Content-Type: multipart/mixed; boundary="-------------0405211152757" This is a multi-part message in MIME format. ---------------0405211152757 Content-Type: text/plain; name="04-158.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-158.comments" 2000 MSC numbers: 34C20, 34C14 ---------------0405211152757 Content-Type: text/plain; name="04-158.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-158.keywords" Convergence of normal forms hamiltonian systems reversible systems ---------------0405211152757 Content-Type: application/postscript; name="delshamsLaz0521.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="delshamsLaz0521.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86e Copyright 2001 Radical Eye Software %%Title: delshamsLaz0521.dvi %%Pages: 26 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o delshamsLaz0521.ps 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Fp(X)33 b Fq(=)25 b(\010\()p Fp(\037)p Fq(\))g(=)g Fp(\037)12 b Fq(+)595 4204 y Fo(b)587 4227 y Fq(\010\()p Fp(\037)p Fq(\))27 b(in)e(suc)m(h)h(a)h(w)m(a)m(y)h(that)f(the)f(new)g(system)48 b(_)-46 b Fp(\037)25 b Fq(=)g(\010)2433 4194 y Fn(\003)2472 4227 y Fp(F)13 b Fq(\()p Fp(\037)p Fq(\))26 b(=:)f Fp(N)10 b Fq(\()p Fp(\037)p Fq(\))26 b(=)f(\003)p Fp(\037)12 b Fq(+)3388 4204 y Fo(b)3364 4227 y Fp(N)e Fq(\()p Fp(\037)p Fq(\))27 b(b)s(ecomes)-83 4340 y(in)d Fm(normal)30 b(form)p Fq(,)e(that)e(is,)g(when)1126 4317 y Fo(b)1102 4340 y Fp(N)36 b Fq(con)m(tains)25 b(only)g(the)h(so-named)g Fm(r)-5 b(esonant)30 b(terms)p Fq(,)d(monomials)d(whose)i(p)s(o)m(w)m (ers)-83 4452 y(are)k(in)m(timately)e(related)h(to)h(the)f(v)m(ector)i Fp(\025)25 b Fq(=)g(\()p Fp(\025)1602 4466 y Fl(1)1642 4452 y Fp(;)15 b(\025)1735 4466 y Fl(2)1775 4452 y Fp(;)g(:)g(:)g(:)h (;)f(\025)2029 4466 y Fk(m)2096 4452 y Fq(\))30 b(formed)f(b)m(y)g(the) g(eigen)m(v)-5 b(alues)29 b(of)g(the)h(matrix)e(\003)h(of)-83 4565 y(system)h(\(1\).)58 4678 y(In)i(this)f(w)m(ork,)i(w)m(e)g(will)c (fo)s(cus)j(our)g(atten)m(tion)h(on)f(analytic)g(v)m(ector)i(\014elds)d (and)g(will)f(b)s(e)h(sp)s(ecially)g(concerned)-83 4791 y(with)37 b(the)i(con)m(v)m(ergence)i(of)e(the)f Fm(normalizing)k(tr)-5 b(ansformation)43 b Fq(\010.)64 b(There)38 b(are)h(t)m(w)m(o)h(w)m (ell-kno)m(wn)e(cases)h(where)-83 4904 y(the)f(con)m(v)m(ergence)h(of)f (the)f(normalizing)e(transformation)i(follo)m(ws)g(just)f(from)h(the)h (prop)s(erties)e(of)h(the)h(v)m(ector)g(of)-83 5017 y(c)m (haracteristic)31 b(exp)s(onen)m(ts)f Fp(\025)h Fq(\(see,)g(for)f (instance,)h([1,)g(Chapter)f(5,)h Fj(x)p Fq(24]\):)-3 5205 y(\()p Fp(i)p Fq(\))46 b(when)33 b Fp(\025)h Fq(b)s(elongs)f(to)i (the)f(P)m(oincar)m(\023)-43 b(e)35 b(domain,)f(that)h(is,)f(the)g(con) m(v)m(ex)i(h)m(ull)31 b(of)k(the)f(set)g Fj(f)q Fp(\025)3284 5219 y Fl(1)3323 5205 y Fp(;)15 b(\025)3416 5219 y Fl(2)3456 5205 y Fp(;)g(:)g(:)g(:)i(;)e(\025)3711 5219 y Fk(m)3778 5205 y Fj(g)34 b Fq(in)144 5318 y(the)d(complex)f(plane)f(do)s(es)h (not)h(con)m(tain)g(the)f(origin;)-35 5505 y(\()p Fp(ii)p Fq(\))47 b(when)25 b Fp(\025)g Fq(b)s(elongs)g(to)h(the)g(complemen)m (tary)f(of)h(this)e(domain,)i(the)g(so-called)f(Siegel's)g(domain,)g (and)g(satis\014es)144 5618 y(a)31 b(Diophan)m(tine)e(condition.)1902 6018 y(1)p eop %%Page: 2 2 2 1 bop -83 191 a Fq(In)33 b(the)h(\014rst)g(case,)i(the)e(Theorem)g (of)g(P)m(oincar)m(\023)-43 b(e{Dulac)36 b(ensures)d(the)h(con)m(v)m (ergence)i(of)f(a)f(normalizing)e(transfor-)-83 304 y(mation)c (conjugating)g(the)g(original)f(system)h(to)h(a)g(system)f(ha)m(ving)g (only)f(a)h(\014nite)g(n)m(um)m(b)s(er)e(of)j(resonan)m(t)f(terms.)40 b(In)-83 417 y(the)29 b(second)g(case,)i(the)e(Diophan)m(tine)f (condition)g(p)s(ermits)f(to)j(b)s(ound)d(the)i(small)f(divisors)e(app) s(earing)i(in)g(the)h(nor-)-83 530 y(malizing)i(transformation)i(and)f (its)h(con)m(v)m(ergence)i(is)e(also)g(deriv)m(ed)f(\(Siegel's)h (Theorem\).)49 b(The)32 b(original)g(system)-83 643 y(is)d(conjugated)j (to)f(its)e(linear)g(part.)58 756 y(Notice)34 b(that)f(in)f(b)s(oth)g (cases)h(of)g(con)m(v)m(ergence)i(the)e(normal)f(form)g(is)g(a)h Fm(p)-5 b(olynomial)36 b Fq(or,)e(in)d(other)i(w)m(ords,)g(the)-83 868 y(n)m(um)m(b)s(er)24 b(of)i(resonan)m(t)g(terms)g(is)e(\014nite.)38 b(Ho)m(w)m(ev)m(er,)29 b(non)c(p)s(olynomial)e(normal)i(forms)g(do)g (arise)g(in)f(some)i(imp)s(ortan)m(t)-83 981 y(families)35 b(of)i(dynamical)e(systems,)k(lik)m(e)d(the)h(Hamiltonian)e(or)h(the)h (rev)m(ersible)f(ones,)j(where)d(the)h(c)m(haracteristic)-83 1094 y(exp)s(onen)m(ts)26 b(alw)m(a)m(ys)i(b)s(elong)d(to)j(the)e (Siegel's)g(domain)g(since)g(they)h(come)g(in)e(pairs)h Fj(f\006)p Fp(\025)p Fj(g)p Fq(.)40 b(In)26 b(these)h(cases,)h(con)m(v) m(er-)-83 1207 y(gence)35 b(results)e(dep)s(end)g(not)h(only)f(on)h (the)h(lo)s(cation)e(of)i(the)f(c)m(haracteristic)h(exp)s(onen)m(ts)f (and)f(their)h(arithmetical)-83 1320 y(prop)s(erties)29 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b(transformation)h(to)i(the)f(one)h(of)f (determining)e(if)h(some)i(analytic)-83 3412 y(scalar-v)-5 b(alued)29 b(functions)846 3388 y Fo(b)852 3412 y Fp(b)891 3426 y Fl(1)960 3412 y Fq(and)1131 3388 y Fo(b)1137 3412 y Fp(b)1176 3426 y Fl(2)1245 3412 y Fq(v)-5 b(anish.)39 b(Finally)-8 b(,)29 b(ev)m(en)i(in)e(the)h(case)h(that)g(these)f (functions)f(do)h(not)g(v)-5 b(anish,)-83 3525 y(some)31 b(in)m(teresting)e(dynamical)g(consequences)i(can)g(b)s(e)f(deriv)m(ed) f(from)h(this)f(pseudo-normal)g(form.)-83 3704 y Fs(Theorem)34 b(1)h(\(Main)g(Theorem\))44 b Fm(Given)32 b(a)h Fq(4)p Fm(-dimensional)i(system)1433 3876 y Fq(_)1397 3899 y Fp(X)e Fq(=)25 b Fp(F)13 b Fq(\()p Fp(X)7 b Fq(\))26 b(=)f(\003)p Fp(X)j Fq(+)2221 3876 y Fo(b)2203 3899 y Fp(F)13 b Fq(\()p Fp(X)7 b Fq(\))p Fp(;)1366 b Fq(\(6\))-83 4093 y Fm(analytic)29 b(ar)-5 b(ound)30 b(the)f(origin,)g(wher)-5 b(e)1264 4070 y Fo(b)1246 4093 y Fp(F)13 b Fq(\()p Fp(X)7 b Fq(\))30 b Fm(denotes)f(terms)g(of)f(or)-5 b(der)30 b(at)f(le)-5 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Fp(B)t Fm(,)e(as)i(in)e Fq(\(4\))p Fm(,)i(in)e(such)h(a)g(way)h(that)f(the)g(e)-5 b(quality)1531 5056 y Fp(D)s Fq(\010)32 b Fp(N)e Fq(+)1920 5033 y Fo(b)1901 5056 y Fp(B)g Fq(=)25 b Fp(F)33 b Fj(\016)21 b Fq(\010)-83 5251 y Fm(holds.)43 b(Mor)-5 b(e)g(over,)34 b(if)e(system)i Fq(\(6\))f Fm(is)g(r)-5 b(e)g(al)34 b(analytic,)g Fq(\010)p Fm(,)e Fp(N)42 b Fm(and)34 b Fp(B)j Fm(ar)-5 b(e)33 b(also)h(r)-5 b(e)g(al)34 b(analytic.)-83 5430 y Fq(Section)i(2)h(is)e(dev)m(oted)j (to)f(the)g(pro)s(of)e(of)i(this)e(theorem,)k(whic)m(h)c(is)g (constructiv)m(e.)60 b(It)36 b(is)g(based)g(on)g(a)h(recurren)m(t)-83 5543 y(sc)m(heme)k(whic)m(h)d(pro)m(vides)h(the)h(co)s(e\016cien)m(ts)h (of)f(\010,)i Fp(N)50 b Fq(and)2086 5520 y Fo(b)2067 5543 y Fp(B)5 b Fq(,)42 b(order)e(b)m(y)g(order.)68 b(Moreo)m(v)m(er,) 45 b(a)40 b(condition)f(for)-83 5656 y(determining)28 b(the)j(radius)d(of)j(con)m(v)m(ergence)i(of)d(these)h(v)m(ector)h (\014elds)d(is)g(pro)m(vided)g(in)g(equation)h(\(64\).)58 5769 y(A)h(\014rst)e(consequence)i(of)g(Theorem)f(1)h(is)e(that)i(for)f (an)h(initial)c Fm(Hamiltonian)32 b Fq(system,)f(\011)p Fi(NF)f Fq(b)s(ecomes)h Fi(BNF)p Fq(.)1902 6018 y(3)p eop %%Page: 4 4 4 3 bop -83 191 a Fs(Prop)s(osition)36 b(H1)46 b Fm(System)30 b Fq(\(6\))g Fm(is)f(Hamiltonian)h(in)f(a)h(neighb)-5 b(orho)g(o)g(d)33 b(of)c(the)g(origin)h(if)f(and)h(only)g(if)3519 168 y Fo(b)3500 191 y Fp(B)k Fm(vanishes)-83 304 y(\(and,)g(ther)-5 b(efor)g(e,)34 b Fq(\011)p Fi(NF)e Fm(b)-5 b(e)g(c)g(omes)34 b Fi(BNF)p Fm(\).)-83 488 y Fq(The)g(pro)s(of)g(is)g(giv)m(en)h(in)f (Section)h(3.)54 b(In)34 b(the)h(case)h(that)g(system)f(\(6\))h(is)e(a) h(2-degrees)h(of)f(freedom)g(Hamiltonian,)-83 601 y(this)29 b(prop)s(osition)f(pro)m(vides)i(a)g(new)g(pro)s(of)g(for)g(the)h (celebrated)f(Moser's-Ly)m(apuno)m(v)i(theorem.)