(m)13 b(z)s Fm(!\0061)3224 3529 y Ft(\010\()p Fp(z)t Ft(\))30 b(exist)113 3649 y(and)k(v)m(erify)f(\010\(+)p Fp(i)p Fo(1)p Ft(\))28 b(=)g Fo(\000)p Ft(\010\()p Fo(\000)p Fp(i)p Fo(1)p Ft(\).)113 3826 y Fr(Prop)s(osition)44 b(A2.2)95 b Fe(The)39 b(spaces)g Fp(A)1661 3790 y Fm(0)1689 3826 y Ft(\()p Fq(T)1791 3790 y Fs(1)1839 3826 y Ft(\))e Fe(and)i Fo(O)2193 3841 y Fs(\006)2293 3826 y Fe(are)f(canonically)f (isomorphic.)57 b(T)-8 b(o)113 3946 y(eac)m(h)35 b Fp(U)j Fo(2)28 b Fp(A)613 3910 y Fm(0)641 3946 y Ft(\()p Fq(T)743 3910 y Fs(1)791 3946 y Ft(\))33 b Fe(corresp)s(onds)i Ft(\010)27 b Fo(2)h(O)1682 3961 y Fs(\006)1778 3946 y Fe(giv)m(en)33 b(b)m(y)1262 4175 y Ft(\010\()p Fp(z)t Ft(\))27 b(=)i Fp(U)11 b Ft(\()p Fp(C)1784 4190 y Fn(z)1829 4175 y Ft(\))28 b Fp(;)44 b Fo(8)p Fp(z)32 b Fo(2)c Fq(C)49 b Fo(n)21 b Fp(K)35 b(;)113 4405 y Fe(where)54 b Fp(C)492 4420 y Fn(z)538 4405 y Ft(\()p Fp(x)p Ft(\))59 b(=)i(cotg)16 b Fp(\031)t Ft(\()p Fp(x)34 b Fo(\000)h Fp(z)t Ft(\))p Fe(.)103 b(Con)m(v)m(ersely)53 b(to)f(eac)m(h)h Ft(\010)60 b Fo(2)g(O)2837 4420 y Fs(\006)2952 4405 y Fe(corresp)s(onds)54 b(the)113 4524 y(h)m(yp)s(erfunction)1030 4689 y Fp(U)11 b Ft(\(\011\))28 b(=)1417 4622 y Fp(i)p 1409 4666 50 4 v 1409 4757 a Ft(2)1487 4554 y Fk(Z)1543 4780 y Fs(\000)1614 4689 y Ft(\010\()p Fp(z)t Ft(\)\011\()p Fp(z)t Ft(\))p Fp(dz)33 b(;)44 b Fo(8)p Ft(\011)28 b Fo(2)g Fp(A)p Ft(\()p Fq(T)2653 4648 y Fs(1)2701 4689 y Ft(\))113 4936 y Fe(where)j Ft(\000)e Fe(is)g(an)m(y)h(piecewise)g Fo(C)1256 4900 y Fs(1)1330 4936 y Fe(path)g(winding)f(around)h(a)f(closed)h(in)m(terv) -6 b(al)29 b Fp(I)36 b Fo(\032)28 b Fq(R)40 b Fe(of)29 b(length)113 5056 y Ft(1)k Fe(in)h(the)f(p)s(ositiv)m(e)g(direction.)44 b(W)-8 b(e)34 b(will)e(also)h(use)h(the)f(notation)1157 5327 y Fp(U)11 b Ft(\()p Fp(x)p Ft(\))27 b(=)1532 5259 y(1)p 1515 5304 85 4 v 1515 5395 a(2)p Fp(i)1611 5327 y Ft([\010\()p Fp(x)22 b Ft(+)g Fp(i)p Ft(0\))g Fo(\000)h Ft(\010\()p Fp(x)f Fo(\000)g Fp(i)p Ft(0\)])113 5584 y Fe(for)34 b(short.)p eop %%Page: 59 59 59 58 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(59)313 383 y(The)33 b(nice)h(fact)g(is)f(that)g(the)g(follo)m(wing)g (diagram)f(comm)m(utes)g(:)1141 827 y Fp(A)1216 791 y Fm(0)1244 827 y Ft(\([0)p Fp(;)17 b Ft(1]\))97 b Fo(\000)-40 b(\000)-21 b(\000)f(\000)g(\000)-40 b(!)100 b(O)2142 791 y Fs(1)2186 827 y Ft(\()p 2225 746 72 4 v Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1]\))1114 1059 y Fk(P)1219 1163 y Fj(Z)1291 951 y Fk(?)1291 1011 y(?)1291 1071 y(?)1291 1131 y(?)1291 1191 y(y)2305 951 y(?)2305 1011 y(?)2305 1071 y(?)2305 1131 y(?)2305 1191 y(y)2400 1059 y(P)2505 1163 y Fj(Z)1185 1431 y Fp(A)1260 1395 y Fm(0)1288 1431 y Ft(\()p Fq(T)1390 1395 y Fs(1)1437 1431 y Ft(\))144 b Fo(\000)-40 b(\000)-21 b(\000)f(\000)g(\000)-40 b(!)314 b(O)2353 1446 y Fs(\006)113 1866 y Ft(the)28 b(horizon)m(tal)f(lines)g (are)g(the)h(ab)s(o)m(v)m(e)f(men)m(tioned)g(isomorphisms,)2687 1792 y Fk(P)2792 1896 y Fn(Z)2883 1866 y Ft(is)g(de\014ned)i(in)e (2.2.2.)113 2117 y Fr(A3.)51 b(App)s(endix)36 b(3)j(:)51 b(Some)38 b(prop)s(erties)f(of)h(the)f(dilogarithm)113 2272 y(A3.1)49 b Ft(The)h(classical)f(dilogarithmic)d(series)k(\(see)g ([Le],)i([O])d(for)g(more)f(information\))g(is)113 2391 y(de\014ned)36 b(b)m(y)1539 2605 y(Li)1629 2620 y Fs(2)1674 2605 y Ft(\()p Fp(z)t Ft(\))28 b(=)1940 2481 y Fs(+)p Fm(1)1938 2511 y Fk(X)1935 2722 y Fn(n)p Fs(=1)2114 2538 y Fp(z)2164 2502 y Fn(n)p 2114 2582 105 4 v 2114 2674 a Fp(n)2174 2645 y Fs(2)3377 2605 y Ft(\()p Fp(A)p Ft(3)p Fp(:)p Ft(1\))113 2912 y(and)34 b(it)f(is)g(con)m(v)m(ergen)m(t)i(for)f Fo(j)p Fp(z)t Fo(j)28 b(\024)g Ft(1.)44 b(Since)35 b Fo(\000)17 b Ft(log)q(\(1)k Fo(\000)i Fp(z)t Ft(\))29 b(=)2437 2837 y Fk(P)2542 2861 y Fs(+)p Fm(1)2542 2941 y Fn(n)p Fs(=1)2726 2872 y Fn(z)2767 2842 y Fj(n)p 2726 2889 90 4 v 2746 2946 a Fn(n)2828 2912 y Ft(,)k(dividing)g(b)m(y)g Fp(z)38 b Ft(and)113 3031 y(in)m(tegrating)28 b(one)h(obtains)f(the)h (analytic)e(con)m(tin)m(uation)i(of)f(the)h(dilogarithm)d(to)i Fq(C)k Fo(n)12 b Ft([1)p Fp(;)17 b Ft(+)p Fo(1)p Ft(\))113 3151 y(b)m(y)34 b(means)f(of)g(the)h(in)m(tegral)f(form)m(ula)812 3460 y(Li)902 3475 y Fs(2)947 3460 y Ft(\()p Fp(z)t Ft(\))28 b(=)g Fo(\000)1302 3325 y Fk(Z)1402 3350 y Fn(z)1358 3551 y Fs(0)1476 3393 y Ft(log)q(\(1)22 b Fo(\000)g Fp(t)p Ft(\))p 1476 3438 414 4 v 1665 3529 a Fp(t)1902 3460 y(dt)28 b Ft(=)2123 3325 y Fk(Z)2222 