-83 785 y Fs(Corollary)j(H2)46 b Fi([Ly)m(apunov,)f(Moser])e Fm(F)-7 b(or)44 b(an)f(analytic)h(Hamiltonian)h(system)e(ar)-5 b(ound)45 b(a)e(sadd)5 b(le-fo)-5 b(cus)44 b(or)f(a)-83 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b(by)d(The)-5 b(or)g(em)35 b(1)e(vanishes)g (identic)-5 b(al)5 b(ly.)-83 4409 y Fq(On)33 b(the)h(other)g(hand,)g (assuming)1146 4385 y Fo(b)1128 4409 y Fp(f)1173 4423 y Fl(3)1243 4409 y Fj(\021)1362 4385 y Fo(b)1345 4409 y Fp(f)1390 4423 y Fl(4)1460 4409 y Fj(\021)d Fq(0)j(in)f(system)h (\(7\))h(\(that)g(is,)f(the)g(origin)e(is)h(a)h(cen)m(ter)h(in)e(the)h (\()p Fp(q)s(;)15 b(p)p Fq(\)-)-83 4522 y(v)-5 b(ariables\),)29 b(taking)h(p)s(olar)e(co)s(ordinates,)i(scaling)e(time)i(if)e (necessary)i(and)f(\014xing)g(an)g(in)m(v)-5 b(arian)m(t)29 b(cycle,)h(w)m(e)g(ha)m(v)m(e)h(a)-83 4635 y(system)f(of)h(the)g(form) 1363 4669 y Fo(8)1363 4751 y(<)1363 4915 y(:)1501 4740 y Fq(_)-41 b Fp(x)83 b Fq(=)149 b Fp(\025x)20 b Fq(+)h Fo(b)-52 b Fp(g)2099 4754 y Fl(1)2139 4740 y Fq(\()p Fp(x;)15 b(y)s(;)g(\022)s Fq(\))1505 4853 y(_)-41 b Fp(y)86 b Fq(=)d Fj(\000)p Fp(\025y)22 b Fq(+)f Fo(b)-52 b Fp(g)2099 4867 y Fl(2)2139 4853 y Fq(\()p Fp(x;)15 b(y)s(;)g(\022)s Fq(\))1509 4947 y(_)1492 4971 y Fp(\022)85 b Fq(=)310 b(1)364 b Fp(;)3771 4856 y Fq(\(10\))-83 5133 y(where)39 b Fp(\015)46 b Fq(=)41 b Fj(f)q Fp(x)25 b Fq(=)g Fp(y)j Fq(=)d(0)p Fj(g)41 b Fq(is)d(no)m(w)i(a)g Fm(hyp)-5 b(erb)g(olic)44 b(p)-5 b(erio)g(dic)43 b(orbit)d Fq(\(of)h(c)m(haracteristic)f(exp)s (onen)m(ts)g Fj(\006)p Fp(\025)p Fq(,)i Fp(\025)f(>)g Fq(0\))-83 5246 y(and)33 b Fo(b)-52 b Fp(g)139 5260 y Fl(1)179 5246 y Fq(,)34 b Fo(b)-51 b Fp(g)281 5260 y Fl(2)353 5246 y Fq(are)33 b(analytic)g(functions)e(of)i Fp(x)p Fq(,)h Fp(y)h Fq(and)e Fp(\022)s Fq(.)47 b(F)-8 b(or)33 b(suc)m(h)g(a)g(system)g(w)m(e)g(ha)m(v)m(e)h(from)f(Prop)s (osition)e(H)r(1)i(the)-83 5359 y(follo)m(wing)c(result.)-83 5543 y Fs(Corollary)35 b(H4)46 b Fi([Moser)40 b([20)q(]])j Fm(Assume)e Fq(\(10\))i Fm(is)f(an)g(analytic)h(Hamiltonian)g(system.) 70 b(Then,)44 b(ther)-5 b(e)42 b(exists)g(a)-83 5656 y(c)-5 b(onver)g(gent)26 b(tr)-5 b(ansformation)29 b(le)-5 b(ading)26 b(system)h Fq(\(10\))f Fm(into)g Fq(\011)p Fi(NF)f Fm(in)g(a)h(neighb)-5 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b Fq(satis\014es)1695 1290 y Fg(G)1766 1252 y Fn(\003)1806 1290 y Fp(F)38 b Fq(=)25 b Fj(\000)p Fp(F)s(;)1616 b Fq(\(11\))-83 1486 y(where)30 b Fg(G)251 1453 y Fn(\003)291 1486 y Fp(F)39 b Fq(=)25 b(\()p Fp(D)s Fg(G)p Fq(\))703 1453 y Fn(\000)p Fl(1)798 1486 y Fp(F)13 b Fq(\()p Fg(G)p Fq(\).)43 b(The)30 b(di\013eomorphism)e Fg(G)j Fq(is)e(commonly)h(called)g(a)h Fm(r)-5 b(eversing)31 b Fq(in)m(v)m(olution)e(of)i(this)-83 1599 y(system)k(and)g(is,)g(in)f (general,)i(non)f(linear.)53 b(In)34 b(this)g(w)m(ork)h(w)m(e)h(are)f (dealing)f(with)f(analytic)i(systems,)i(so)e(w)m(e)g(will)-83 1712 y(consider)29 b(analytic)g(in)m(v)m(olutions)f Fg(G)p Fq(.)41 b(A)29 b(set)i Fp(S)j Fq(whic)m(h)29 b(is)f(in)m(v)-5 b(arian)m(t)29 b(under)g(the)g(action)h(of)g Fg(G)g Fq(\(that)h(is,)e Fg(G)p Fq(\()p Fp(S)5 b Fq(\))26 b Fj(\022)f Fp(S)5 b Fq(\))-83 1824 y(is)28 b(called)h Fg(G)p Fq(-)p Fm(symmetric)i Fq(or,)f(simply)-8 b(,)28 b Fm(symmetric)i Fq(if)f(there)g(is)g(no)g 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b(and)h(the)h(rev)m(ersible.)49 b(Indeed,)34 b(w)m(e)g(can)g(summarize) f(the)g(previous)g(statemen)m(ts)i(in)d(the)i(follo)m(wing)-83 4240 y(theorem.)-83 4421 y Fs(Theorem)g(2)46 b Fm(L)-5 b(et)33 b(us)f(c)-5 b(onsider)34 b(an)f(analytic)h(system)1747 4594 y Fq(_)1711 4617 y Fp(X)f Fq(=)25 b Fp(F)13 b Fq(\()p Fp(X)7 b Fq(\))1633 b(\(12\))-83 4812 y Fm(and)35 b(assume)f(that)i (one)e(of)g(the)g(fol)5 b(lowing)35 b(thr)-5 b(e)g(e)36 b(situations)f(holds)h(\(c)-5 b(orr)g(esp)g(onding)37 b(to)d(dimensions)i Fq(2)p Fm(,)e Fq(3)g Fm(and)h Fq(4)p Fm(,)-83 4925 y(r)-5 b(esp)g(e)g(ctively\),)-3 5106 y Fq(\()p Fp(i)p Fq(\))46 b Fp(X)33 b Fq(=)25 b(\()p Fp(q)s(;)15 b(p)p Fq(\))26 b Fj(2)f Fh(R)719 5073 y Fl(2)797 5106 y Fm(and)34 b(the)f(origin)g(is)f(a)h(line)-5 b(ar)34 b(c)-5 b(enter)33 b(e)-5 b(quilibrium)33 b(p)-5 b(oint)34 b(\(like)e(in)h(system)g Fq(\(8\))p Fm(\).)-35 5291 y Fq(\()p Fp(ii)p Fq(\))47 b Fp(X)33 b Fq(=)25 b(\()p Fp(x;)15 b(y)s(;)g(\022)s Fq(\))26 b Fj(2)f Fh(R)815 5258 y Fl(2)881 5291 y Fj(\002)20 b Fh(T)34 b Fm(and)f Fp(\015)e Fq(=)25 b Fj(f)p Fp(x)g Fq(=)g Fp(y)j Fq(=)d(0)p Fj(g)34 b Fm(is)e(a)h(hyp)-5 b(erb)g(olic)35 b(p)-5 b(erio)g(dic)34 b(orbit)g(\(like)e(in)h(system)g Fq(\(10\))p Fm(\).)-66 5475 y Fq(\()p Fp(iii)p Fq(\))47 b Fp(X)35 b Fq(=)27 b(\()p Fp(x;)15 b(y)s(;)g(q)s(;)g(p)p Fq(\))29 b Fj(2)e Fh(R)909 5442 y Fl(4)988 5475 y Fm(and)34 b(the)h(origin)f(is)g(a)g(sadd)5 b(le-c)-5 b(enter)35 b(or)f(sadd)5 b(le-fo)-5 b(cus)35 b(e)-5 b(quilibrium)34 b(p)-5 b(oint)35 b(\(like)f(in)144 5588 y(system)g Fq(\(7\))p Fm(\).)-83 5769 y(Then,)e(in)f(a)h(neighb)-5 b(orho)g(o)g(d)35 b(of)c(the)h(c)-5 b(orr)g(esp)g(onding)34 b(critic)-5 b(al)33 b(element,)e(the)h(fol)5 b(lowing)32 b(statements)h(ar)-5 b(e)32 b(e)-5 b(quivalent)1902 6018 y Fq(5)p eop %%Page: 6 6 6 5 bop -3 191 a Fq(\()p Fp(i)p Fq(\))46 b Fm(System)34 b Fq(\(12\))g Fm(is)e(Hamiltonian)j(\(with)e(r)-5 b(esp)g(e)g(ct)34 b(to)g(some)f(suitable)g Fq(2)p Fm(-form)g Fp(!)s Fm(\).)-35 373 y Fq(\()p Fp(ii)p Fq(\))47 b Fm(System)34 b Fq(\(12\))g Fm(is)e(r)-5 b(eversible)33 b(\(with)h(r)-5 b(esp)g(e)g(ct)34 b(to)f(some)h(suitable)e(r)-5 b(eversing)33 b(involution)h Fg(G)p Fm(\).)-66 554 y Fq(\()p Fp(iii)p Fq(\))47 b Fm(The)33 b(analytic)h(ve)-5 b(ctor)33 b(\014eld)1157 531 y Fo(b)1137 554 y Fp(B)k Fm(\(as)d(in)e Fq(\(4\))p Fm(\))i(pr)-5 b(ovide)g(d)35 b(by)e(The)-5 b(or)g(em)34 b Fq(1)f Fm(vanishes.)58 727 y Fq(This)i(lo)s(cal)h(dualit)m(y)g(around)f(critical)h(elemen)m (ts)h(b)s(et)m(w)m(een)h(Hamiltonian)d(and)h(rev)m(ersible)g(systems)g (is)g(quite)-83 840 y(common.)61 b(As)37 b(an)g(example,)h(see)g(for)f (instance)f([18)q(],)k(where)c(it)h(is)f(pro)m(v)m(ed)h(this)f(equiv)-5 b(alence)36 b(in)g(the)h(case)h(of)f(a)-83 953 y(nonsemisimple)32 b(1)j(:)f(1)i(resonance,)i(whic)m(h)c(o)s(ccurs)h(when)g(t)m(w)m(o)i (pairs)d(of)i(purely)e(imaginary)g(eigen)m(v)-5 b(alues)36 b(of)f(the)-83 1066 y(linearized)i(system)h(collide.)64 b(Nev)m(ertheless,)41 b(there)e(exist)f(also)h(coun)m(ter)g(examples)f (of)h(suc)m(h)f(equiv)-5 b(alence.)65 b(F)-8 b(or)-83 1179 y(instance,)33 b(see)h(the)f(one)g(giv)m(en)g(at)g([23)r(],)g (where)g(it)f(is)g(giv)m(en)h(a)g(class)g(of)g(area)g(preserving)f (mappings,)g(with)f(linear)-83 1292 y(part)f(the)h(iden)m(tit)m(y)-8 b(,)30 b(whic)m(h)f(are)i(not)g(rev)m(ersible.)58 1405 y(Bey)m(ond)40 b(the)f(consequences)g(pro)m(vided)f(b)m(y)g(Theorem)h (1)g(in)f(the)h(Hamiltonian)e(or)i(rev)m(ersible)e(framew)m(orks,)-83 1518 y(this)32 b(\011)p Fi(NF)p Fq(-approac)m(h)i(can)f(b)s(e)g(useful) e(to)j(\014nd)e(out)h(isolated)g(p)s(erio)s(dic)e(orbits)h(and)g (related)i(in)m(v)-5 b(arian)m(t)32 b(manifolds)-83 1631 y(in)37 b(other)i(situations.)64 b(F)-8 b(or)39 b(instance,)i(in)c([10) q(])i(it)f(is)g(sho)m(wn)g(that)h(for)f(the)h(cen)m(ter-fo)s(cus)h (problem)c(\(case)k(\()p Fp(i)p Fq(\))g(in)-83 1743 y(Theorem)27 b(2\))h(eac)m(h)g(zero)g(of)f(the)g(analytic)g(function)1739 1719 y Fo(b)1745 1743 y Fp(b)p Fq(,)h(de\014ned)e(in)g(\(9\),)j(giv)m (es)e(rise)f(to)i(a)g(limit)d(cycle)i(of)g(system)h(\(8\))-83 1856 y(close)j(to)g(the)f(origin.)58 1969 y(No)m(w,)39 b(consider)c(system)i(\(12\))h(with)d(the)h(origin)f(b)s(eing)g(a)i (saddle-cen)m(ter)f(equilibrium)c(p)s(oin)m(t)j(\(case)j(\()p Fp(iii)p Fq(\))g(in)-83 2082 y(Theorem)20 b(2\).)38 b(Let)21 b Fp(N)31 b Fq(and)877 2059 y Fo(b)858 2082 y Fp(B)t Fq(,)23 b(as)e(in)e(\(2)p Fp(b)p Fq(\),)j(\(4)p Fp(b)p Fq(\),)i(b)s(e)c(the)g(analytic)g(v)m(ector)j(\014elds)c(pro)m(vided)g (b)m(y)h(Theorem)g(1.)38 b(Assume)-83 2195 