3350 y Fn(z)2178 3551 y Fs(0)2285 3320 y Fk(\022)2358 3325 y(Z)2458 3350 y Fn(t)2414 3551 y Fs(0)2581 3393 y Fp(d\020)p 2521 3438 223 4 v 2521 3529 a Ft(1)22 b Fo(\000)h Fp(\020)2756 3320 y Fk(\023)2858 3393 y Fp(dt)p 2858 3438 88 4 v 2884 3529 a(t)3377 3460 y Ft(\()p Fp(A)p Ft(3)p Fp(:)p Ft(2\))113 3766 y(whic)m(h)30 b(w)m(e)f(will)f(use)h(as)g(a)g(de\014nition)g(of)g (the)g(dilogarithm.)40 b(Note)29 b(that)f([1)p Fp(;)17 b Ft(+)p Fo(1)p Ft(\))27 b(is)i(a)f(branc)m(h)113 3886 y(cut.)313 4008 y(Since)1388 4181 y(Li)1478 4196 y Fs(2)1523 4181 y Ft(\()p Fp(z)t Ft(\))g(=)g Fp(z)1851 4045 y Fk(Z)1951 4070 y Fs(1)1906 4272 y(0)2062 4113 y Ft(log)18 b Fp(t)p 2024 4158 259 4 v 2024 4249 a(tz)27 b Fo(\000)22 b Ft(1)2294 4181 y Fp(dt)113 4428 y Ft(one)34 b(ob)m(viously)f(has)g(that)1317 4620 y(Li)1407 4635 y Fs(2)1468 4479 y Fk(\022)1554 4553 y Ft(1)p 1554 4597 51 4 v 1554 4688 a Fp(z)1616 4479 y Fk(\023)1717 4620 y Ft(=)28 b Fo(\000)1916 4484 y Fk(Z)2016 4509 y Fs(1)1972 4711 y(0)2103 4553 y Ft(log)17 b Fp(t)p 2089 4597 209 4 v 2089 4688 a(z)27 b Fo(\000)c Fp(t)2310 4620 y(dt)k(;)924 b Ft(\()p Fp(A)p Ft(3)p Fp(:)p Ft(3\))113 4886 y(whic)m(h)34 b(sho)m(ws)h(that)d(Li)990 4901 y Fs(2)1052 4805 y Fk(\000)1110 4847 y Fs(1)p 1109 4863 41 4 v 1109 4920 a Fn(z)1162 4805 y Fk(\001)1241 4886 y Ft(is)h(the)h(Cauc)m(h)m(y{Hilb)s(ert)f(transform)g(of)g(the)h(real)f (function)1248 5180 y Fp(')1313 5195 y Fs(0)1358 5180 y Ft(\()p Fp(t)p Ft(\))27 b(=)1604 5069 y Fk(n)1687 5132 y Fo(\000)17 b Ft(log)g Fp(t)100 b Ft(if)33 b Fp(t)28 b Fo(2)g Ft([0)p Fp(;)17 b Ft(1])1687 5252 y(0)325 b(elsewhere)3377 5180 y(\()p Fp(A)p Ft(3)p Fp(:)p Ft(4\))113 5458 y(Note)33 b(also)g(that)1337 5584 y(Im)o(Li)1546 5599 y Fs(2)1590 5584 y Ft(\()p Fp(t)22 b Fo(\006)h Fp(i)p Ft(0\))k(=)h Fo(\006)p Fp(\031)21 b Ft(log)c Fp(t)28 b(;)943 b Ft(\()p Fp(A)p Ft(3)p Fp(:)p Ft(5\))p eop %%Page: 60 60 60 59 bop 113 144 a Ft(60)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y(where)34 b Fp(t)28 b Fo(2)g Ft([1)p Fp(;)17 b Ft(+)p Fo(1)p Ft(\).)43 b(Moreo)m(v)m(er)1127 614 y Fo(j)p Ft(Li)1244 629 y Fs(2)1289 614 y Ft(\()p Fp(z)t Ft(\))p Fo(j)27 b Ft(=)i Fo(O)s Ft(\(log)1827 569 y Fs(2)1888 614 y Fo(j)p Fp(z)t Fo(j)p Ft(\))f(as)p Fo(j)p Fp(z)t Fo(j)g(!)f Ft(+)p Fo(1)h Fp(:)733 b Ft(\()p Fp(A)p Ft(3)p Fp(:)p Ft(6\))113 873 y Fr(A3.2)34 b Fl(Euler's)h (functional)g(e)-5 b(quations)950 1162 y Ft(Li)1040 1177 y Fs(2)1085 1162 y Ft(\()p Fp(z)t Ft(\))22 b(+)h(Li)1425 1177 y Fs(2)1486 1022 y Fk(\022)1572 1095 y Ft(1)p 1572 1139 51 4 v 1572 1231 a Fp(z)1634 1022 y Fk(\023)1735 1162 y Ft(=)28 b Fo(\000)1929 1095 y Ft(1)p 1929 1139 50 4 v 1929 1231 a(2)1992 1162 y(\(log\()p Fo(\000)p Fp(z)t Ft(\)\))2403 1121 y Fs(2)2470 1162 y Fo(\000)2582 1095 y Fp(\031)2643 1059 y Fs(2)p 2582 1139 106 4 v 2609 1231 a Ft(6)2727 1162 y Fp(;)622 b Ft(\()p Fp(A)p Ft(3)p Fp(:)p Ft(7\))888 1445 y(Li)978 1460 y Fs(2)1023 1445 y Ft(\()p Fp(z)t Ft(\))22 b(+)h(Li)1363 1460 y Fs(2)1408 1445 y Ft(\(1)e Fo(\000)i Fp(z)t Ft(\))28 b(=)g Fo(\000)17 b Ft(log)h Fp(z)j Ft(log)q(\(1)g Fo(\000)i Fp(z)t Ft(\))g(+)2709 1378 y Fp(\031)2770 1342 y Fs(2)p 2709 1422 V 2737 1514 a Ft(6)2854 1445 y Fp(;)495 b Ft(\()p Fp(A)p Ft(3)p Fp(:)p Ft(8\))113 1710 y(where)34 b(z)g(v)-6 b(aries)33 b(in)h Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(+)p Fo(1)p Ft(])31 b(and)j Fq(C)48 b Fo(n)22 b Ft(\(\()p Fo(\0001)p Fp(;)17 b Ft(0])k Fo([)h Ft([1)p Fp(;)17 b Ft(+)p Fo(1)p Ft(\)\))32 b(resp)s(ectiv)m(ely)-8 b(.)113 1858 y Fr(A3.3)34 b Fl(Sp)-5 b(e)g(cial)35 b(values)808 2147 y Ft(Li)898 2162 y Fs(2)943 2147 y Ft(\(1\))27 b(=)1215 2080 y Fp(\031)1276 2043 y Fs(2)p 1215 2124 V 1242 2215 a Ft(6)1360 2147 y Fp(;)71 b Ft(Li)1549 2162 y Fs(2)1594 2147 y Ft(\()p Fo(\000)p Ft(1\))27 b(=)h Fo(\000)2020 2080 y Fp(\031)2081 2043 y Fs(2)p 2021 2124 V 2024 2215 a Ft(12)2166 2147 y Fp(;)659 2404 y Ft(2Li)798 2419 y Fs(2)843 2404 y Ft(\(1)p Fp(=)p Ft(2\))f(=)1215 2336 y Fp(\031)1276 2300 y Fs(2)p 1215 2381 V 1242 2472 a Ft(6)1354 2404 y Fo(\000)c Ft(\(log)17 b(2\))1727 2363 y Fs(2)1799 2404 y Fp(;)71 b Ft(Li)1988 2419 y Fs(2)2033 2404 y Ft(\(2)22 b Fo(\006)g Fp(i)p Ft(0\))28 b(=)2511 2336 y Fp(\031)2572 2300 y Fs(2)p 2511 2381 V 2538 2472 a Ft(4)2650 2404 y Fo(\006)23 b Fp(\031)t(i)17 b Ft(log)f(2)28 b Fp(:)113 2778 y Fr(A4.)51 b(App)s(endix)36 b(4)j(:)51 b(Ev)m(en)38 b(Brjuno)g(functions)113 2927 y Ft(In)26 b([MMY])g(w)m(e)g(also)f(considered)j(an)d(ev)m(en)i(v) m(ersion)f(of)g(the)g(Brjuno)g(function)h(and)f(w)m(e)g(pro)m(v)m(ed) 113 3046 y(that)g(this)g(di\013ers)h(from)e(the)i(one)f(considered)i (here)f(b)m(y)f(a)g(1)p Fp(=)p Ft(2{H\177)-50 b(older)25 b(con)m(tin)m(uous)j(function.)113 3166 y(In)f(this)e(app)s(endix)i(w)m (e)f(explicit)f(the)h(relation)g(among)e(the)j(t)m(w)m(o)e(asso)s (ciated)h(complex)f(Brjuno)113 3285 y(functons.)113 3469 y Fr(A4.1)34 b Ft(Let)f Fp(\033)k Ft(denote)d(the)f(matrix)1468 3328 y Fk(\022)1558 3409 y Fo(\000)p Ft(1)100 b(1)1597 3528 y(0)138 b(1)1852 3328 y Fk(\023)1958 3469 y Ft(whic)m(h)34 b(corresp)s(onds)h(to)e Fp(x)28 b Fo(7!)f Ft(1)22 b Fo(\000)g Fp(x)p Ft(.)313 3648 y(A)m(t)35 b(the)i Fl(r)-5 b(e)g(al)38 b(level)p Ft(,)f(replacing)f(p)s(erio)s(dic)g(ev)m(en)g(functions)i (with)d(functions)j(on)e([0)p Fp(;)17 b Ft(1)p Fp(=)p Ft(2])113 3768 y(and)58 b(n)m(ull)g(outside)g(this)g(in)m(terv)-6 b(al,)63 b(the)58 b(op)s(erator)f Fp(T)2237 3783 y Fn(ev)s(en)2467 3768 y Ft(acting)g(on)g Fp(L)3020 3731 y Fs(2)3065 3768 y Ft(\([0)p Fp(;)17 b Ft(1)p Fp(=)p Ft(2]\))55 b(\(for)113 3887 y(example\))32 b(can)i(b)s(e)g(written)f(explicitely)f(as)751 4178 y Fp(T)809 4193 y Fn(ev)s(en)981 4178 y Fp(f)11 b Ft(\()p Fp(x)p Ft(\))27 b(=)1322 4083 y Fk(X)1308 4297 y Fn(m)p Fm(\025)p Fs(2)1497 4178 y Fp(xf)1630 4037 y Fk(\022)1719 4110 y Ft(1)p 1715 4155 57 4 v 1715 4246 a Fp(x)1806 4178 y Fo(\000)c Fp(m)1993 4037 y Fk(\023)2089 4178 y Ft(+)2203 4083 y Fk(X)2189 4297 y Fn(m)p Fm(\025)p Fs(3)2378 4178 y Fp(xf)2511 4037 y Fk(\022)2584 4178 y Fp(m)g Fo(\000)2809 4110 y Ft(1)p 2805 4155 V 2805 4246 a Fp(x)2874 4037 y Fk(\023)2992 4178 y Fp(:)313 4529 y Ft(A)m(t)44 b(the)g Fl(c)-5 b(omplex)47 b(level)d Ft(\(i.e.)77 b(after)44 b(the)h(iden)m(ti\014cation)f(of)h Fo(A)2759 4493 y Fm(0)2786 4529 y Ft(\([0)p Fp(;)17 b Ft(1)p Fp(=)p Ft(2]\))42 b(to)i Fo(O)3422 4493 y Fs(1)3467 4529 y Ft(\()p 3506 4449 72 4 v Fq(C)55 b Fo(n)113 4649 y Ft([0)p Fp(;)17 b Ft(1)p Fp(=)p Ft(2]\)\))31 b(one)j(gets)1095 4788 y Fp(T)1153 4803 y Fn(ev)s(en)1325 4788 y Fp(')28 b Ft(=)1538 4694 y Fk(X)1523 4908 y Fn(m)p Fm(\025)p Fs(2)1713 4788 y Fp(L)1781 4806 y Fn(g)s Fs(\()p Fn(m)p Fs(\))1960 4788 y Fp(')22 b Ft(+)2161 4694 y Fk(X)2147 4908 y Fn(m)p Fm(\025)p Fs(2)2336 4788 y Fp(L)2404 4806 y Fn(g)2446 4786 y Ff(0)2472 4806 y Fs(\()p Fn(m)p Fs(\))2610 4788 y Fp(')113 5151 y Ft(where)34 b Fp(g)453 5115 y Fm(0)481 5151 y Ft(\()p Fp(m)p Ft(\))27 b(=)778 5010 y Fk(\022)907 5091 y Ft(0)243 b(1)868 5211 y Fo(\000)p Ft(1)100 b Fp(m)23 b Ft(+)f(1)1371 5010 y Fk(\023)1472 5151 y Ft(=)28 b Fp(g)t Ft(\()p Fp(m)p Ft(\))p Fp(\033)i(:)113 5354 y Fr(A4.2)k Ft(W)-8 b(e)33 b(w)m(an)m(t)h(to)e(consider)787 5584 y(\(1)22 b Fo(\000)g Fp(T)1055 5599 y Fn(ev)s(en)1228 5584 y Ft(\))1267 5543 y Fm(\000)p Fs(1)1418 5584 y Ft(:)44 b Fo(O)1572 5543 y Fs(1)1616 5584 y Ft(\()p 1655 5504 V Fq(C)k Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1)p Fp(=)p Ft(2]\))26 b Fo(!)h(O)2396 5543 y Fs(1)2441 5584 y Ft(\()p 2480 5504 V Fq(C)47 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1)p Fp(=)p Ft(2]\))p eop %%Page: 61 61 61 60 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(61)113 383 y(and)32 b(then)g(one)f(will)f(ha)m(v)m(e)h(to)g(mak)m(e)f (the)h(resulting)g(function)h(ev)m(en)g(and)f(p)s(erio)s(dic,)g(th)m (us)h(one)113 503 y(will)h(tak)m(e)584 558 y Fk(X)626 772 y Fn(Z)728 652 y Ft(\(1)21 b(+)i Fp(L)1006 667 y Fn(\033)1059 652 y Ft(\)\(1)f Fo(\000)g Fp(T)1366 667 y Fn(ev)s(en)1539 652 y Ft(\))1578 611 y Fm(\000)p Fs(1)1729 652 y Ft(:)27 b Fo(O)1866 611 y Fs(1)1911 652 y Ft(\()p 1950 572 72 4 v Fq(C)48 b Fo(n)22 b Ft([0)p Fp(;)17 b Ft(1)p Fp(=)p Ft(2]\))25 b Fo(!)i(O)2687 667 y Fn(ev)s(en)2860 652 y Ft(\()p Fq(H)20 b Fp(=)p Fq(Z)p Ft(\))31 b Fp(:)313 891 y Ft(Note)h(that)h Fp(Z)d Fo(t)22 b Fp(Z)7 b(\033)31 b Ft(=)e Fp(Z)g Fo(t)22 b Fp(\033)t(Z)7 b Ft(.)313 1010 y(When)44 b(w)m(e)g(expand)f(\(1)29 b(+)g Fp(L)1417 1025 y Fn(\033)1470 1010 y Ft(\)\(1)f Fo(\000)h Fp(T)1790 1025 y Fn(ev)s(en)1963 1010 y Ft(\))2002 974 y Fm(\000)p Fs(1)2152 1010 y Ft(w)m(e)43 b(obtain)g(a)g(sum)2941 936 y Fk(P)3063 1010 y Fp(L)3131 1025 y Fn(g)3220 1010 y Ft(where)h(the)113 1130 y(matrices)33 b Fp(g)j Ft(ha)m(v)m(e)e(the)f (form)1271 1280 y Fp(g)e Ft(=)d Fp(")1501 1295 y Fs(0)1546 1280 y Fp(g)t Ft(\()p Fp(i)1671 1295 y Fs(1)1715 1280 y Ft(\))p Fp(")1800 1295 y Fs(1)1845 1280 y Fp(g)t