y(this)30 b(system)i(\(12\))h(is)d(not)i(lo)s(cally)e(Hamiltonian)g(\(neither)g (rev)m(ersible,)h(therefore\).)45 b(Equiv)-5 b(alen)m(tly)d(,)30 b(functions)3823 2171 y Fo(b)3829 2195 y Fp(b)3868 2209 y Fl(1)3907 2195 y Fq(,)-89 2284 y Fo(b)-83 2308 y Fp(b)-44 2322 y Fl(2)26 2308 y Fq(in)f(equation)h(\(4)p Fp(b)p Fq(\))h(do)g(not)f(v)-5 b(anish)29 b(sim)m(ultaneously)-8 b(.)39 b(Then)29 b(the)i(transformed)f(system)g(b)s(ecomes)h(of)f(the)h (form)1370 2494 y(_)-46 b Fp(\037)25 b Fq(=)g Fp(N)10 b Fq(\()p Fp(\037)p Fq(\))21 b(+)f(\()p Fp(D)s Fq(\010\()p Fp(\037)p Fq(\)\))2190 2453 y Fn(\000)p Fl(1)2319 2472 y Fo(b)2300 2494 y Fp(B)t Fq(\()p Fp(\037)p Fq(\))-83 2681 y(or,)31 b(more)f(precisely)-8 b(,)500 2813 y Fo(0)500 2973 y(B)500 3028 y(B)500 3086 y(@)646 2860 y Fq(_)627 2884 y Fp(\030)641 2997 y Fq(_)-41 b Fp(\021)639 3109 y Fq(_)e Fp(\026)639 3222 y Fq(_)i Fp(\027)718 2813 y Fo(1)718 2973 y(C)718 3028 y(C)718 3086 y(A)822 3050 y Fq(=)918 2813 y Fo(0)918 2973 y(B)918 3028 y(B)918 3086 y(@)1121 2881 y Fp(\030)t(a)1213 2895 y Fl(1)1252 2881 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026)1474 2848 y Fl(2)1535 2881 y Fq(+)20 b Fp(\027)1677 2848 y Fl(2)1716 2881 y Fq(\))1046 2994 y Fj(\000)p Fp(\021)s(a)1213 3008 y Fl(1)1252 2994 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026)1474 2961 y Fl(2)1535 2994 y Fq(+)20 b Fp(\027)1677 2961 y Fl(2)1716 2994 y Fq(\))1114 3107 y Fp(\027)6 b(a)1213 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y Fl(2)2689 3117 y Fq(\()p Fp(\030)t(\021)s(;)c (\026)2911 3084 y Fl(2)2972 3117 y Fq(+)k Fp(\027)3113 3084 y Fl(2)3153 3117 y Fq(\))2560 3241 y Fp(\027)2605 3217 y Fo(b)2611 3241 y Fp(b)2650 3255 y Fl(2)2689 3241 y Fq(\()p Fp(\030)t(\021)s(;)c(\026)2911 3208 y Fl(2)2972 3241 y Fq(+)k Fp(\027)3113 3208 y Fl(2)3153 3241 y Fq(\))3229 2786 y Fo(1)3229 2946 y(C)3229 3000 y(C)3229 3055 y(C)3229 3113 y(A)3324 3050 y Fp(:)422 b Fq(\(13\))-83 3446 y(Assume)25 b(that)436 3422 y Fo(b)442 3446 y Fp(b)481 3460 y Fl(2)546 3446 y Fq(do)s(es)h(not)g(v)-5 b(anish)25 b(iden)m(tically)f(but)h (there)h(exists,)h(at)f(least,)i(a)e(non-zero)h(v)-5 b(alue)25 b Fp(I)3326 3460 y Fn(\003)3391 3446 y Fp(>)g Fq(0)h(satisfying)-89 3535 y Fo(b)-83 3559 y Fp(b)-44 3573 y Fl(2)-5 3559 y Fq(\(0)p Fp(;)15 b(I)155 3573 y Fn(\003)196 3559 y Fq(\))29 b(=)g(0.)48 b(If)32 b(w)m(e)h(tak)m(e)h (initial)c(conditions)h Fp(\030)1652 3526 y Fl(0)1721 3559 y Fq(=)d Fp(\021)1868 3526 y Fl(0)1937 3559 y Fq(=)h(0)k(in)e 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Fq(\))25 b Fj(6)p Fq(=)g(0.)41 b(That)31 b(is,)1515 4334 y(\000)1572 4348 y Fn(\003)1637 4334 y Fq(=)1733 4260 y Fo(\010)1786 4334 y Fp(\026)1841 4296 y Fl(2)1900 4334 y Fq(+)20 b Fp(\027)2042 4296 y Fl(2)2106 4334 y Fq(=)25 b Fp(I)2242 4348 y Fn(\003)2282 4260 y Fo(\011)-83 4521 y Fq(is)41 b(a)h(h)m(yp)s(erb)s(olic)d(p)s(erio)s(dic)g(orbit)i (of)h(system)g(\(14\))i(with)c(p)s(erio)s(d)g(2)p Fp(\031)s(=a)2495 4535 y Fl(2)2535 4521 y Fq(\(0)p Fp(;)15 b(I)2695 4535 y Fn(\003)2736 4521 y Fq(\))42 b(and)f(c)m(haracteristic)i(exp)s(onen)m (t)-83 4633 y Fp(a)-35 4647 y Fl(1)4 4633 y Fq(\(0)p Fp(;)15 b(I)164 4647 y Fn(\003)205 4633 y Fq(\))26 b(=)f Fp(\025)20 b Fq(+)g Fp(O)s Fq(\()p Fp(I)673 4647 y Fn(\003)712 4633 y Fq(\).)41 b(Consequen)m(tly)-8 b(,)1093 4820 y(\000)25 b(=)g(\010\(\000)1429 4834 y Fn(\003)1468 4820 y Fq(\))h(=)1625 4746 y Fo(\010)1678 4820 y Fq(\010\(0)p Fp(;)15 b Fq(0)p Fp(;)g(\026;)g(\027)6 b Fq(\))27 b(:)f Fp(\026)2263 4782 y Fl(2)2322 4820 y Fq(+)20 b Fp(\027)2464 4782 y Fl(2)2528 4820 y Fq(=)25 b Fp(I)2664 4834 y Fn(\003)2704 4746 y Fo(\011)-83 5006 y Fq(is)38 b(a)i(h)m(yp)s(erb)s(olic)d(p)s(erio)s(dic) f(orbit)j(of)g(system)h(\(12\).)69 b(It)39 b(is)f(also)i(straigh)m (tforw)m(ard)f(to)h(parameterize)f(the)h(corre-)-83 5119 y(sp)s(onding)32 b(\(lo)s(cal\))j(stable)f(and)g(unstable)g(in)m(v)-5 b(arian)m(t)34 b(manifolds)e(of)j(\000.)54 b(Namely)-8 b(,)36 b(there)f(exists)f Fp(\016)i(>)d Fq(0,)j(giv)m(en)f(b)m(y)-83 5232 y(the)c(radius)d(of)j(con)m(v)m(ergence)h(of)f(the)f(\011)p Fi(NF)p Fq(,)h(suc)m(h)f(that)669 5415 y Fp(W)768 5382 y Fl(s)755 5442 y(lo)r(c)847 5415 y Fq(\(\000\))84 b(=)1212 5341 y Fo(\010)1265 5415 y Fq(\010\(0)p Fp(;)15 b(\021)1499 5382 y Fl(0)1539 5415 y Fp(e)1581 5382 y Fn(\000)p Fk(ta)1698 5391 y Ff(1)1734 5382 y Fl(\(0)p Fk(;I)1847 5390 y Fe(\003)1883 5382 y Fl(\))1915 5415 y Fp(;)g(\026;)g(\027)6 b Fq(\))25 b(:)2212 5337 y Fo(\014)2212 5392 y(\014)2242 5415 y Fp(\021)2290 5382 y Fl(0)2330 5337 y Fo(\014)2330 5392 y(\014)2386 5415 y Fp(<)g(\016)n(;)46 b(\026)2646 5382 y Fl(2)2706 5415 y Fq(+)19 b Fp(\027)2847 5382 y Fl(2)2912 5415 y Fq(=)25 b Fp(I)3048 5429 y Fn(\003)3087 5341 y Fo(\011)3155 5415 y Fp(;)669 5590 y(W)768 5557 y Fl(u)755 5618 y(lo)r(c)847 5590 y Fq(\(\000\))84 b(=)1212 5516 y Fo(\010)1265 5590 y Fq(\010\()p Fp(\030)1410 5557 y Fl(0)1449 5590 y Fp(e)1491 5557 y Fk(ta)1553 5566 y Ff(1)1589 5557 y Fl(\(0)p Fk(;I)1702 5565 y Fe(\003)1738 5557 y Fl(\))1769 5590 y Fp(;)15 b Fq(0)p Fp(;)g(\026;)g(\027)6 b Fq(\))27 b(:)2153 5513 y Fo(\014)2153 5567 y(\014)2183 5590 y Fp(\021)2231 5557 y Fl(0)2271 5513 y Fo(\014)2271 5567 y(\014)2326 5590 y Fp(<)e(\016)n(;)47 b(\026)2587 5557 y Fl(2)2646 5590 y Fq(+)20 b Fp(\027)2788 5557 y Fl(2)2853 5590 y Fq(=)k Fp(I)2988 5604 y Fn(\003)3028 5516 y Fo(\011)3096 5590 y Fp(:)3771 5501 y Fq(\(15\))-83 5769 y(W)-8 b(e)32 b(\014nish)27 b(this)j(in)m(tro)s(duction)e (summarizing)g(this)i(result.)1902 6018 y(6)p eop %%Page: 7 7 7 6 bop -83 191 a Fs(Corollary)35 b(3)46 b Fm(Consider)31 b(system)f Fq(\(12\))i Fm(wher)-5 b(e)31 b(the)f(origin)g(is)g(a)g (sadd)5 b(le-c)-5 b(enter)31 b(e)-5 b(quilibrium)30 b(p)-5 b(oint)31 b(\(c)-5 b(ase)30 b Fq(\()p Fp(iii)p Fq(\))i Fm(in)-83 304 y(The)-5 b(or)g(em)38 b Fq(2)p Fm(\))f(and)g(let)f Fp(N)46 b Fm(and)1046 281 y Fo(b)1027 304 y Fp(B)5 b Fm(,)36 b(as)h(in)f Fq(\(2)p Fp(b)p Fq(\))f(,)g(\(4)p Fp(b)p Fq(\))p Fm(,)i(b)-5 b(e)37 b(the)f(analytic)h(ve)-5 b(ctor)37 b(\014elds)g(pr)-5 b(ovide)g(d)38 b(by)e(The)-5 b(or)g(em)38 b(1.)-83 417 y(Assume)h(that)h(the)f(\(analytic\))i (function)e Fp(I)44 b Fj(7!)1607 393 y Fo(b)1613 417 y Fp(b)1652 431 y Fl(2)1691 417 y Fq(\(0)p Fp(;)15 b(I)7 b Fq(\))p Fm(,)42 b(de\014ne)-5 b(d)40 b(in)f(a)h(neighb)-5 b(orho)g(o)g(d)42 b(of)d(the)h(origin,)h(do)-5 b(es)40 b(not)-83 530 y(vanish)32 b(identic)-5 b(al)5 b(ly)32 b(\(so)g(system)g Fq(\(12\))h Fm(is)e(neither)h(Hamiltonian)g(nor)g(r) -5 b(eversible\).)42 b(Thus,)32 b(every)f(p)-5 b(ositive)32 b(zer)-5 b(o)32 b(of)-89 619 y Fo(b)-83 643 y Fp(b)-44 657 y Fl(2)-5 643 y Fq(\(0)p Fp(;)15 b Fj(\003)p Fq(\))40 b Fm(gives)c(rise)i(to)g(a)f(hyp)-5 b(erb)g(olic)40 b(p)-5 b(erio)g(dic)39 b(orbit)e(of)h(system)g Fq(\(12\))p Fm(.)57 b(Mor)-5 b(e)g(over,)40 b(p)-5 b(ar)g(ameterizations)41 b(for)d(the)-83 756 y(\(lo)-5 b(c)g(al\))35 b(stable)e(and)h(unstable)f (invariant)g(manifolds)i(asso)-5 b(ciate)g(d)35 b(to)e(this)h(p)-5 b(erio)g(dic)34 b(orbit)f(ar)-5 b(e)34 b(given)e(by)g Fq(\(15\))p Fm(.)-83 1042 y Fr(2)135 b(Pro)t(of)45 b(of)g(the)g(Main)g (Theorem)-83 1248 y Fd(2.1)112 b(The)38 b(formal)e(solution:)49 b(a)38 b(\014rst)f(approac)m(h)-83 1420 y Fq(It)f(is)g(w)m(orth)g (noting)g(that)h(b)s(oth)f(cases,)j(the)d(origin)f(b)s(eing)g(a)i (saddle-fo)s(cus)e(or)h(b)s(eing)f(a)i(saddle-cen)m(ter,)i(can)d(b)s(e) -83 1533 y(treated)27 b(formally)e(with)g(the)i(same)g(argumen)m(t.)40 b(Moreo)m(v)m(er,)29 b(w)m(e)e(will)d(deal)i(\014rst)f(with)g(the)i (case)h(of)e(a)h Fm(c)-5 b(omplex)28 b Fq(\011)p Fi(NF)-83 1646 y Fq(and)i(will)e(deriv)m(e)i(subsequen)m(tly)f(the)i(case)g(of)g (a)g Fm(r)-5 b(e)g(al)32 b Fq(\011)p Fi(NF)p Fq(.)f(Indeed,)f(let)g(us) g(assume)g(that)h(w)m(e)h(ha)m(v)m(e)f(complexi\014ed)-83 1759 y(the)c(original)f(v)-5 b(ariables)26 b(in)g(suc)m(h)h(a)g(w)m(a)m (y)i(that)f(the)f(new)g(\(complex\))h(matrix)e(\003)h(is)g(diagonal.)39 b(Under)26 b(this)g(common)-83 1871 y(approac)m(h,)40 b(w)m(e)f(will)c(refer)i(often)h(to)h Fj(f)q(\006)p Fp(\025)1407 1885 y Fl(1)1446 1871 y Fp(;)30 b Fj(\006)p Fp(\025)1625 1885 y Fl(2)1664 1871 y Fj(g)39 b Fq(as)f(the)g(c)m(haracteristic)g (exp)s(onen)m(ts)g(of)g(the)g(origin,)g(meaning)-83 1984 y Fj(f\006)p Fp(\025)20 b Fj(\006)g Fq(i)p Fp(\013)p Fj(g)42 b Fq(in)e(the)i(\014rst)f(case)h(and)f Fj(f\006)p Fp(\025;)31 b Fj(\006)p Fq(i)p Fp(\013)p Fj(g)41 b Fq(in)f(the)i (second)f(one,)k(resp)s(ectiv)m(ely)-8 b(,)45 b(alw)m(a)m(ys)d(with)e Fp(\025;)15 b(\013)45 b(>)e Fq(0.)