Ft(\()p Fp(i)1970 1295 y Fs(2)2013 1280 y Ft(\))17 b Fp(:)g(:)g(:)e(g)t Ft(\()p Fp(i)2327 1295 y Fn(r)2370 1280 y Ft(\))p Fp(")2455 1295 y Fn(r)113 1430 y Ft(with)33 b Fp(r)e Fo(\025)d Ft(0,)33 b Fp(i)666 1445 y Fn(k)743 1430 y Fo(\025)28 b Ft(2)33 b(and)g Fp(")1170 1445 y Fn(k)1248 1430 y Fo(2)28 b(f)p Ft(1)p Fp(;)17 b(\033)t Fo(g)p Ft(.)313 1549 y(Note)41 b(that)h Fp(\033)t(g)t Ft(\()p Fp(i)p Ft(\))f(=)i Fp(g)t Ft(\(1\))p Fp(g)t Ft(\()p Fp(i)26 b Fo(\000)i Ft(1\))42 b(for)h(all)e Fp(i)i Fo(\025)g Ft(2,)h(th)m(us)f(all)e(matrices)g(of)i (the)f(form)113 1669 y Fp(")159 1684 y Fs(0)205 1669 y Fp(g)t Ft(\()p Fp(i)330 1684 y Fs(1)373 1669 y Ft(\))p Fp(")458 1684 y Fs(1)520 1669 y Fp(:)17 b(:)g(:)e(g)t Ft(\()p Fp(i)778 1684 y Fn(r)821 1669 y Ft(\))33 b(b)s(elong)g(to)g (the)h(monoid)e Fo(M)p Ft(.)113 1808 y Fr(A4.3)i Ft(Let)f Fp(r)d Fo(\025)f Ft(0,)1167 1928 y Fo(M)1287 1887 y Fs(\()p Fn(r)r Fs(\))1421 1928 y Ft(=)f Fo(f)p Fp(g)t Ft(\()p Fp(j)1708 1943 y Fs(1)1751 1928 y Ft(\))17 b Fp(:)g(:)g(:)d(g)t Ft(\()p Fp(j)2071 1943 y Fn(r)2114 1928 y Ft(\))28 b Fp(;)44 b(j)2294 1943 y Fn(k)2371 1928 y Fo(\025)28 b Ft(1)p Fo(g)113 2093 y Ft(and)k(let)470 2067 y Fk(c)446 2093 y Fo(M)566 2056 y Fs(\()p Fn(r)r Fs(\))703 2093 y Ft(denote)g(the)f(part)g(of)g Fo(M)1636 2056 y Fs(\()p Fn(r)r Fs(\))1773 2093 y Ft(made)f(of)h(matrices)f(whic)m(h)i(can)f(b)s (e)g(written)g(as)g(a)113 2212 y(pro)s(duct)j Fp(")533 2227 y Fs(0)578 2212 y Fp(g)t Ft(\()p Fp(i)703 2227 y Fs(1)747 2212 y Ft(\))p Fp(")832 2227 y Fs(1)894 2212 y Fp(:)17 b(:)g(:)e(g)t Ft(\()p Fp(i)1152 2227 y Fn(s)1193 2212 y Ft(\).)44 b(W)-8 b(e)34 b(ha)m(v)m(e)f(the)h(follo)m(wing)113 2387 y Fr(Lemma)58 b(A4.1)f Fe(Eac)m(h)51 b(matrix)c Fp(g)59 b Fo(2)1687 2362 y Fk(c)1663 2387 y Fo(M)1783 2351 y Fs(\()p Fn(r)r Fs(\))1939 2387 y Fe(can)50 b(b)s(e)h(uniquely)f (written)f(as)h(a)g(pro)s(duct)113 2506 y Fp(")159 2521 y Fs(0)205 2506 y Fp(g)t Ft(\()p Fp(i)330 2521 y Fs(1)373 2506 y Ft(\))p Fp(")458 2521 y Fs(1)520 2506 y Fp(:)17 b(:)g(:)e(g)t Ft(\()p Fp(i)778 2521 y Fn(s)819 2506 y Ft(\))p Fe(.)44 b(Moreo)m(v)m(er)34 b(one)g(has)f Fo(M)1848 2470 y Fs(\(0\))1983 2506 y Ft(=)2112 2481 y Fk(c)2088 2506 y Fo(M)2208 2470 y Fs(\(0\))2342 2506 y Ft(=)28 b Fo(f)p Ft(1)p Fo(g)33 b Fe(and)h(for)f(all)f Fp(r)f(>)d Ft(0)1265 2671 y Fo(M)1385 2630 y Fs(\()p Fn(r)r Fs(\))1513 2671 y Fo(n)1609 2646 y Fk(c)1585 2671 y Fo(M)1705 2630 y Fs(\()p Fn(r)r Fs(\))1838 2671 y Ft(=)1967 2646 y Fk(c)1944 2671 y Fo(M)2064 2630 y Fs(\()p Fn(r)r Fm(\000)p Fs(1\))2272 2671 y Fp(g)t Ft(\(1\))e Fp(:)113 2841 y Fl(Pr)-5 b(o)g(of.)42 b Ft(Uniqueness)h(is)e(eviden)m(t)i(\(just)f(consider)g(the)g(\014rst)g (place)g(at)g(whic)m(h)g(the)g(pro)s(duct)113 2961 y Fp(")159 2976 y Fs(0)205 2961 y Fp(g)t Ft(\()p Fp(i)330 2976 y Fs(1)373 2961 y Ft(\))p Fp(")458 2976 y Fs(1)520 2961 y Fp(:)17 b(:)g(:)e(g)t Ft(\()p Fp(i)778 2976 y Fn(s)819 2961 y Ft(\))46 b(di\013ers)h(from)e Fp(")1508 2925 y Fm(0)1508 2986 y Fs(0)1553 2961 y Fp(g)t Ft(\()p Fp(i)1678 2925 y Fm(0)1678 2986 y Fs(1)1722 2961 y Ft(\))p Fp(")1807 2925 y Fm(0)1807 2986 y Fs(1)1868 2961 y Fp(:)17 b(:)g(:)e(g)t Ft(\()p Fp(i)2126 2925 y Fm(0)2126 2988 y Fn(s)2164 2968 y Ff(0)2195 2961 y Ft(\)\).)81 b(The)46 b(second)i(assertion)e(follo)m(ws)113 3098 y(easily)39 b(from)g(the)h(remark)e(that)1407 3073 y Fk(c)1383 3098 y Fo(M)1503 3062 y Fs(\()p Fn(r)r Fs(\))1649 3098 y Ft(is)h(indeed)i (made)e(of)h(matrices)e Fp(g)k Ft(=)c Fp(g)t Ft(\()p Fp(j)3211 3113 y Fs(1)3255 3098 y Ft(\))17 b Fp(:)g(:)g(:)d(g)t Ft(\()p Fp(j)3575 3113 y Fn(r)3618 3098 y Ft(\))113 3218 y(whic)m(h)34 b(end)h(with)e(an)g(ev)m(en)h(n)m(um)m(b)s(er)g(of)f Fp(g)t Ft(\(1\)'s.)1649 b Fd(\003)313 3392 y Ft(Let)515 3367 y Fk(c)491 3392 y Fo(M)27 b Ft(=)i Fo(t)810 3407 y Fn(r)r Fm(\025)p Fs(0)980 3367 y Fk(c)956 3392 y Fo(M)1076 3356 y Fs(\()p Fn(r)r Fs(\))1215 3392 y Ft(and)1409 3318 y Fk(P)1544 3430 y(b)1514 3435 y Fm(M)1641 3392 y Ft(=)1746 3318 y Fk(P)1851 3435 y Fn(g)s Fm(2)1976 3430 y Fk(b)1947 3435 y Fm(M)2061 3392 y Fp(L)2129 3407 y Fn(g)2175 3392 y Ft(.)44 b(One)35 b(clearly)d(has)899 3633 y(\(1)22 b(+)g Fp(L)1177 3648 y Fn(\033)1231 3633 y Ft(\))g Fo(\016)g Ft(\(1)f Fo(\000)i Fp(T)1632 3648 y Fn(ev)s(en)1804 3633 y Ft(\))1843 3592 y Fm(\000)p Fs(1)1978 3633 y Ft(=)2083 3492 y Fk(\022)2173 3538 y(X)2228 3776 y(b)2198 3781 y Fm(M)2333 3492 y Fk(\023)2429 3633 y Fo(\016)f Ft(\(1)f(+)i Fp(L)2779 