-83 2097 y(Moreo)m(v)m(er,)37 b(it)d(is)e(not)j(di\016cult)c(to)k(c)m(hec)m(k)h(that)e(with)f(suc)m (h)g(uni\014ed)f(notation)i(the)g(v)m(ector)h(\014elds)e Fp(N)43 b Fq(and)3678 2074 y Fo(b)3659 2097 y Fp(B)38 b Fq(tak)m(e)-83 2210 y(the)33 b(same)h(form)e(\(2)p Fp(a)p Fq(\))i(and)f(\(4)p Fp(a)p Fq(\),)i(resp)s(ectiv)m(ely)-8 b(,)33 b(in)f(b)s(oth)g(cases.)50 b(This)31 b(will)g(b)s(e)h(their)g (formal)g(asp)s(ect)i(along)f(this)-83 2323 y(pro)s(of)d(if)f(nothing)g (against)i(is)e(explicitly)f(said.)58 2436 y(The)35 b(sk)m(etc)m(h)h (of)g(the)f(pro)s(of)g(follo)m(ws)f(the)h(standard)g(pattern:)50 b(\014rst,)36 b(w)m(e)g(will)c(lo)s(ok)j(for)g(a)g(formal)g(solution)e (of)-83 2549 y(equation)1532 2662 y Fp(D)s Fq(\010)d Fp(N)g Fq(+)1919 2639 y Fo(b)1900 2662 y Fp(B)g Fq(=)25 b Fp(F)33 b Fj(\016)21 b Fq(\010)1453 b(\(16\))-83 2829 y(b)m(y)30 b(means)h(of)f(a)h(recurren)m(t)f(sc)m(heme,)i(that)f(will)d (consist)i(on)g(t)m(w)m(o)i(steps,)e(an)h(initial)c(approac)m(h)k(and)f (a)h(\014nal)e(re\014ne-)-83 2942 y(men)m(t.)47 b(Later)33 b(on,)h(it)e(will)d(b)s(e)j(in)m(tro)s(duced)f(a)i(norm)f(whic)m(h)f (will)f(allo)m(w)i(us)f(to)i(establish)e(the)i(con)m(v)m(ergence)i(of)d (the)-83 3055 y(functions)d(in)m(v)m(olv)m(ed.)58 3167 y(Th)m(us,)h(let)g(us)g(start)g(with)g(the)g(\014rst)g(part.)40 b(W)-8 b(e)32 b(recall)d(that)2177 3144 y Fo(b)2159 3167 y Fp(G)i Fq(denotes)g(that)g Fp(G)g Fq(is)e(formed)h(b)m(y)g(formal)g (p)s(o)m(w)m(er)-83 3280 y(series)f(b)s(eginning)d(with)i(terms)i(of)f (order)g(at)h(least)g(2.)40 b(No)m(w,)31 b(since)e(the)g(linear)f(part) h(of)h Fp(F)13 b Fq(\()p Fp(X)7 b Fq(\))26 b(=)f(\003)p Fp(X)h Fq(+)3581 3257 y Fo(b)3563 3280 y Fp(F)13 b Fq(\()p Fp(X)7 b Fq(\))30 b(\(or)-83 3393 y(shorter,)38 b Fp(F)48 b Fq(=)34 b(\003)25 b(+)670 3370 y Fo(b)652 3393 y Fp(F)13 b Fq(\))37 b(is)e(in)g(normal)g(form,)j(w)m(e)e(ha)m(v)m(e)i(that)e (the)h(linear)d(part)i(of)h Fp(N)46 b Fq(is)35 b(just)h(\003)g (\(notice)h(that)g(\003)-83 3506 y(represen)m(ts)32 b(also)g(the)g (complex)g(matrix)g(of)g(eigen)m(v)-5 b(alues)32 b Fj(\006)p Fp(\025)2046 3520 y Fl(1)2085 3506 y Fq(,)h Fj(\006)p Fp(\025)2267 3520 y Fl(2)2306 3506 y Fq(\))f(.)46 b(W)-8 b(riting)32 b(\010)c(=)f(id)21 b(+)3169 3483 y Fo(b)3162 3506 y Fq(\010)31 b(and)h Fp(N)38 b Fq(=)28 b(\003)21 b(+)3848 3483 y Fo(b)3825 3506 y Fp(N)9 b Fq(,)-83 3619 y(equation)30 b(\(16\))i(b)s(ecomes)1303 3732 y Fp(D)1388 3709 y Fo(b)1381 3732 y Fq(\010)d Fp(N)i Fj(\000)19 b Fq(\003)1741 3709 y Fo(b)1733 3732 y Fq(\010)26 b(=)1938 3709 y Fo(b)1920 3732 y Fp(F)34 b Fj(\016)20 b Fq(\010)g Fj(\000)2278 3709 y Fo(b)2254 3732 y Fp(N)30 b Fj(\000)2467 3709 y Fo(b)2448 3732 y Fp(B)5 b(:)1224 b Fq(\(17\))-83 3899 y(Assume)24 b(that)g(w)m(e)h(already)e(kno)m(w)1119 3876 y Fo(b)1111 3899 y Fq(\010,)1251 3876 y Fo(b)1228 3899 y Fp(N)33 b Fq(and)1524 3876 y Fo(b)1505 3899 y Fp(B)28 b Fq(up)23 b(to)i(some)g(order)e Fp(K)31 b Fq(and)23 b(let)h(us)g(see)g(whic)m(h)f(di\016culties)f(in)m(v)m(olv)m(es)-83 4012 y(the)32 b(computation)h(of)f(the)g(terms)h(of)f(order)g Fp(K)c Fq(+)21 b(1)33 b(of)1860 3989 y Fo(b)1852 4012 y Fq(\010.)46 b(F)-8 b(rom)33 b(equation)f(\(17\))i(w)m(e)f(realize)e (that)i(w)m(e)g(only)e(ha)m(v)m(e)-83 4125 y(to)g(consider)e(the)i (terms)f(up)g(to)h(order)f Fp(K)c Fq(+)20 b(1)31 b(of)g(equation)1562 4329 y Fp(D)1647 4306 y Fo(b)1640 4329 y Fq(\010)f Fp(N)g Fj(\000)20 b Fq(\003)2001 4306 y Fo(b)1993 4329 y Fq(\010)25 b(=)2201 4306 y Fo(b)2180 4329 y Fp(H)7 b(:)1483 b Fq(\(18\))-83 4533 y(where)200 4510 y Fo(b)179 4533 y Fp(H)32 b Fq(=)401 4510 y Fo(b)383 4533 y Fp(F)e Fj(\016)18 b Fq(\010)29 b(only)f(con)m(tains)h(terms)g(up)f(to)i(order)f Fp(K)35 b Fq(of)2133 4510 y Fo(b)2126 4533 y Fq(\010)o(.)40 b(The)29 b(terms)g(in)2823 4510 y Fo(b)2799 4533 y Fp(N)39 b Fq(and)3105 4510 y Fo(b)3086 4533 y Fp(B)34 b Fq(of)29 b(order)f Fp(K)d Fq(+)17 b(1)29 b(will)-83 4646 y(b)s(e)h(determined)f(later.)40 b(By)31 b(direct)f(computation,)h(writing)831 4827 y Fo(b)823 4850 y Fq(\010)25 b(=)g(\()1054 4826 y Fo(b)1045 4850 y Fp(\036)1099 4813 y Fl(\(1\))1194 4850 y Fp(;)1244 4826 y Fo(b)1234 4850 y Fp(\036)1288 4813 y Fl(\(2\))1383 4850 y Fp(;)1433 4826 y Fo(b)1423 4850 y Fp(\036)1477 4813 y Fl(\(3\))1572 4850 y Fp(;)1622 4826 y Fo(b)1612 4850 y Fp(\036)1666 4813 y Fl(\(4\))1761 4850 y Fq(\))p Fp(;)2040 4827 y Fo(b)2018 4850 y Fp(H)32 b Fq(=)25 b(\()2255 4826 y Fo(b)2257 4850 y Fp(h)2309 4813 y Fl(\(1\))2404 4850 y Fp(;)2443 4826 y Fo(b)2444 4850 y Fp(h)2496 4813 y Fl(\(2\))2592 4850 y Fp(;)2631 4826 y Fo(b)2632 4850 y Fp(h)2684 4813 y Fl(\(3\))2779 4850 y Fp(;)2818 4826 y Fo(b)2819 4850 y Fp(h)2871 4813 y Fl(\(4\))2966 4850 y Fq(\))p Fp(;)-83 5055 y Fq(with)436 5143 y Fo(b)426 5167 y Fp(\036)480 5130 y Fl(\()p Fk(i)p Fl(\))564 5167 y Fq(\()p Fp(\030)t(;)15 b(\021)s(;)g(\026;)g(\027)6 b Fq(\))27 b(=)1075 5081 y Fo(X)1221 5167 y Fp(\036)1275 5120 y Fl(\()p Fk(i)p Fl(\))1275 5197 y Fk(j)t(k)r(`m)1472 5167 y Fp(\030)1516 5130 y Fk(j)1552 5167 y Fp(\021)1600 5130 y Fk(k)1643 5167 y Fp(\026)1698 5130 y Fk(`)1731 5167 y Fp(\027)1782 5130 y Fk(m)1848 5167 y Fp(;)1978 5143 y Fo(b)1980 5167 y Fp(h)2032 5130 y Fl(\()p Fk(i)p Fl(\))2115 5167 y Fq(\()p Fp(\030)t(;)15 b(\021)s(;)g(\026;)g(\027)6 b Fq(\))27 b(=)2626 5081 y Fo(X)2773 5167 y Fp(h)2825 5120 y Fl(\()p Fk(i)p Fl(\))2825 5197 y Fk(j)t(k)r(`m)3021 5167 y Fp(\030)3065 5130 y Fk(j)3102 5167 y Fp(\021)3150 5130 y Fk(k)3193 5167 y Fp(\026)3248 5130 y Fk(`)3281 5167 y Fp(\027)3332 5130 y Fk(m)3398 5167 y Fp(;)-83 5348 y Fq(for)k Fp(i)d Fq(=)f(1)p Fp(;)15 b(:)g(:)g(:)i(;)e Fq(4,)33 b(and)e(using)f(that)i Fp(N)42 b Fq(starts)32 b(with)e(\()p Fp(\030)t(\025)1889 5362 y Fl(1)1929 5348 y Fp(;)15 b Fj(\000)p Fp(\021)s(\025)2141 5362 y Fl(1)2181 5348 y Fp(;)g(\026\025)2329 5362 y Fl(2)2369 5348 y Fp(;)g Fj(\000)p Fp(\027)6 b(\025)2584 5362 y Fl(2)2623 5348 y Fq(\))32 b(the)g(terms)f(up)g(to)h(order)f Fp(K)d Fq(+)21 b(1)31 b(of)-83 5461 y(equation)f(\(18\))i(come)f(from)f(the)h(system,) 553 5587 y Fo(\020)608 5688 y Fp(\030)662 5664 y Fo(b)652 5688 y Fp(\036)706 5640 y Fl(\()p Fk(i)p Fl(\))706 5717 y Fk(\030)809 5688 y Fj(\000)20 b Fp(\021)958 5664 y Fo(b)948 5688 y Fp(\036)1002 5650 y Fl(\()p Fk(i)p Fl(\))1002 5710 y Fk(\021)1086 5587 y Fo(\021)1155 5688 y Fp(a)1203 5702 y Fl(1)1242 5688 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))22 b(+)1662 5587 y Fo(\020)1717 5688 y Fp(\026)1781 5664 y Fo(b)1772 5688 y Fp(\036)1826 5650 y Fl(\()p Fk(i)p Fl(\))1826 5710 y Fk(\026)1929 5688 y Fj(\000)e Fp(\027)2080 5664 y Fo(b)2071 5688 y Fp(\036)2125 5650 y Fl(\()p Fk(i)p Fl(\))2125 5710 y Fk(\027)2208 5587 y Fo(\021)2277 5688 y Fp(a)2325 5702 y Fl(2)2365 5688 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))21 b Fj(\000)f Fp(\025)2838 5650 y Fn(\003)2838 5710 y Fk(i)2887 5664 y Fo(b)2877 5688 y Fp(\036)2931 5650 y Fl(\()p Fk(i)p Fl(\))3040 5688 y Fq(=)3134 5664 y Fo(b)3136 5688 y Fp(h)3188 5650 y Fl(\()p Fk(i)p Fl(\))3271 5688 y Fp(;)1902 6018 y Fq(7)p eop %%Page: 8 8 8 7 bop -83 191 a Fq(for)27 b Fp(i)f Fq(=)f(1)p Fp(;)15 b Fq(2)p Fp(;)g(:)g(:)g(:)j(;)d Fq(4)28 b(and)f Fp(\025)839 158 y Fn(\003)839 215 y Fl(1)879 191 y Fp(;)15 b(\025)972 158 y Fn(\003)972 215 y Fl(2)1012 191 y Fp(;)g(\025)1105 158 y Fn(\003)1105 215 y Fl(3)1144 191 y Fp(;)g(\025)1237 158 y Fn(\003)1237 215 y Fl(4)1305 191 y Fq(equal)27 b(to)h Fp(\025)1703 205 y Fl(1)1743 191 y Fp(;)15 b Fj(\000)p Fp(\025)1907 205 y Fl(1)1946 191 y Fp(;)g(\025)2039 205 y Fl(2)2107 191 y Fq(and)27 b Fj(\000)p Fp(\025)2405 205 y Fl(2)2444 191 y Fq(,)h(resp)s(ectiv)m(ely)-8 b(.)40 b(Therefore,)28 b(the)g(terms)f(of)-83 312 y(order)j Fp(K)d Fq(+)20 b(1)30 b(of)537 289 y Fo(b)529 312 y Fq(\010)g(come)h (from)594 595 y Fp(\036)648 547 y Fl(\(1\))648 624 y Fk(j)t(k)r(`m)897 595 y Fq(=)1465 517 y Fp(h)1517 469 y Fl(\(1\))1517 547 y Fk(j)t(k)r(`m)p 1061 574 1027 4 v 1061 657 a Fp(\025)1114 671 y Fl(1)1154 657 y Fq(\()p Fp(j)26 b Fj(\000)20 b Fp(k)j Fj(\000)d Fq(1\))h(+)f Fp(\025)1749 671 y Fl(2)1788 657 y Fq(\()p Fp(`)h Fj(\000)f Fp(m)p Fq(\))2098 595 y Fp(;)197 b Fq(if)60 b Fp(j)31 b Fj(6)p Fq(=)25 b Fp(k)e Fq(+)d(1)31 b(or)60 b Fp(`)25 b Fj(6)p Fq(=)g Fp(m;)594 889 y(\036)648 841 y Fl(\(2\))648 918 y Fk(j)t(k)r(`m)897 889 y Fq(=)1465 811 y Fp(h)1517 763 y Fl(\(2\))1517 840 y Fk(j)t(k)r(`m)p 1061 868 V 1061 951 a Fp(\025)1114 965 y Fl(1)1154 951 y Fq(\()p Fp(j)h Fj(\000)20 b Fp(k)j Fq(+)d(1\))h(+)f Fp(\025)1749 965 y Fl(2)1788 951 y Fq(\()p Fp(`)h Fj(\000)f Fp(m)p Fq(\))2098 889 y Fp(;)197 b Fq(if)60 b Fp(k)28 b Fj(6)p Fq(=)d Fp(j)h Fq(+)20 b(1)31 b(or)60 b Fp(`)25 b Fj(6)p Fq(=)g Fp(m;)531 b Fq(\(19\))594 1182 y Fp(\036)648 1134 y Fl(\(3\))648 1212 y Fk(j)t(k)r(`m)897 1182 y Fq(=)1465 1105 y Fp(h)1517 1057 y Fl(\(3\))1517 1134 y Fk(j)t(k)r(`m)p 1061 1161 V 1061 1245 a Fp(\025)1114 1259 y Fl(1)1154 1245 y Fq(\()p Fp(j)26 b Fj(\000)20 b Fp(k)s Fq(\))h(+)e Fp(\025)1592 1259 y Fl(2)1632 1245 y Fq(\()p Fp(`)h Fj(\000)g Fp(m)g Fj(\000)g Fq(1\))2098 1182 y Fp(;)197 b Fq(if)60 b Fp(j)31 b Fj(6)p Fq(=)25 b Fp(k)33 b Fq(or)61 b Fp(`)25 b Fj(6)p Fq(=)g Fp(m)20 b Fq(+)g(1)p Fp(;)594 1476 y(\036)648 1428 y Fl(\(4\))648 1506 y Fk(j)t(k)r(`m)897 1476 y Fq(=)1465 1398 y Fp(h)1517 1351 y Fl(\(4\))1517 1428 y Fk(j)t(k)r(`m)p 1061 1455 V 1061 1539 a Fp(\025)1114 1553 y Fl(1)1154 1539 y Fq(\()p Fp(j)26 b Fj(\000)20 b Fp(k)s Fq(\))h(+)e Fp(\025)1592 1553 y Fl(2)1632 1539 y Fq(\()p Fp(`)h Fj(\000)g Fp(m)g Fq(+)g(1\))2098 1476 y Fp(;)197 b Fq(if)60 b Fp(j)31 b Fj(6)p Fq(=)25 b Fp(k)33 b Fq(or)61 b Fp(m)25 b Fj(6)p Fq(=)g Fp(`)20 b Fq(+)g(1)p Fp(:)-83 1724 y Fq(It)30 b(is)g(clear)g(from)g(these)h(equations)f(that)h(terms)f(of)h(the)f (form)1279 1852 y Fo(0)1279 2012 y(B)1279 2067 y(B)1279 2121 y(B)1279 2176 y(B)1279 2234 y(@)1415 1939 y Fp(\030)1474 1871 y Fo(P)1586 1939 y Fp(\036)1640 1891 y Fl(\(1\))1640 1968 y Fk(k)r Fl(+1)p Fk(;k)r(mm)1986 1939 y Fq(\()p Fp(\030)t(\021)s Fq(\))2148 1906 y Fk(k)2192 1939 y Fq(\()p Fp(\026\027)6 b Fq(\))2368 1906 y Fk(m)1483 2079 y Fp(\021)1546 2011 y Fo(P)1657 2079 y Fp(\036)1711 2031 y Fl(\(2\))1711 2108 y Fk(j;j)t Fl(+1)p Fk(;``)1994 2079 y Fq(\()p Fp(\030)t(\021)s Fq(\))2156 2046 y Fk(j)2193 2079 y Fp(\026\027)g Fq(\))2334 2046 y Fk(`)1400 2219 y Fp(\026)1470 2151 y Fo(P)1581 2219 y Fp(\036)1635 2171 y Fl(\(3\))1635 2248 y Fk(k)r(k)r(;m)p Fl(+1)p Fk(;m)2001 2219 y Fq(\()p Fp(\030)t(\021)s Fq(\))2163 2186 y Fk(k)2207 2219 y Fq(\()p Fp(\026\027)g Fq(\))2383 2186 y Fk(m)1464 2359 y Fp(\027)1530 2291 y Fo(P)1641 2359 y Fp(\036)1695 2311 y Fl(\(4\))1695 2388 y Fk(j)t(j;`;`)p Fl(+1)1977 2359 y Fq(\()p Fp(\030)t(\021)s Fq(\))2139 2326 y Fk(j)2177 2359 y Fq(\()p Fp(\026\027)g Fq(\))2353 2326 y Fk(`)2491 1852 y Fo(1)2491 2012 y(C)2491 2067 y(C)2491 2121 y(C)2491 2176 y(C)2491 2234 y(A)3771 2144 y Fq(\(20\))-83 2564 y(cannot)24 b(b)s(e)e(determined)g(and)h(remain)f (in)g(principle)e(arbitrary)-8 b(.)37 b(In)23 b(terms)g(of)g(simply)e (linear)h(algebra)h(this)f(amoun)m(ts)-83 2677 y(to)33 b(sa)m(y)g(that)f(the)h(transformation)e(\010)h(is)f(completely)h (determined)f(once)h(it)g(has)g(b)s(een)f(\014xed)h(its)f(pro)5 b(jection)32 b(on)g(a)-83 2790 y(suitable)d(v)m(ectorial)i(subspace,)f (called)f Fm(r)-5 b(esonant)35 b(subsp)-5 b(ac)g(e)p Fq(.)-83 3033 y Fd(2.2)112 b(De\014nition)36 b(of)i(the)f(pro)6 b(jections)-83 3205 y Fq(The)35 b(t)m(yp)s(e)i(of)f(co)s(e\016cien)m (ts)g(app)s(earing)f(in)g(expression)f(\(20\))k(and)d(the)h(remarks)g (ab)s(o)m(v)m(e)h(motiv)-5 b(ate)37 b(the)f(follo)m(wing)-83 3318 y(de\014nition.)-83 3505 y Fs(De\014nition)f(4)46 b Fm(Given)32 b(a)h(formal)i(series)d Fp(h)p Fq(\()p Fp(\030)t(;)15 b(\021)s(;)g(\026;)g(\027)6 b Fq(\))28 b(=)1957 3437 y Fo(P)2068 3505 y Fp(h)2120 3520 y Fk(j)t(k)r(`m)2319 3505 y Fp(\030)2363 3472 y Fk(j)2399 3505 y Fp(\021)2447 3472 y Fk(k)2490 3505 y Fp(\026)2545 3472 y Fk(`)2578 3505 y Fp(\027)2629 3472 y Fk(m)2696 3505 y Fm(,)k(we)h(de\014ne)g(the) g(pr)-5 b(oje)g(ctions)1055 3728 y Fp(P)1113 3742 y Fl(1)1153 3728 y Fp(h)83 b Fq(:=)g Fp(\030)1627 3642 y Fo(X)1526 3840 y Fk(k)r Fn(\025)p Fl(0)p Fk(;)32 b(m)p Fn(\025)p Fl(1)1875 3728 y Fp(h)1927 3743 y Fk(k)r Fl(+1)p Fk(;k)r(mm)2275 3728 y Fq(\()p Fp(\030)t(\021)s Fq(\))2437 3691 y Fk(k)2481 3728 y Fq(\()p Fp(\026\027)6 b Fq(\))2657 3691 y Fk(m)2724 3728 y Fp(;)1055 3990 y(P)1113 4004 y Fl(2)1153 3990 y Fp(h)83 b Fq(:=)g Fp(\021)1612 3904 y Fo(X)1531 4102 y Fk(j)t Fn(\025)p Fl(0)p Fk(;)31 b(`)p Fn(\025)p Fl(1)1839 3990 y Fp(h)1891 4005 y Fk(j;j)t Fl(+1)p Fk(;``)2176 3990 y Fq(\()p Fp(\030)t(\021)s Fq(\))2338 3953 y Fk(j)2376 3990 y Fq(\()p Fp(\026\027)6 b Fq(\))2552 3953 y Fk(`)2585 3990 y Fp(;)1055 4252 y(P)1113 4266 y Fl(3)1153 4252 y Fp(h)83 b Fq(:=)g Fp(\026)1638 4166 y Fo(X)1537 4363 y Fk(k)r Fn(\025)p Fl(1)p Fk(;)32 b(m)p Fn(\025)p Fl(0)1886 4252 y Fp(h)1938 4267 y Fk(k)r(k)r(;m)p Fl(+1)p Fk(;m)2306 4252 y Fq(\()p Fp(\030)t(\021)s Fq(\))2468 4215 y Fk(k)2512 4252 y Fq(\()p Fp(\026\027)6 b Fq(\))2688 4215 y Fk(m)2754 4252 y Fp(;)1055 4514 y(P)1113 4528 y Fl(4)1153 4514 y Fp(h)83 b Fq(:=)g Fp(\027)1614 4428 y Fo(X)1533 4625 y Fk(j)t Fn(\025)p Fl(1)p Fk(;)32 b(`)p Fn(\025)p Fl(0)1842 4514 y Fp(h)1894 4529 y Fk(j)t(j)t(`;`)p Fl(+1)2163 4514 y Fq(\()p Fp(\030)t(\021)s Fq(\))2325 4477 y Fk(j)2363 4514 y Fq(\()p Fp(\026\027)6 b Fq(\))2539 4477 y Fk(`)2572 4514 y Fp(:)-83 4830 y Fm(Mor)-5 b(e)g(over,)34 b(if)e Fp(H)g Fq(=)25 b(\()p Fp(h)724 4797 y Fl(\(1\))819 4830 y Fp(;)15 b(h)911 4797 y Fl(\(2\))1006 4830 y Fp(;)g(h)1098 4797 y Fl(\(3\))1193 4830 y Fp(;)g(h)1285 4797 y Fl(\(4\))1380 4830 y Fq(\))33 b Fm(is)g(a)g(\(formal\))h(ve)-5 b(ctor)34 b(\014eld)f(we)g(de\014ne)739 5034 y Fj(P)7 b Fp(H)33 b Fq(:=)26 b(\()p Fp(P)1133 5048 y Fl(1)1173 5034 y Fp(h)1225 4997 y Fl(\(1\))1319 5034 y Fp(;)15 b(P)1417 5048 y Fl(2)1458 5034 y Fp(h)1510 4997 y Fl(\(2\))1604 5034 y Fp(;)g(P)1702 5048 y Fl(3)1743 5034 y Fp(h)1795 4997 y Fl(\(3\))1889 5034 y Fp(;)g(P)1987 5048 y Fl(4)2027 5034 y Fp(h)2079 4997 y Fl(\(4\))2174 5034 y Fq(\))p Fp(;)202 b Fj(R)p Fp(H)32 b Fq(:=)25 b Fp(H)i Fj(\000)20 b(P)7 b Fp(H)r(:)-83 5238 y Fq(As)29 b(it)f(has)h(b)s(een)f(noticed)g(b)s(efore,)h Fj(P)1204 5215 y Fo(b)1196 5238 y Fq(\010)g(corresp)s(onds)f(to)h(the)g (terms)g(whic)m(h)f(remain)f(arbitrary)h(from)g(the)h(solution)-83 5351 y(of)g(equation)g(\(18\).)42 b(Moreo)m(v)m(er,)32 b(v)m(ector)e(\014elds)e Fp(N)39 b Fq(and)1849 5328 y Fo(b)1830 5351 y Fp(B)34 b Fq(are)29 b(in)m(v)-5 b(arian)m(t)29 b(under)e(the)i(action)h(of)f Fj(P)7 b Fq(.)41 b(This)27 b(prop)s(ert)m(y)-83 5464 y(will)f(b)s(e)j(used)f(in)g(the)h(solution)e (of)j(equation)f(\(17\).)41 b(In)29 b(this)f(sense,)h(w)m(e)h(ha)m(v)m (e)g(the)f(follo)m(wing)e(lemma,)j(whose)e(pro)s(of)-83 5577 y(is)h(omitted)i(since)f(it)f(consists)h(on)h(straigh)m(tforw)m (ard)f(computations.)1902 6018 y(8)p eop %%Page: 9 9 9 8 bop -83 191 a Fs(Lemma)33 b(5)45 b Fm(Given)33 b Fp(N)i Fq(=)25 b(\003)20 b(+)1053 168 y Fo(b)1029 191 y Fp(N)43 b Fm(of)33 b(the)g(form)g Fq(\(2a\))p Fm(,)h(the)f(op)-5 b(er)g(ator)36 b Fj(L)2416 205 y Fk(N)2515 191 y Fm(de\014ne)-5 b(d)34 b(as)1497 359 y Fj(L)1560 373 y Fk(N)1627 359 y Fq(\011)25 b(:=)g Fp(D)s Fq(\011)32 b Fp(N)e Fj(\000)20 b Fq(\003\011)1418 b(\(21\))-83 527 y Fm(satis\014es)33 b(the)g(fol)5 b(lowing)34 b(pr)-5 b(op)g(erties.)