3648 y Fn(\033)2832 3633 y Ft(\))113 3900 y(and)34 b(b)m(y)f(the)h(previous)g(Lemma)234 4010 y Fk(X)259 4224 y Fm(M)406 4104 y Ft(=)533 4010 y Fk(X)511 4224 y Fn(g)s Fm(2M)716 4104 y Fp(L)784 4119 y Fn(g)858 4104 y Ft(=)963 4010 y Fk(X)965 4224 y 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Ft(b)s(elongs)e(to)e(the)h(monoid)1871 5287 y Fk(c)1847 5312 y Fo(M)g Ft(if)g(and)g(only)f(if)h Fp(d)c Fo(\025)h Ft(2)p Fp(b)f(>)g Ft(0,)k Fp(c)c Fo(\025)h Ft(2)p Fp(a)f Fo(\025)g Ft(0,)113 5445 y Fp(d)j Fo(\025)g Fp(Gc)p Ft(,)j(where)g Fp(G)d Ft(=)1028 5348 y Fm(p)p 1094 5348 40 4 v 1094 5406 a Fs(5)o(+1)p 1028 5422 207 4 v 1111 5480 a(2)1246 5445 y Ft(.)56 b(Moreo)m(v)m(er)38 b(the)g(decomp)s(osition)e Fp(g)i Ft(=)d Fp(")2852 5460 y Fs(0)2897 5445 y Fp(g)t Ft(\()p Fp(i)3022 5460 y Fs(1)3066 5445 y Ft(\))p Fp(")3151 5460 y Fs(1)3213 5445 y Fp(:)17 b(:)g(:)d(g)t Ft(\()p Fp(i)3470 5460 y Fn(r)3514 5445 y Ft(\))37 b(is)113 5565 y(unique.)p eop %%Page: 62 62 62 61 bop 113 144 a Ft(62)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y Fr(A5.)51 b(App)s(endix)36 b(5)j(:)51 b(The)37 b(real)g(Brjuno)i(function)d(as)i(a)g(co)s(cycle)113 531 y Ft(In)e(this)f(App)s(endix)h(w)m(e)f(sho)m(w)h(ho)m(w)g(to)e(in)m (terpret)i(the)g(real)f(Brjuno)g(function)i(as)e(a)g(co)s(cycle)113 650 y(under)d(the)e(action)g(of)g(PGL\(2)p Fp(;)17 b Fq(Z)p Ft(\))26 b(on)k Fq(R)21 b Fo(n)16 b Fq(Q)h Ft(.)49 b(T)-8 b(o)29 b(this)h(purp)s(ose)h(w)m(e)g(\014rst)f(recall)g(some)f (basic)113 770 y(de\014nitions)44 b(tak)m(en)g(from)e(the)i(cohomology) d(of)i(groups.)75 b(W)-8 b(e)43 b(refer)h(to)f([Ja])g(and)g([Se])g(for) 113 889 y(more)33 b(information)f(and)h(the)h(pro)s(ofs.)113 1037 y Fr(A5.1)39 b(Group)f(cohomology)313 1156 y Ft(Let)24 b Fp(G)h Ft(b)s(e)f(a)g(group)h(and)g Fp(M)34 b Ft(an)25 b(ab)s(elian)f(group)g(with)g(a)g(left)h Fp(G)p Ft({action,)g(i.e.)41 b(a)24 b(structure)113 1276 y(of)34 b(a)f(left)g Fq(Z)553 1240 y Fn(G)618 1276 y Ft({mo)s(dule.)43 b(Recall)33 b(that)g(for)g Fp(n)28 b Fo(\025)g Ft(0,)33 b(one)g(de\014nes)158 1395 y(\(i\))49 b(the)37 b(ab)s(elian)f(group)h(of)g Fp(n)p Ft({co)s(c)m(hains)g Fp(C)1830 1359 y Fn(n)1885 1395 y Ft(\()p Fp(G;)17 b(M)11 b Ft(\),)36 b(whose)h(elemen)m(ts)g(are) g(applications)313 1515 y(from)32 b Fp(G)626 1479 y Fn(n)714 1515 y 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2776 y Fs(0)2280 2812 y Ft(\()p Fp(G;)17 b(M)11 b Ft(\))55 b(=)i Fp(Z)2853 2776 y Fs(0)2898 2812 y Ft(\()p Fp(G;)17 b(M)11 b Ft(\))55 b(=)i Fo(f)p Fp(m)g Fo(2)113 2931 y Fp(M)h Ft(;)64 b Fp(g)t(m)52 b Ft(=)g Fp(m)c Ft(for)33 b(all)47 b Fp(g)55 b Fo(2)d Fp(G)p Fo(g)47 b Ft(with)h Fp(M)11 b Ft(.)86 b(An)48 b(application)f Fp(c)99 b Ft(:)52 b Fp(G)g Fo(!)f Fp(M)58 b Ft(is)48 b(a)f(1{)113 3051 y(co)s(cycle)h(i\013)f Fp(c)p Ft(\()p Fp(g)734 3066 y Fs(0)778 3051 y Fp(g)826 3066 y Fs(1)870 3051 y Ft(\))52 b(=)g Fp(g)1138 3066 y Fs(0)1182 3051 y Fp(c)p Ft(\()p Fp(g)1312 3066 y Fs(1)1356 3051 y Ft(\))31 b(+)h Fp(c)p Ft(\()p Fp(g)1665 3066 y Fs(0)1709 3051 y Ft(\))48 b Fp(;)f Ft(and)h(a)f(1{cob)s(oundary)h(i\013)f Fp(c)p Ft(\()p Fp(g)t Ft(\))j(=)i Fp(g)35 b Fo(\001)c Fp(m)h Fo(\000)113 3170 y Fp(m)i Ft(for)g(some)e Fp(m)i Ft(and)g(all)e Fp(g)f Fo(2)d Fp(G:)113 3318 y Fr(A5.2)39 b(Automorphic)e(factors,)h (co)s(cycles)g(and)f(cob)s(oundaries.)313 3437 y Ft(Let)23 b Fp(G)g 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Ft(\))26 b(=)i Fp(\037)p Ft(\()p Fp(g)1538 4981 y Fm(\000)p Fs(1)1644 5022 y Fp(;)17 b(x)p Ft(\))p Fp(')p Ft(\()p Fp(g)1941 4981 y Fm(\000)p Fs(1)2068 5022 y Fo(\001)22 b Fp(x)p Ft(\))g Fo(\000)h Fp(')p Ft(\()p Fp(x)p Ft(\))k Fo(8)p Fp(x)h Fo(2)g Fp(X)35 b(:)113 5243 y Ft(A)27 b(1{)p Fl(c)-5 b(o)g(cycle)26 b Ft(is)g(an)h(application)f Fp(c)44 b Ft(:)28 b Fp(G)g Fo(!)f Fp(M)1856 5207 y Fn(X)1958 5243 y Ft(v)m(erifying)f Fp(g)2409 5258 y Fs(0)2462 5243 y Fo(\001)9 b Fp(c)p Ft(\()p Fp(g)2629 5258 y Fs(1)2672 5243 y Ft(\))g Fo(\000)g Fp(c)p Ft(\()p Fp(g)2936 5258 y Fs(0)2980 5243 y Fp(g)3028 5258 y Fs(1)3072 5243 y Ft(\))g(+)g Fp(c)p Ft(\()p Fp(g)3336 5258 y Fs(0)3380 5243 y Ft(\))27 b(=)i(0)e Fp(;)113 5363 y Ft(i.e.,)33 b(letting)h(\024)-52 b Fp(c)p Ft(\()p Fp(g)t Ft(\))26 b(=)j Fp(c)p Ft(\()p Fp(g)1056 5327 y Fm(\000)p Fs(1)1161 5363 y Ft(\))k(:)518 5584 y(\024)-52 b Fp(c)p Ft(\()p Fp(g)646 5599 y Fs(0)690 5584 y Fp(g)738 5599 y Fs(1)782 5584 y Ft(\))28 b(=)g Fp(c)p Ft(\()p Fp(g)1088 5542 y Fm(\000)p Fs(1)1084 5611 y(1)1194 5584 y Fp(g)1246 5542 y Fm(\000)p Fs(1)1242 5611 y(0)1352 5584 y Ft(\))f(=)h Fp(c)p Ft(\()p Fp(g)1657 5542 y Fm(\000)p Fs(1)1653 5611 y(1)1763 5584 y Ft(\))22 b(+)h Fp(g)1976 5542 y Fm(\000)p Fs(1)1972 5611 y(1)2082 5584 y Fp(c)p Ft(\()p Fp(g)2216 5542 y Fm(\000)p Fs(1)2212 5611 y(0)2322 5584 y Ft(\))k(=)k(\024)-53 b Fp(c)p Ft(\()p Fp(g)2623 5599 y Fs(1)2667 5584 y Ft(\))22 b(+)h Fp(g)2880 5542 y Fm(\000)p Fs(1)2876 5611 y(1)2988 5584 y Ft(\024)-52 b Fp(c)p Ft(\()p Fp(g)3116 5599 y Fs(0)3160 5584 y Ft(\))27 b Fp(;)p eop %%Page: 63 63 63 62 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(63)113 383 y(or,)33 b(equiv)-6 b(alen)m(tly)e(,)34 b(\024)-52 b Fp(c)p Ft(\()p Fp(g)966 398 y Fs(0)1010 383 y Fp(g)1058 398 y Fs(1)1102 383 y Fp(;)17 b(x)p Ft(\))27 b(=)h Fp(\037)p Ft(\()p Fp(g)1524 398 y Fs(1)1568 383 y Fp(;)17 b(x)p Ft(\))r(\024)-52 b Fp(c)o Ft(\()p Fp(g)1838 398 y Fs(0)1882 383 y Fp(;)17 b(g)1975 398 y Fs(1)2041 383 y Fo(\001)22 b Fp(x)p Ft(\))g(+)i(\024)-52 b Fp(c)p Ft(\()p Fp(g)2438 398 y Fs(1)2482 383 y Fp(;)17 b(x)p Ft(\))27 b Fo(8)p Fp(x)h Fo(2)g Fp(X)35 b(:)113 539 y Fr(A5.3)k(Action)e(of)h(PGL)p Ft(\(2)p Fp(;)17 b Fq(Z)p Ft(\))35 b Fr(on)j Fq(R)33 b Fo(n)22 b Fq(Q)11 b Fr(.)313 662 y Ft(Let)40 b(us)i(consider)f Fp(G)f Ft(=)h(PGL\(2)p Fp(;)17 b Fq(Z)p Ft(\))36 b(and)41 b Fp(X)48 b Ft(=)40 b Fq(R)e Fo(n)27 b Fq(Q)11 b Ft(,)48 b(the)41 b(action)f(b)s(eing)h (giv)m(en)g(b)m(y)113 781 y(the)e(homographies.)59 b(The)39 b(transformations)e Fp(T)14 b Ft(\()p Fp(x)p Ft(\))36 b(=)g Fp(x)26 b Ft(+)g(1)38 b(and)h Fp(S)6 b Ft(\()p Fp(x)p Ft(\))35 b(=)i Fp(x)3146 745 y Fm(\000)p Fs(1)3291 781 y Ft(generate)113 901 y(PGL)q(\(2)p Fp(;)17 b Fq(Z)p Ft(\).)40 b(One)34 b(has)g(the)f(follo)m(wing)g(more)f(precise)j (result)e(:)113 1057 y Fr(Prop)s(osition)39 b(A5.1)34 b Fl(L)-5 b(et)37 b Fp(g)31 b Fo(2)e Fl(PGL)q Ft(\(2)p Fp(;)17 b Fq(Z)p Ft(\))32 b Fl(and)k(let)g Fp(x)2220 1072 y Fs(0)2294 1057 y Fo(2)29 b Fq(R)k Fo(n)22 b 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Ft(generate)313 3170 y(PGL\(2)p Fp(;)17 b Fq(Z)p Ft(\))29 b(this)k(implies)g(the)g (existence)h(in)f(all)g(p)s(ossible)g(cases.)313 3293 y(W)-8 b(e)47 b(can)g(no)m(w)g(pro)m(v)m(e)g(uniqueness.)86 b(It)46 b(is)h(su\016cien)m(t)h(to)e(sho)m(w)h(that)g(if)f Fp(r)53 b(>)e Ft(0)46 b(and)113 3413 y Fp(g)161 3428 y Fs(1)206 3413 y Fp(;)17 b(:)g(:)g(:)d(;)j(g)476 3428 y Fn(r)546 3413 y Fo(2)28 b(f)p Fp(S;)17 b(T)8 b(;)17 b(T)979 3376 y Fm(\000)p Fs(1)1086 3413 y Fo(g)p Ft(,)32 b Fp(x)1253 3428 y Fs(0)1326 3413 y Fo(2)c Fq(R)33 b Fo(n)22 b Fq(Q)50 b Ft(v)m(erify)756 3661 y Fp(g)804 3676 y Fn(r)864 3661 y Fp(:)17 b(:)g(:)e(g)1045 3676 y Fs(1)1117 3661 y Ft(=)28 b(1)g Fp(;)116 b Ft(and)67 b Fp(x)1728 3676 y Fn(i)p Fm(\000)p Fs(1)1890 3661 y Fp(>)29 b Ft(0)k(if)g Fp(g)2218 3676 y Fn(i)2278 3661 y Ft(=)28 b Fp(S)34 b Ft(\(1)27 b Fo(\024)h Fp(i)g Fo(\024)h Fp(r)s Ft(\))j Fp(;)113 3910 y Ft(then)h(there)g(exists)g(1)27 b Fp(<)h(i)g Fo(\024)g Fp(r)36 b Ft(suc)m(h)d(that)f Fp(g)1784 3925 y Fn(i)1817 3910 y Fp(g)1865 3925 y Fn(i)p Fm(\000)p Fs(1)2027 3910 y Ft(=)d(1.)43 b(W)-8 b(e)33 b(pro)m(v)m(e)g(this)f(b)m(y)h(con)m(tradiction)f(:)113 4030 y(let)h Fp(r)j Ft(b)s(e)e(minimal,)c Fp(r)g(>)e Ft(0.)313 4152 y(If)46 b Fp(x)482 4167 y Fs(0)574 4152 y Fp(<)j Ft(0)c(one)h(m)m(ust)f(ha)m(v)m(e)h Fp(r)51 b Fo(\025)d Ft(2,)g Fp(g)1886 4167 y Fs(1)1979 4152 y Ft(=)g Fp(T)60 b Ft(and)46 b Fp(g)2476 4167 y Fn(r)2567 4152 y Ft(=)j Fp(T)2765 4116 y Fm(\000)p Fs(1)2872 4152 y Ft(,)f(th)m(us)f Fp(r)j Fo(\025)f Ft(3)c(and)113 4272 y Fp(g)161 4287 y Fn(r)r Fm(\000)p Fs(1)324 4272 y Fp(:)17 b(:)g(:)d(g)504 4287 y Fs(2)576 4272 y Ft(=)28 b(1,)33 b(whic)m(h)h(con)m(tradicts)g(the)g(minimalit)m(y)29 b(of)34 b Fp(r)s Ft(.)313 4395 y(One)j(is)f(led)g(to)f(assume)h Fp(x)1320 4410 y Fs(0)1397 4395 y Fp(>)d Ft(0.)52 b(Clearly)34 b(one)j(m)m(ust)e(ha)m(v)m(e)i Fp(x)2710 4410 y Fn(i)2775 4395 y Fp(>)c Ft(0)i(for)i(all)e Fp(i)d Fo(2)h Ft([0)p Fp(;)17 b(r)s Ft(].)113 4514 y(Let)49 b Fp(i)341 4529 y Fs(1)440 4514 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y Fo(\000)g Ft(1\))45 b Fp(>)h Ft(1.)76 b(Therefore)46 b Fp(x)2177 4888 y Fn(i)2205 4899 y Fj(l)2234 4888 y Fm(\000)p Fs(1)2386 4873 y Fp(>)g Ft(1)e(for)g(all)f(1)i Fo(\024)h Fp(l)h Fo(\024)f Fp(k)s Ft(.)76 b(But)113 4992 y(then)347 4918 y Fk(Q)441 4942 y Fn(r)441 5022 y Fs(1)562 4948 y Fn(dg)642 4958 