-3 685 y Fq(\()p Fp(i)p Fq(\))46 b Fj(L)207 699 y Fk(N)274 685 y Fq(\011)32 b Fm(is)h(line)-5 b(ar)34 b(with)f(r)-5 b(esp)g(e)g(ct)34 b(to)g Fq(\011)e Fm(and)h Fp(N)10 b Fm(,)33 b(that)h(is)761 853 y Fj(L)824 867 y Fk(N)906 780 y Fo(\000)948 853 y Fq(\011)20 b(+)g(\011)1201 816 y Fn(0)1224 780 y Fo(\001)1290 853 y Fq(=)25 b Fj(L)1449 867 y Fk(N)1516 853 y Fq(\011)20 b(+)g Fj(L)1761 867 y Fk(N)1828 853 y Fq(\011)1899 816 y Fn(0)2294 853 y Fj(L)2357 868 y Fk(N)7 b Fl(+)p Fk(N)2538 850 y Fe(0)2564 853 y Fq(\011)25 b(=)g Fj(L)2819 867 y Fk(N)2886 853 y Fq(\011)20 b(+)g Fj(L)3131 868 y Fk(N)3194 850 y Fe(0)3220 853 y Fq(\011)p Fp(:)-35 1053 y Fq(\()p Fp(ii)p Fq(\))47 b Fj(L)207 1067 y Fk(N)307 1053 y Fm(pr)-5 b(eserves)33 b(or)-5 b(der,)35 b(that)e(is,)g Fj(L)1341 1067 y Fk(N)1408 1053 y Fq(\011)f Fm(and)i Fq(\011)e Fm(start)i(with)f(terms)h(in)e Fq(\()p Fp(\030)t(;)15 b(\021)s(;)g(\026;)g(\027)6 b Fq(\))34 b Fm(of)f(the)g(same)h(or)-5 b(der.)-66 1228 y Fq(\()p Fp(iii)p Fq(\))47 b Fm(The)33 b(pr)-5 b(oje)g(ctions)35 b Fj(P)40 b Fm(and)34 b Fj(R)e Fm(c)-5 b(ommute)34 b(with)f Fj(L)1824 1242 y Fk(N)1891 1228 y Fm(,)g(that)h(is,)963 1397 y Fj(P)23 b Fq(\()p Fj(L)1147 1411 y Fk(N)1214 1397 y Fq(\011\))i(=)g Fj(L)1504 1411 y Fk(N)1586 1397 y Fq(\()q Fj(P)7 b Fq(\011\))15 b Fp(;)388 b Fj(R)15 b Fq(\()p Fj(L)2416 1411 y Fk(N)2483 1397 y Fq(\011\))25 b(=)g Fj(L)2773 1411 y Fk(N)2855 1397 y Fq(\()q Fj(R)p Fq(\011\))15 b Fp(:)-83 1600 y Fd(2.3)112 b(The)38 b(recurren)m(t)f(sc)m(heme)-83 1771 y Fq(Let)31 b(us)g(come)h(bac)m(k)f(to)h(the)f(solution)f(of)h (equation)g(\(17\).)44 b(Ha)m(ving)31 b(in)f(mind)e(the)k(de\014nition) c(of)j(the)h(op)s(erator)f Fj(L)3866 1785 y Fk(N)-83 1884 y Fq(it)f(can)h(b)s(e)e(written)h(as)1453 1997 y Fj(L)1516 2011 y Fk(N)1591 1974 y Fo(b)1583 1997 y Fq(\010)25 b(=)1788 1974 y Fo(b)1770 1997 y Fp(F)33 b Fj(\016)21 b Fq(\010)f Fj(\000)2127 1974 y Fo(b)2104 1997 y Fp(N)30 b Fj(\000)2317 1974 y Fo(b)2298 1997 y Fp(B)t(;)1375 b Fq(\(22\))-83 2146 y(whic)m(h)39 b(is)h(of)g(t)m(yp)s(e)h(\(18\))h (pro)m(vided)d(w)m(e)h(tak)m(e)1575 2123 y Fo(b)1553 2146 y Fp(H)49 b Fq(=)1809 2123 y Fo(b)1791 2146 y Fp(F)40 b Fj(\016)27 b Fq(\010)g Fj(\000)2175 2123 y Fo(b)2151 2146 y Fp(N)37 b Fj(\000)2378 2123 y Fo(b)2359 2146 y Fp(B)t Fq(.)71 b(In)39 b(a)i(\014rst)f(approac)m(h)h(to)g(this)e(kind)f (of)-83 2259 y(equations)i(w)m(e)i(ha)m(v)m(e)g(sho)m(wn)e(that)h(they) g(could)f(b)s(e)g(solv)m(ed)g(recurren)m(tly)g(for)h(those)g(terms)g (in)e(\010)j(=)g(id)26 b(+)3761 2236 y Fo(b)3753 2259 y Fq(\010)40 b(of)-83 2372 y(t)m(yp)s(e)i Fj(R)218 2349 y Fo(b)210 2372 y Fq(\010,)j(remaining)c(those)h(of)g(the)h(form)e Fj(P)1620 2349 y Fo(b)1612 2372 y Fq(\010)i(arbitrary)-8 b(.)75 b(This)40 b(fact)j(suggests)g(the)g(idea)e(of)i(splitting)c(the) -83 2485 y(transformation)27 b(w)m(e)i(are)f(lo)s(oking)f(for,)h(\010,) g(in)m(to)g(id)14 b(+)h Fj(P)1849 2462 y Fo(b)1841 2485 y Fq(\010)h(+)f Fj(R)2094 2462 y Fo(b)2086 2485 y Fq(\010,)29 b(to)f(determine)f Fj(R)2821 2462 y Fo(b)2813 2485 y Fq(\010)h(from)f(equation)h(\(22\))i(and)d(to)-83 2597 y(c)m(ho)s(ose)k(a)g(suitable)e(v)-5 b(alue)30 b(for)g Fj(P)1072 2574 y Fo(b)1064 2597 y Fq(\010.)-83 2755 y Fs(Remark)k(6)46 b Fm(In)26 b(Normal)i(F)-7 b(orm)28 b(the)-5 b(ory)28 b(it)e(is)g(standar)-5 b(d)30 b(to)c(set)h Fj(P)2210 2732 y Fo(b)2202 2755 y Fq(\010)f(=)e(0)j Fm(in)f(or)-5 b(der)28 b(to)f(simplify)g(the)g(c)-5 b(omputations.)-83 2868 y(However,)33 b(it)f(c)-5 b(ould)34 b(b)-5 b(e)32 b(useful)h(to)g(take)g(advantage)h(of)f(this)g(fr)-5 b(e)g(e)g(dom)35 b(in)d(some)h(c)-5 b(oncr)g(ete)34 b(situations.)-83 3026 y Fq(Applying)28 b Fj(R)i Fq(on)m(to)i(equation)e(\(22\),)1183 3216 y Fj(R)1275 3116 y Fo(\020)1329 3216 y Fj(L)1392 3230 y Fk(N)1467 3193 y Fo(b)1459 3216 y Fq(\010)1525 3116 y Fo(\021)1605 3216 y Fq(=)25 b Fj(R)1793 3116 y Fo(\020)1865 3193 y(b)1847 3216 y Fp(F)13 b Fq(\(\010\))2054 3116 y Fo(\021)2129 3216 y Fj(\000)20 b(R)2321 3193 y Fo(b)2297 3216 y Fp(N)30 b Fj(\000)20 b(R)2587 3193 y Fo(b)2568 3216 y Fp(B)5 b(;)-83 3428 y Fq(using)34 b(Lemma)h(5)g(and)g (taking)g(in)m(to)g(accoun)m(t)h(that)g Fj(R)1868 3405 y Fo(b)1844 3428 y Fp(N)43 b Fq(=)33 b Fj(R)2160 3405 y Fo(b)2141 3428 y Fp(B)38 b Fq(=)33 b(0)i(if)2544 3405 y Fo(b)2520 3428 y Fp(N)45 b Fq(and)2839 3405 y Fo(b)2820 3428 y Fp(B)39 b Fq(are)d(assumed)e(to)i(b)s(e)e(of)h(the)-83 3541 y(form)30 b(\(2)p Fp(a)p Fq(\))h(and)f(\(4)p Fp(a)p Fq(\),)i(resp)s(ectiv)m(ely)-8 b(,)30 b(w)m(e)h(obtain)f(the)h (equation)1442 3732 y Fj(L)1505 3746 y Fk(N)1587 3631 y Fo(\020)1641 3732 y Fj(R)1726 3709 y Fo(b)1718 3732 y Fq(\010)1784 3631 y Fo(\021)1863 3732 y Fq(=)25 b Fj(R)2051 3631 y Fo(\020)2124 3709 y(b)2106 3732 y Fp(F)13 b Fq(\(\010\))2313 3631 y Fo(\021)2383 3732 y Fp(:)1363 b Fq(\(23\))-83 3922 y(On)39 b(the)g(other)h(hand,)h(applying)d(no)m(w)h Fj(P)47 b Fq(on)m(to)41 b(\(22\),)j(taking)39 b(again)h(in)m(to)g (accoun)m(t)h(Lemma)e(5,)k(the)d(fact)g(that)-83 4035 y Fj(P)11 4012 y Fo(b)-13 4035 y Fp(N)c Fq(=)216 4012 y Fo(b)192 4035 y Fp(N)10 b Fq(,)30 b Fj(P)420 4012 y Fo(b)400 4035 y Fp(B)h Fq(=)615 4012 y Fo(b)596 4035 y Fp(B)k Fq(and)29 b(c)m(ho)s(osing)i Fj(P)1324 4012 y Fo(b)1316 4035 y Fq(\010)25 b Fj(\021)g Fq(0,)31 b(it)f(follo)m(ws)g (that)1533 4198 y Fo(b)1509 4221 y Fp(N)g Fq(+)1722 4198 y Fo(b)1703 4221 y Fp(B)g Fq(=)25 b Fj(P)1984 4120 y Fo(\020)2056 4198 y(b)2038 4221 y Fp(F)13 b Fq(\(\010\))2245 4120 y Fo(\021)2315 4221 y Fp(:)1431 b Fq(\(24\))-83 4407 y(A)32 b(usual)e(w)m(a)m(y)i(to)g(deal)f(with)f(suc)m(h)h(kind)f (of)i(equations)f(is)f(to)j(consider)d(it)h(as)h(a)g(\014xed)e(p)s(oin) m(t)h(problem.)42 b(Th)m(us,)31 b(w)m(e)-83 4520 y(can)g(set)g Fj(P)304 4497 y Fo(b)296 4520 y Fq(\010)25 b Fj(\021)g Fq(0,)31 b(tak)m(e)h(initial)c(v)-5 b(alues)1267 4688 y(\010)1333 4651 y Fl(\(1\))1452 4688 y Fq(=)25 b(id)o Fp(;)106 b(N)1838 4651 y Fl(\(1\))1958 4688 y Fq(=)25 b(\003)p Fp(;)2267 4665 y Fo(b)2248 4688 y Fp(B)2322 4651 y Fl(\(1\))2441 4688 y Fq(=)g(0)1189 b(\(25\))-83 4856 y(and)30 b(obtain,)g(recurren)m(tly)-8 b(,)734 5025 y(\010)800 4987 y Fl(\()p Fk(K)5 b Fl(+1\))1039 5025 y Fq(=)25 b(id)19 b(+)g Fj(R)1406 5002 y Fo(b)1398 5025 y Fq(\010)1464 4987 y Fl(\()p Fk(K)5 b Fl(+1\))1678 5025 y Fp(;)106 b(N)1892 4987 y Fl(\()p Fk(K)5 b Fl(+1\))2131 5025 y Fq(=)25 b(\003)20 b(+)2425 5002 y Fo(b)2401 5025 y Fp(N)2484 4987 y Fl(\()p Fk(K)5 b Fl(+1\))2697 5025 y Fp(;)2848 5002 y Fo(b)2828 5025 y Fp(B)2902 4987 y Fl(\()p Fk(K)g Fl(+1\))3771 5025 y Fq(\(26\))-83 5193 y(from)30 b(equations)1139 5386 y Fj(L)1202 5412 y Fk(N)1265 5392 y Ff(\()p Fc(K)s Ff(\))1391 5285 y Fo(\020)1445 5386 y Fj(R)1530 5363 y Fo(b)1522 5386 y Fq(\010)1588 5324 y Fl(\()p Fk(K)5 b Fl(+1\))1801 5285 y Fo(\021)1938 5386 y Fq(=)83 b Fj(R)2184 5285 y Fo(\020)2257 5363 y(b)2239 5386 y Fp(F)2325 5285 y Fo(\020)2379 5386 y Fq(\010)2445 5349 y Fl(\()p Fk(K)5 b Fl(\))2568 5285 y Fo(\021\021)3771 5386 y Fq(\(27\))1185 5560 y Fo(b)1161 5583 y Fp(N)1244 5546 y Fl(\()p Fk(K)g Fl(+1\))1478 5583 y Fq(+)1588 5560 y Fo(b)1568 5583 y Fp(B)1642 5546 y Fl(\()p Fk(K)g Fl(+1\))1938 5583 y Fq(=)83 b Fj(P)2178 5482 y Fo(\020)2250 5560 y(b)2232 5583 y Fp(F)2319 5482 y Fo(\020)2373 5583 y Fq(\010)2439 5546 y Fl(\()p Fk(K)5 b Fl(\))2562 5482 y Fo(\021\021)2685 5583 y Fp(:)1061 b Fq(\(28\))-83 5769 y(W)-8 b(e)32 b(will)27 b(see)k(no)m(w)g(ho)m(w)f(these)h(t)m(w)m(o)h(equations)e(can)g(b)s(e)g (solv)m(ed)g(formally)-8 b(.)1902 6018 y(9)p eop %%Page: 10 10 10 9 bop -83 191 a Fs(2.3.1)105 b(Solution)36 b(of)f(a)g Fj(L)917 205 y Fk(N)983 191 y Fq(\()p