y Fj(i)p 514 4970 207 4 v 514 5027 a Fn(dx)600 5037 y Fj(i)p Ff(\000)p Fi(1)733 4992 y Ft(\()p Fp(x)829 5007 y Fn(i)p Fm(\000)p Fs(1)963 4992 y Ft(\))39 b Fp(<)h Ft(1,)h(in)f(con)m(tradiction)g(with)g(the)g(assumption)g Fp(g)3007 5007 y Fn(r)3067 4992 y Fp(:)17 b(:)g(:)e(g)3248 5007 y Fs(1)3331 4992 y Ft(=)40 b(1.)64 b(If)113 5130 y(there)41 b(exists)f Fp(l)i Ft(suc)m(h)g(that)d Fp(x)1240 5145 y Fn(i)1268 5156 y Fj(l)1297 5145 y Fm(\000)p Fs(1)1444 5130 y Fp(<)g Ft(1)h(one)h(p)s(erm)m(utes)f(circularly)g(all)f Fp(g)2920 5145 y Fn(i)2993 5130 y Ft(and)i Fp(x)3251 5145 y Fn(i)3324 5130 y Ft(\(mo)s(d)15 b Fp(r)s Ft(\))113 5249 y(un)m(til)42 b(one)h(is)f(bac)m(k)g(to)f(the)i(case)f(previously) g(considered.)72 b(Finally)41 b(if)h Fp(x)2973 5264 y Fn(i)3001 5275 y Fj(l)3030 5264 y Fm(\000)p Fs(1)3179 5249 y Fp(<)h Ft(1)e(for)i(all)113 5369 y(1)49 b Fo(\024)g Fp(l)i Fo(\024)e Fp(k)g Ft(then)885 5294 y Fk(Q)980 5319 y Fn(r)980 5399 y Fs(1)1100 5324 y Fn(dg)1180 5334 y Fj(i)p 1053 5346 V 1053 5403 a Fn(dx)1139 5413 y Fj(i)p Ff(\000)p Fi(1)1271 5369 y Ft(\()p Fp(x)1367 5384 y Fn(i)p Fm(\000)p Fs(1)1502 5369 y Ft(\))f Fp(>)h Ft(1,)g(whic)m(h)e(is)f (again)f(in)h(con)m(tradiction)g(with)f(the)113 5499 y(assumption)33 b Fp(g)688 5514 y Fn(r)749 5499 y Fp(:)17 b(:)g(:)d(g)929 5514 y Fs(1)1001 5499 y Ft(=)28 b(1.)2395 b Fd(\003)p eop %%Page: 64 64 64 63 bop 113 144 a Ft(64)2405 b(Marmi,)31 b(Moussa,)j(Y)-8 b(o)s(ccoz)113 383 y Fr(Corollary)49 b(A5.2)100 b Fe(Let)42 b Fp(A)g Fe(b)s(e)h(an)f(ab)s(elian)g(ring,)i(and)f(the)g(maps)e Fp(T)57 b Fe(and)42 b Fp(S)48 b Fe(suc)m(h)c(that)113 503 y Fp(t)55 b Ft(:)38 b Fq(R)g Fo(n)26 b Fq(Q)55 b Fo(!)38 b Fp(A)773 466 y Fm(\003)819 503 y Fe(,)j Fp(s)54 b Ft(:)38 b(\(0)p Fp(;)17 b Ft(1\))25 b Fo(\\)i Ft(\()p Fq(R)37 b Fo(n)26 b Fq(Q)12 b Ft(\))44 b Fo(!)37 b Fp(A)1976 466 y Fm(\003)2022 503 y Fe(.)63 b(There)40 b(exists)g(a)f(unique)h (automorphic)113 622 y(factor)34 b Fp(\037)f Fe(suc)m(h)i(that)1078 854 y Fp(\037)p Ft(\()p Fp(T)8 b(;)17 b(x)p Ft(\))28 b(=)g Fp(t)p Ft(\()p Fp(x)p Ft(\))f Fe(for)33 b(all)16 b Fp(x)28 b Fo(2)g Fq(R)33 b Fo(n)22 b Fq(Q)45 b Ft(=)28 b Fp(X)35 b(;)1084 1003 y(\037)p Ft(\()p Fp(S;)17 b(x)p Ft(\))27 b(=)h Fp(s)p Ft(\()p Fp(x)p Ft(\))e Fe(for)34 b(all)16 b Fp(x)27 b Fo(2)h Fp(X)i Fo(\\)22 b Ft(\(0)p Fp(;)17 b Ft(1\))27 b Fp(:)113 1277 y Fl(Pr)-5 b(o)g(of.)42 b Ft(Let)g Fp(s)p Ft(\()p Fp(x)p Ft(\))e(=)i(\()p Fp(s)p Ft(\()p Fp(x)1129 1241 y Fm(\000)p Fs(1)1235 1277 y Ft(\)\))1313 1241 y Fm(\000)p Fs(1)1461 1277 y Ft(for)g(all)f Fp(x)g Fo(2)h Fp(X)35 b Fo(\\)28 b Ft(\(1)p Fp(;)17 b Ft(+)p Fo(1)p Ft(\).)67 b(The)42 b(map)f Fp(s)g Ft(is)g(therefore)113 1396 y(de\014ned)36 b(on)d Fp(X)c Fo(\\)23 b Ft(\(0)p Fp(;)17 b Ft(+)p Fo(1)p Ft(\))32 b(and)i(one)f(m)m(ust)g(ha)m(v)m(e) 1125 1634 y Fp(\037)p Ft(\()p Fp(S;)17 b(x)p Ft(\))27 b(=)h Fp(s)p Ft(\()p Fp(x)p Ft(\))f(for)33 b(all)16 b Fp(x)27 b Fo(2)h Fp(X)r(;)17 b(x)27 b(>)h Ft(0)g Fp(:)113 1871 y Ft(a\))34 b(The)h Fl(uniqueness)f Ft(of)g Fp(\037)h Ft(follo)m(ws)f(from)f(the)i Fl(existenc)-5 b(e)33 b Ft(in)h(the)h(previous)g(prop)s(osition)f(:)46 b(if)113 1990 y Fp(g)31 b Fo(2)d Ft(PGL\(2)p Fp(;)17 b Fq(Z)p Ft(\))30 b(and)j Fp(x)1016 2005 y Fs(0)1089 1990 y Fo(2)28 b Fp(X)40 b Ft(one)34 b(m)m(ust)f(ha)m(v)m(e)1349 2308 y Fp(\037)p Ft(\()p Fp(g)t(;)17 b(x)1604 2323 y Fs(0)1647 2308 y Ft(\))27 b(=)1864 2183 y Fn(r)1819 2213 y Fk(Y)1819 2425 y Fn(i)p Fs(=1)1964 2308 y Fp(\037)p Ft(\()p Fp(g)2113 2323 y Fn(i)2146 2308 y Fp(;)17 b(x)2248 2323 y Fn(i)p Fm(\000)p Fs(1)2382 2308 y Ft(\))1108 b(\(1\))113 2632 y(where)34 b Fp(g)449 2647 y Fs(1)494 2632 y Fp(;)17 b(:)g(:)g(:)d(;)j(g)764 2647 y 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Fn(i)p Fm(\000)p Fs(1)2021 3531 y Fo(6)p Ft(=)31 b(1)j(for)h(all)f(1)c Fp(<)g(i)g Fo(\024)h Fp(r)s Ft(\))j(to)g(de\014ne)i Fp(\037)f Ft(and)113 3650 y(one)f(m)m(ust)f(c)m(hec)m(k)h(that)1207 3770 y Fp(\037)p Ft(\()p Fp(g)1360 3729 y Fm(0)1387 3770 y Fp(g)t(;)17 b(x)1541 3785 y Fs(0)1584 3770 y Ft(\))27 b(=)h Fp(\037)p Ft(\()p Fp(g)1908 3729 y Fm(0)1935 3770 y Fp(;)17 b(g)t(x)2089 3785 y Fs(0)2132 3770 y Ft(\))p Fp(\037)p Ft(\()p Fp(g)t(;)g(x)2426 3785 y Fs(0)2469 3770 y Ft(\))28 b Fp(:)113 3959 y Ft(Let)34 b Fp(y)341 3974 y Fs(0)413 3959 y Ft(=)28 b Fp(x)575 3974 y Fn(r)647 3959 y Ft(=)g Fp(g)t(x)861 3974 y Fs(0)933 3959 y Fp(;)99 b(g)1112 3922 y Fm(0)1167 3959 y Ft(=)28 b Fp(g)1324 