Fj(R)1106 168 y Fo(b)1096 191 y Fq(\011\))25 b(=)g Fj(R)1421 168 y Fo(b)1400 191 y Fp(H)7 b Fs(-t)m(yp)s(e)35 b(equation)-83 363 y Fq(Assuming)30 b(that)j(w)m(e)f(kno)m(w)g(the)g(co)s(e\016cien)m(ts)g (of)g Fp(N)42 b Fq(and)31 b Fj(R)2025 340 y Fo(b)2003 363 y Fp(H)39 b Fq(up)31 b(to)i(a)f(giv)m(en)g(order)f Fp(K)7 b Fq(,)32 b(the)g(co)s(e\016cien)m(ts)h(of)f Fj(R)3872 340 y Fo(b)3862 363 y Fq(\011)-83 475 y(of)f(the)f(same)h(order)f(will) d(b)s(e)j(determined)f(from)1571 670 y Fj(L)1634 684 y Fk(N)1716 569 y Fo(\020)1770 670 y Fj(R)1857 647 y Fo(b)1847 670 y Fq(\011)1918 569 y Fo(\021)1998 670 y Fq(=)c Fj(R)2192 647 y Fo(b)2171 670 y Fp(H)7 b(:)1492 b Fq(\(29\))-83 882 y(Indeed,)30 b(writing)900 1012 y Fj(R)987 989 y Fo(b)977 1012 y Fq(\011)25 b(=)1169 911 y Fo(\020)1239 988 y(b)1223 1012 y Fp( )1282 1026 y Fl(1)1322 1012 y Fp(;)1379 988 y Fo(b)1362 1012 y Fp( )1421 1026 y Fl(2)1461 1012 y Fp(;)1518 988 y Fo(b)1501 1012 y Fp( )1560 1026 y Fl(3)1600 1012 y Fp(;)1657 988 y Fo(b)1640 1012 y Fp( )1699 1026 y Fl(4)1739 911 y Fo(\021)1809 1012 y Fp(;)197 b Fj(R)2129 989 y Fo(b)2108 1012 y Fp(H)32 b Fq(=)2312 911 y Fo(\020)2365 988 y(b)2366 1012 y Fp(h)2418 1026 y Fl(1)2458 1012 y Fp(;)2497 988 y Fo(b)2498 1012 y Fp(h)2550 1026 y Fl(2)2590 1012 y Fp(;)2629 988 y Fo(b)2630 1012 y Fp(h)2682 1026 y Fl(3)2723 1012 y Fp(;)2762 988 y Fo(b)2763 1012 y Fp(h)2815 1026 y Fl(4)2855 911 y Fo(\021)2924 1012 y Fp(;)-83 1192 y Fq(where)446 1281 y Fo(b)430 1305 y Fp( )489 1319 y Fk(w)546 1305 y Fq(\()p Fp(\030)t(;)15 b(\021)s(;)g(\026;)g(\027)6 b Fq(\))27 b(=)1057 1218 y Fo(X)1203 1305 y Fp( )1265 1257 y Fl(\()p Fk(w)r Fl(\))1262 1334 y Fk(j)t(k)r(`m)1444 1305 y Fp(\030)1488 1267 y Fk(j)1524 1305 y Fp(\021)1572 1267 y Fk(k)1616 1305 y Fp(\026)1671 1267 y Fk(`)1703 1305 y Fp(\027)1754 1267 y Fk(m)1821 1305 y Fp(;)2041 1281 y Fo(b)2043 1305 y Fp(h)2095 1319 y Fk(w)2152 1305 y Fq(\()p Fp(\030)t(;)15 b(\021)s(;)g(\026;)g(\027)6 b Fq(\))27 b(=)2663 1218 y Fo(X)2808 1281 y(b)2809 1305 y Fp(h)2861 1257 y Fl(\()p Fk(w)r Fl(\))2861 1334 y Fk(j)t(k)r(`m)3043 1305 y Fp(\030)3087 1267 y Fk(j)3123 1305 y Fp(\021)3171 1267 y Fk(k)3215 1305 y Fp(\026)3270 1267 y Fk(`)3302 1305 y Fp(\027)3353 1267 y Fk(m)-83 1480 y Fq(for)42 b Fp(w)48 b Fq(=)d(1)p Fp(;)15 b(:)g(:)g(:)i(;)e Fq(4,)46 b(and)c(taking)g(in)m(to)g(accoun)m (t)i(that)f Fp(N)10 b Fq(\()p Fp(\030)t(;)15 b(\021)s(;)g(\026;)g(\027) 6 b Fq(\))44 b(has)e(the)g(form)g(\(2)p Fp(a)p Fq(\),)47 b(with)41 b Fp(a)3480 1494 y Fk(i)3508 1480 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))46 b(=)-83 1593 y Fp(\025)-30 1607 y Fk(i)18 1593 y Fq(+)19 b Fo(b)-50 b Fp(a)157 1607 y Fk(i)186 1593 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\),)31 b(it)f(follo)m(ws)f(that)i(the)g(left-hand)e (side)h(of)g(\(29\))i(is)e(equiv)-5 b(alen)m(t)29 b(to)390 1709 y Fo(0)390 1869 y(B)390 1924 y(B)390 1978 y(B)390 2033 y(B)390 2088 y(B)390 2142 y(B)390 2200 y(@)511 1810 y Fq(\(\()p Fp(j)d Fj(\000)20 b Fp(k)j Fj(\000)d Fq(1\))p Fp(\025)1029 1824 y Fl(1)1090 1810 y Fq(+)g(\()p Fp(`)g Fj(\000)g Fp(m)p Fq(\))p Fp(\025)1533 1824 y Fl(2)1573 1810 y Fq(\))g(+)1719 1709 y Fo(\020)1773 1810 y Fp(\030)1834 1786 y Fo(b)1817 1810 y Fp( )1876 1825 y Fl(1)p Fk(;\030)1989 1810 y Fj(\000)g Fp(\021)2145 1786 y Fo(b)2128 1810 y Fp( )2187 1824 y Fl(1)p Fk(;\021)2285 1709 y Fo(\021)2353 1810 y(b)-50 b Fp(a)2402 1824 y Fl(1)2462 1810 y Fq(+)2553 1709 y Fo(\020)2607 1810 y Fp(\026)2678 1786 y Fo(b)2662 1810 y Fp( )2721 1824 y Fl(1)p Fk(;\026)2843 1810 y Fj(\000)20 b Fp(\027)3001 1786 y Fo(b)2985 1810 y Fp( )3044 1824 y Fl(1)p Fk(;\027)3142 1709 y Fo(\021)3210 1810 y(b)-50 b Fp(a)3259 1824 y Fl(2)511 1974 y Fq(\(\()p Fp(j)26 b Fj(\000)20 b Fp(k)j Fq(+)d(1\))p Fp(\025)1029 1988 y Fl(1)1090 1974 y Fq(+)g(\()p Fp(`)g Fj(\000)g Fp(m)p Fq(\))p Fp(\025)1533 1988 y Fl(2)1573 1974 y Fq(\))g(+)1719 1873 y Fo(\020)1773 1974 y Fp(\030)1834 1950 y Fo(b)1817 1974 y Fp( )1876 1989 y Fl(2)p Fk(;\030)1989 1974 y Fj(\000)g Fp(\021)2145 1950 y Fo(b)2128 1974 y Fp( )2187 1988 y Fl(2)p Fk(;\021)2285 1873 y Fo(\021)2353 1974 y(b)-50 b Fp(a)2402 1988 y Fl(1)2462 1974 y Fq(+)2553 1873 y Fo(\020)2607 1974 y Fp(\026)2678 1950 y Fo(b)2662 1974 y Fp( )2721 1988 y Fl(2)p Fk(;\026)2843 1974 y Fj(\000)20 b Fp(\027)3001 1950 y Fo(b)2985 1974 y Fp( )3044 1988 y Fl(2)p Fk(;\027)3142 1873 y Fo(\021)3210 1974 y(b)-50 b Fp(a)3259 1988 y Fl(2)511 2138 y Fq(\(\()p Fp(j)26 b Fj(\000)20 b Fp(k)s Fq(\))p Fp(\025)873 2152 y Fl(1)933 2138 y Fq(+)g(\()p Fp(`)h Fj(\000)f Fp(m)g Fj(\000)f Fq(1\))p Fp(\025)1532 2152 y Fl(2)1573 2138 y Fq(\))h(+)1719 2037 y Fo(\020)1773 2138 y Fp(\030)1834 2114 y Fo(b)1817 2138 y Fp( )1876 2153 y Fl(3)p Fk(;\030)1989 2138 y Fj(\000)g Fp(\021)2145 2114 y Fo(b)2128 2138 y Fp( )2187 2152 y Fl(3)p Fk(;\021)2285 2037 y Fo(\021)2353 2138 y(b)-50 b Fp(a)2402 2152 y Fl(1)2462 2138 y Fq(+)2553 2037 y Fo(\020)2607 2138 y Fp(\026)2678 2114 y Fo(b)2662 2138 y Fp( )2721 2152 y Fl(3)p Fk(;\026)2843 2138 y Fj(\000)20 b Fp(\027)3001 2114 y Fo(b)2985 2138 y Fp( )3044 2152 y Fl(3)p Fk(;\027)3142 2037 y Fo(\021)3210 2138 y(b)-50 b Fp(a)3259 2152 y Fl(2)511 2301 y Fq(\(\()p Fp(j)26 b Fj(\000)20 b Fp(k)s Fq(\))p Fp(\025)873 2315 y Fl(1)933 2301 y Fq(+)g(\()p Fp(`)h Fj(\000)f Fp(m)g Fq(+)f(1\))p Fp(\025)1532 2315 y Fl(2)1573 2301 y Fq(\))h(+)1719 2200 y Fo(\020)1773 2301 y Fp(\030)1834 2277 y Fo(b)1817 2301 y Fp( )1876 2316 y Fl(4)p Fk(;\030)1989 2301 y Fj(\000)g Fp(\021)2145 2277 y Fo(b)2128 2301 y Fp( )2187 2315 y Fl(4)p Fk(;\021)2285 2200 y Fo(\021)2353 2301 y(b)-50 b Fp(a)2402 2315 y Fl(1)2462 2301 y Fq(+)2553 2200 y Fo(\020)2607 2301 y Fp(\026)2678 2277 y Fo(b)2662 2301 y Fp( )2721 2315 y Fl(4)p Fk(;\026)2843 2301 y Fj(\000)20 b Fp(\027)3001 2277 y Fo(b)2985 2301 y Fp( )3044 2315 y Fl(4)p Fk(;\027)3142 2200 y Fo(\021)3210 2301 y(b)-50 b Fp(a)3259 2315 y Fl(2)3340 1709 y Fo(1)3340 1869 y(C)3340 1924 y(C)3340 1978 y(C)3340 2033 y(C)3340 2088 y(C)3340 2142 y(C)3340 2200 y(A)3435 2056 y Fp(:)-83 2513 y Fq(W)-8 b(e)32 b(can)e(refer)g(to)h(this)f(v)m(ector)i(\014eld,)d(in)g(short,)h (as)1278 2624 y Fo(\020)1332 2725 y Fp(L)1394 2677 y Fl(\(1\))1394 2754 y Fk(N)1504 2701 y Fo(b)1488 2725 y Fp( )1547 2739 y Fl(1)1587 2725 y Fp(;)g(L)1704 2677 y Fl(\(2\))1704 2754 y Fk(N)1815 2701 y Fo(b)1798 2725 y Fp( )1857 2739 y Fl(2)1897 2725 y Fp(;)h(L)2015 2677 y Fl(\(3\))2015 2754 y Fk(N)2125 2701 y Fo(b)2109 2725 y Fp( )2168 2739 y Fl(3)2207 2725 y Fp(;)g(L)2325 2677 y Fl(\(4\))2325 2754 y Fk(N)2435 2701 y Fo(b)2419 2725 y Fp( )2478 2739 y Fl(4)2518 2624 y Fo(\021)-83 2937 y Fq(and)f(write)f(its)h(comp)s(onen)m(ts,)h(in)e(formal)h(p)s(o)m(w)m (er)g(series)g(expansion,)f(as)721 3142 y Fp(L)783 3094 y Fl(\()p Fk(w)r Fl(\))783 3171 y Fk(N)910 3118 y Fo(b)894 3142 y Fp( )953 3156 y Fk(w)1010 3142 y Fq(\()p Fp(\030)t(;)15 b(\021)s(;)g(\026;)g(\027)6 b Fq(\))27 b(=)1664 3056 y Fo(X)1521 3254 y Fk(j)t Fl(+)p Fk(k)r Fl(+)p Fk(`)p Fl(+)p Fk(m)p Fn(\025)p Fl(2)1956 3142 y Fq(~)-48 b Fp(g)1999 3094 y Fl(\()p Fk(w)r Fl(\))1996 3172 y Fk(j)t(k)r(`m)2162 3142 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))15 b Fp( )2547 3094 y Fl(\()p Fk(w)r Fl(\))2544 3172 y Fk(j)t(k)r(`m)2727 3142 y Fp(\030)2771 3105 y Fk(j)2808 3142 y Fp(\021)2856 3105 y Fk(k)2899 3142 y Fp(\026)2954 3105 y Fk(`)2987 3142 y Fp(\027)3038 3105 y Fk(m)3104 3142 y Fp(;)642 b Fq(\(30\))-83 3428 y(b)s(eing)604 3552 y(~)-48 b Fp(g)647 3504 y Fl(\()p Fk(w)r Fl(\))644 3582 y Fk(j)t(k)r(`m)811 3552 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))26 b(:=)f Fp(\015)1318 3504 y Fl(\()p Fk(w)r Fl(\))1313 3582 y Fk(j)t(k)r(`m)1480 3552 y Fq(\()p Fp(\025)p Fq(\))c(+)f(\()p Fp(j)26 b Fj(\000)20 b Fp(k)s Fq(\))14 b Fo(b)-50 b Fp(a)2052 3566 y Fl(1)2092 3552 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))21 b(+)f(\()p Fp(`)g Fj(\000)g Fp(m)p