3922 y Fm(0)1320 3983 y Fn(s)1378 3959 y Fp(:)17 b(:)g(:)e(g)1563 3922 y Fm(0)1559 3983 y Fs(1)1631 3959 y Fp(;)33 b Ft(follo)m(wing)g(the)g(previous)h(prop)s(osition,)e (and)i(let)735 4255 y Fp(g)787 4214 y Fm(00)865 4255 y Ft(=)28 b Fp(g)1022 4214 y Fm(0)1049 4255 y Fp(g)j(;)99 b(g)1307 4214 y Fm(00)1303 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Ft(0)33 b(if)g Fp(g)e Ft(=)d Fp(S)6 b Ft(,)33 b(whic)m(h)h(is)f (immediate)e(to)i(c)m(hec)m(k.)351 b Fd(\003)p eop %%Page: 65 65 65 64 bop 113 144 a Ft(Complex)32 b(Brjuno)i(F)-8 b(unctions)2270 b(65)113 383 y Fr(Corollary)46 b(A5.3)97 b Fe(Let)40 b Fp(A)g Fe(b)s(e)g(an)g(ab)s(elian)f(ring,)i Fp(\037)f Fe(an)g(automorphic)e(factor,)k Fp(M)50 b Fe(a)39 b Fp(A)p Fe({)113 503 y(mo)s(dule,)33 b Fp(M)601 466 y Fn(X)709 503 y Fe(with)g(the)g(structure)i(of)e Fq(Z)1717 466 y Fs([)p Fn(G)p Fs(])1827 503 y Fe({mo)s(dule)f(de\014ned)j(b)m(y)f Fp(\037)p Fe(.)44 b(Let)1442 739 y Ft(\024)-52 b Fp(c)1483 754 y Fn(T)1590 739 y Ft(:)27 b Fp(X)35 b Fo(!)28 b Fp(M)1447 889 y Ft(\024)-52 b Fp(c)1488 904 y Fn(S)1590 889 y Ft(:)27 b Fp(X)j Fo(\\)22 b Ft(\(0)p Fp(;)17 b Ft(1\))26 b Fo(!)i Fp(M)113 1126 y Fe(denote)34 b(t)m(w)m(o)g(maps.)43 b(There)34 b(exists)f(a)g(unique)h(co)s(cycle)h Ft(\024)-52 b Fp(c)45 b Ft(:)27 b Fp(G)22 b Fo(\002)h Fp(M)38 b Fo(!)27 b Fp(M)44 b Fe(suc)m(h)35 b(that)1077 1362 y Ft(\024)-52 b Fp(c)p Ft(\()p Fp(T)14 b Ft(;)j Fp(x)p Ft(\))26 b(=)31 b(\024)-52 b Fp(c)1545 1377 y Fn(T)1607 1362 y Ft(\()p Fp(x)p Ft(\))28 b Fe(for)33 b(all)16 b Fp(x)27 b Fo(2)h Fp(X)1082 1512 y Ft(\024)-52 b Fp(c)p Ft(\()p Fp(S)6 b Ft(;)17 b Fp(x)p Ft(\))26 b(=)31 b(\024)-52 b Fp(c)1545 1527 y Fn(S)1602 1512 y Ft(\()p Fp(x)p Ft(\))27 b Fe(for)34 b(all)16 b Fp(x)27 b Fo(2)h Fp(X)i Fo(\\)22 b Ft(\(0)p Fp(;)17 b Ft(1\))27 b Fp(:)113 1781 y Fl(Pr)-5 b(o)g(of.)34 b Ft(One)h(m)m(ust)e (ha)m(v)m(e)872 2023 y(\024)-52 b Fp(c)p Ft(\()p Fp(T)1024 1982 y Fm(\000)p Fs(1)1131 2023 y Ft(;)17 b Fp(x)p Ft(\))26 b(=)i Fo(\000)p Fp(\037)p Ft(\()p Fp(T)1653 1982 y Fm(\000)p Fs(1)1761 2023 y Fp(;)17 b(x)p Ft(\))r(\024)-52 b Fp(c)1945 2038 y Fn(T)2007 2023 y Ft(\()p Fp(x)22 b Fo(\000)g Ft(1\))27 b(for)34 b(all)e Fp(x)c Fo(2)g Fp(X)113 2265 y Ft(and)824 2388 y(\024)-52 b Fp(c)p Ft(\()p Fp(S)6 b Ft(;)17 b Fp(x)p Ft(\))27 b(=)h Fo(\000)p Fp(\037)p Ft(\()p Fp(S;)17 b(x)p Ft(\))r(\024)-52 b Fp(c)1667 2403 y Fn(S)1724 2388 y Ft(\()p Fp(x)1820 2347 y Fm(\000)p Fs(1)1927 2388 y Ft(\))27 b(for)34 b(all)e Fp(x)c Fo(2)g Fp(X)35 b(;)44 b(x)28 b(>)g Ft(1)f Fp(:)113 2580 y Ft(Moreo)m(v)m(er,)34 b(if)f Fp(g)e Ft(=)d Fp(g)902 2595 y Fn(r)962 2580 y Fp(:)17 b(:)g(:)e(g)1143 2595 y Fs(1)1221 2580 y Ft(and)33 b Fp(x)1471 2595 y Fs(0)1549 2580 y Ft(are)g(giv)m(en)h(as)f(in)g(the)h (prop)s(osition)977 2897 y(\024)-52 b Fp(c)p Ft(\()p Fp(g)t Ft(;)17 b Fp(x)1211 2912 y Fs(0)1254 2897 y Ft(\))28 b(=)1478 2773 y Fn(r)1426 2802 y Fk(X)1433 3015 y Fn(i)p Fs(=1)1570 2897 y Ft(\()r(\024)-52 b Fp(c)o Ft(\()p Fp(g)1738 2912 y Fn(i)1771 2897 y Fp(;)17 b(x)1873 2912 y Fn(i)p Fm(\000)p Fs(1)2007 2897 y Ft(\))p Fp(\037)p Ft(\()p Fp(g)2195 2912 y Fn(i)p Fm(\000)p Fs(1)2346 2897 y Fp(:)g(:)g(:)e(g) 2527 2912 y Fs(1)2571 2897 y Fp(;)i(x)2673 2912 y Fs(0)2717 2897 y Ft(\)\))706 b(\(1)3590 2861 y Fm(0)3618 2897 y Ft(\))113 3226 y(from)34 b(whic)m(h)h(the)g(uniqueness)h(follo)m(ws.)46 b(The)35 b(pro)s(of)g(of)f(existence)h(is)f(the)h(same)e(as)i(the)f (one)113 3346 y(giv)m(en)g(for)f(Corollary)f(A5.2.)2381 b Fd(\003)113 3529 y Fr(A5.4)39 b(The)e(real)g(Brjuno)i(function)d(as)i (a)g(co)s(cycle.)313 3649 y Ft(Let)i Fp(A)e Ft(=)h Fq(R)5 b Ft(,)48 b Fp(t)p Ft(\()p Fp(x)p Ft(\))38 b(=)h(1)h(and)g Fp(s)p Ft(\()p Fp(x)p Ft(\))e(=)h Fp("x)1923 3613 y Fn(\027)2013 3649 y Ft(with)g Fp(")h Fo(2)f(f\000)p Ft(1)p Fp(;)17 b Ft(+1)p Fo(g)p Ft(,)40 b Fp(\027)46 b Fo(2)39 b Fq(R)51 b Ft(and)40 b(apply)113 3769 y(Corollary)32 b(A5.2.)43 b(Then)1026 4006 y Fp(\037)p Ft(\()p Fp(T)1199 3964 y Fn(n)1253 4006 y Fp(;)17 b(x)p Ft(\))27 b(=)h(1)g Fp(;)71 b Ft(for)34 b(all)15 b Fp(n)28 b Fo(2)g Fq(Z)d Fp(;)44 b(x)28 b Fo(2)g Fp(X)1091 4155 y(\037)p Ft(\()p Fp(S;)17 b(x)p Ft(\))27 b(=)h Fp("x)1629 4114 y Fn(\027)1707 4155 y Fp(;)71 b Ft(for)34 b(all)16 b Fp(x)27 b Fo(2)h Fp(X)35 b(;)44 b(x)28 b(>)g Ft(0)g Fp(:)113 4404 y Ft(If)34 b Fp(x)270 4419 y Fs(0)342 4404 y Fo(2)28 b Ft(\(0)p Fp(;)17 b Ft(1\),)32 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