Fq(\))14 b Fo(b)-49 b Fp(a)2875 3566 y Fl(2)2914 3552 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))p Fp(;)-83 3714 y Fq(with)779 3971 y Fp(\015)831 3923 y Fl(\()p Fk(w)r Fl(\))826 4000 y Fk(j)t(k)r(`m)992 3971 y Fq(\()p Fp(\025)p Fq(\))26 b(:=)1263 3730 y Fo(8)1263 3812 y(>)1263 3839 y(>)1263 3866 y(<)1263 4030 y(>)1263 4057 y(>)1263 4085 y(:)1385 3801 y Fq(\()p Fp(j)g Fj(\000)20 b Fp(k)j Fj(\000)d Fq(1\))p Fp(\025)1868 3815 y Fl(1)1929 3801 y Fq(+)g(\()p Fp(`)g Fj(\000)g Fp(m)p Fq(\))p Fp(\025)2372 3815 y Fl(2)2676 3801 y Fq(if)30 b Fp(w)e Fq(=)d(1)1385 3914 y(\()p Fp(j)h Fj(\000)20 b Fp(k)j Fq(+)d(1\))p Fp(\025)1868 3928 y Fl(1)1929 3914 y Fq(+)g(\()p Fp(`)g Fj(\000)g Fp(m)p Fq(\))p Fp(\025)2372 3928 y Fl(2)2676 3914 y Fq(if)30 b Fp(w)e Fq(=)d(2)1385 4027 y(\()p Fp(j)h Fj(\000)20 b Fp(k)s Fq(\))p Fp(\025)1712 4041 y Fl(1)1772 4027 y Fq(+)g(\()p Fp(`)h Fj(\000)e Fp(m)h Fj(\000)g Fq(1\))p Fp(\025)2371 4041 y Fl(2)2676 4027 y Fq(if)30 b Fp(w)e Fq(=)d(3)1385 4140 y(\()p Fp(j)h Fj(\000)20 b Fp(k)s Fq(\))p Fp(\025)1712 4154 y Fl(1)1772 4140 y Fq(+)g(\()p Fp(`)h Fj(\000)e Fp(m)h Fq(+)g(1\))p Fp(\025)2371 4154 y Fl(2)2676 4140 y Fq(if)30 b Fp(w)e Fq(=)d(4)p Fp(:)-83 4320 y Fq(Notice,)37 b(from)d(equation)g(\(30\),)j(that)e Fj(L)1313 4334 y Fk(N)1415 4320 y Fq(acts)g(on)g Fj(R)1824 4297 y Fo(b)1814 4320 y Fq(\011)f Fm(multiplying)h Fq(eac)m(h)h(co)s (e\016cien)m(t)f Fp( )3088 4335 y Fk(j)t(k)r(`m)3289 4320 y Fq(b)m(y)f(a)h(function)e(of)-83 4433 y(the)26 b(pro)s(ducts)e Fp(\030)t(\021)29 b Fq(and)c Fp(\026\027)6 b Fq(.)39 b(T)-8 b(o)26 b(tak)m(e)h(adv)-5 b(an)m(tage)28 b(of)e(this)e(feature)i(w)m(e)h(will)c(express)i(our)g(formal)g(series) g(expansions)-83 4546 y(in)h(a)h(more)h(con)m(v)m(enien)m(t)g(w)m(a)m (y)g(whic)m(h)e(will)f(highligh)m(t)g(those)i(terms)h(of)f(the)g(form)g (\()p Fp(\030)t(\021)s Fq(\))2860 4513 y Fk(p)2928 4546 y Fq(and)f(\()p Fp(\026\027)6 b Fq(\))3277 4513 y Fk(q)3316 4546 y Fq(.)39 b(A)28 b(similar)c(idea)-83 4659 y(w)m(as)31 b(suggested)g(in)e([9)q(].)41 b(In)29 b(our)h(case)h(it)f(w)m(orks)h (as)f(follo)m(ws.)40 b(F)-8 b(or)31 b(an)m(y)g(comp)s(onen)m(t)2848 4635 y Fo(b)2832 4659 y Fp( )2891 4673 y Fk(w)2978 4659 y Fq(of)f Fj(R)3168 4636 y Fo(b)3158 4659 y Fq(\011)g(w)m(e)h(ha)m(v)m (e)535 4845 y Fo(b)519 4869 y Fp( )578 4883 y Fk(w)635 4869 y Fq(\()p Fp(\030)t(;)15 b(\021)s(;)g(\026;)g(\027)6 b Fq(\))27 b(=)1146 4783 y Fo(X)1292 4869 y Fp( )1354 4821 y Fl(\()p Fk(w)r Fl(\))1351 4898 y Fk(j)t(k)r(`m)1533 4869 y Fp(\030)1577 4831 y Fk(j)1613 4869 y Fp(\021)1661 4831 y Fk(k)1704 4869 y Fp(\026)1759 4831 y Fk(`)1792 4869 y Fp(\027)1843 4831 y Fk(m)1935 4869 y Fq(=)2030 4783 y Fo(X)2177 4869 y Fp( )2239 4821 y Fl(\()p Fk(w)r Fl(\))2236 4898 y Fk(j)t(k)r(`m)2418 4869 y Fp(\030)2462 4831 y Fk(j)t Fn(\000)p Fk(k)2607 4869 y Fq(\()p Fp(\030)t(\021)s Fq(\))2770 4827 y Fk(k)2827 4869 y Fp(\026)2882 4831 y Fk(`)p Fn(\000)p Fk(m)3048 4869 y Fq(\()p Fp(\026\027)6 b Fq(\))3224 4827 y Fk(m)3306 4869 y Fp(:)440 b Fq(\(31\))-83 5072 y(De\014ning)26 b Fp(p)f Fq(=)g Fp(j)18 b Fj(\000)13 b Fp(k)s Fq(,)27 b Fp(q)h Fq(=)d Fp(`)13 b Fj(\000)g Fp(m)25 b Fq(and)h(taking)h(in)m(to)g(accoun)m(t)h(that)f Fp(j)18 b Fq(+)13 b Fp(k)i Fq(+)e Fp(`)g Fq(+)g Fp(m)24 b Fj(\025)h Fq(2,)j Fp(p)13 b Fq(+)g Fp(k)27 b Fj(\025)e Fq(0)i(and)f Fp(q)16 b Fq(+)d Fp(m)24 b Fj(\025)h Fq(0,)-83 5185 y(this)k(expansion)g(is)h(equiv)-5 b(alen)m(t)30 b(to)1514 5220 y Fo(X)1490 5417 y Fk(p;q)r Fn(2)p Fb(Z)1685 5306 y Fp( )1747 5269 y Fl(\()p Fk(w)r Fl(\))1744 5329 y Fk(pq)1859 5306 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))15 b Fp(\030)2226 5269 y Fk(p)2267 5306 y Fp(\026)2322 5269 y Fk(q)3771 5306 y Fq(\(32\))-83 5560 y(where)944 5673 y Fp( )1006 5635 y Fl(\()p Fk(w)r Fl(\))1003 5695 y Fk(pq)1118 5673 y Fq(\()p Fp(\030)t(\021)s(;)g(\026\027)6 b Fq(\))27 b(=)1655 5586 y Fo(X)1549 5788 y Fl(\()p Fk(k)r(;m)p Fl(\))p Fn(2)p Fk(Q)1827 5796 y Fc(pq)1908 5673 y Fp( )1970 5625 y Fl(\()p Fk(w)r Fl(\))1967 5702 y Fk(p)p Fl(+)p Fk(k)r(;k)r(;q)r Fl(+)p Fk(m;m)2442 5673 y Fq(\()p Fp(\030)t(\021)s Fq(\))2605 5631 y Fk(k)2662 5673 y Fq(\()q Fp(\026\027)6 b Fq(\))2839 5631 y Fk(m)3771 5673 y Fq(\(33\))1879 6018 y(10)p eop %%Page: 11 11 11 10 bop -83 191 a Fq(and)419 335 y Fp(Q)491 349 y Fk(pq)589 335 y Fq(:=)711 207 y Fo(\032)779 335 y Fq(\()p Fp(k)s(;)15 b(m)p Fq(\))26 b Fj(2)f Fq(\()p Fh(N)33 b Fj([)20 b(f)p Fq(0)p Fj(g)q Fq(\))1505 293 y Fl(2)1569 335 y Fq(:)1721 278 y Fp(k)87 b Fj(\025)82 b Fq(max)16 b Fj(f)p Fq(0)p Fp(;)f Fj(\000)p Fp(p)p Fj(g)1692 391 y Fp(m)83 b Fj(\025)f Fq(max)16 b Fj(f)p Fq(0)p Fp(;)f Fj(\000)p Fp(q)s Fj(g)2527 335 y Fp(;)31 b(k)23 b Fq(+)d Fp(m)25 b Fj(\025)g Fq(1)c Fj(\000)3112 273 y Fp(p)f Fq(+)g Fp(q)p 3112 314 201 4 v 3189 397 a Fq(2)3322 207 y Fo(\033)3406 335 y Fp(:)-83 572 y Fq(In)30 b(the)g(same)h(w)m(a)m(y)-8 b(,)32 b(for)e Fj(R)854 549 y Fo(b)833 572 y Fp(H)37 b Fq(w)m(e)31 b(get)1183 753 y Fo(b)1185 777 y Fp(h)1237 791 y Fk(w)1294 777 y Fq(\()p Fp(\030)t(;)15 b(\021)s(;)g(\026;)g(\027)6 b Fq(\))27 b(=)1829 690 y Fo(X)1805 888 y Fk(p;q)r Fn(2)p Fb(Z)2000 777 y Fp(h)2052 739 y Fl(\()p Fk(w)r Fl(\))2052 799 y Fk(pq)2164 777 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))15 b Fp(\030)2531 739 y Fk(p)2572 777 y Fp(\026)2627 739 y Fk(q)-83 1064 y Fq(where)892 1177 y Fp(h)944 1140 y Fl(\()p Fk(w)r Fl(\))944 1200 y Fk(pq)1056 1177 y Fq(\()p Fp(\030)t(;)g(\021)s(;)g(\026;)g(\027)6 b Fq(\))27 b(=)1674 1091 y Fo(X)1567 1292 y Fl(\()p Fk(k)r(;m)p Fl(\))p Fn(2)p Fk(Q)1845 1300 y Fc(pq)1927 1177 y Fp(h)1979 1129 y Fl(\()p Fk(w)r Fl(\))1979 1206 y Fk(p)p Fl(+)p Fk(k)r(;k)r(;q)r Fl(+)p Fk(m;m)2453 1177 y Fq(\()q Fp(\030)t(\021)s Fq(\))2616 1135 y Fk(k)2674 1177 y Fq(\()q Fp(\026\027)6 b Fq(\))2850 1135 y Fk(m)2932 1177 y Fp(:)814 b Fq(\(34\))-83 1446 y(With)30 b(this)f(notation)i(form)m(ula)e(\(30\))j(b)s(ecomes) 1273 1569 y Fo(X)1249 1766 y Fk(p;q)r Fn(2)p Fb(Z)1444 1655 y Fp(g)1490 1618 y Fl(\()p Fk(w)r Fl(\))1487 1678 y Fk(pq)1602 1655 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))15 b Fp( )1987 1618 y Fl(\()p Fk(w)r Fl(\))1984 1678 y Fk(pq)2100 1655 y Fq(\()p Fp(\030)t(\021)s(;)g(\026\027)6 b Fq(\))15 b Fp(\030)2467 1618 y Fk(p)2508 1655 y Fp(\026)2563 1618 y Fk(q)-83 1943 y Fq(where)30 b(no)m(w)905 2056 y Fp(g)951 2018 y Fl(\()p Fk(w)r Fl(\))948 2078 y Fk(pq)1063 2056 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))26 b(:=)f(\000)1575 2018 y Fl(\()p Fk(w)r Fl(\))1575 2078 y Fk(pq)1686 2056 y Fq(\()p Fp(\025)p Fq(\))c(+)f Fp(p)14 b Fo(b)-50 b Fp(a)2030 2070 y Fl(1)2070 2056 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))21 b(+)f Fp(q)c Fo(b)-49 b Fp(a)2597 2070 y Fl(2)2636 2056 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))-83 2219 y(b)s(eing)1008 2493 y(\000)1065 2456 y Fl(\()p Fk(w)r Fl(\))1065 2516 y Fk(pq)1177 2493 y Fq(\()p Fp(\025)p Fq(\))26 b(:=)1447 2252 y Fo(8)1447 2334 y(>)1447 2361 y(>)1447 2389 y(<)1447 2552 y(>)1447 2579 y(>)1447 2607 y(:)1569 2324 y Fq(\()p Fp(p)21 b Fj(\000)e Fq(1\))p Fp(\025)1894 2338 y Fl(1)1955 2324 y Fq(+)h Fp(q)s(\025)2143 2338 y Fl(2)2447 2324 y Fq(if)29 b Fp(w)f Fq(=)d(1)1569 2437 y(\()p Fp(p)c Fq(+)e(1\))p Fp(\025)1894 2451 y Fl(1)1955 2437 y Fq(+)h Fp(q)s(\025)2143 2451 y Fl(2)2447 2437 y Fq(if)29 b Fp(w)f Fq(=)d(2)1569 2549 y Fp(p\025)1668 2563 y Fl(1)1728 2549 y Fq(+)20 b(\()p Fp(q)j Fj(\000)d Fq(1\))p Fp(\025)2142 2563 y Fl(2)2447 2549 y Fq(if)29 b Fp(w)f Fq(=)d(3)1569 2662 y Fp(p\025)1668 2676 y Fl(1)1728 2662 y Fq(+)20 b(\()p Fp(q)j Fq(+)d(1\))p Fp(\025)2142 2676 y Fl(2)2447 2662 y Fq(if)29 b Fp(w)f Fq(=)d(4)q Fp(:)3771 2493 y Fq(\(35\))-83 2823 y(Th)m(us,)30 b(equalit)m(y)f(\(29\))j(giv)m(es)f (rise)f(to)h(the)f(equations)1275 3035 y Fp(L)1337 2987 y Fl(\()p Fk(w)r Fl(\))1337 3063 y Fk(N)1465 3011 y Fo(b)1449 3035 y Fp( )1508 3049 y Fk(w)1565 3035 y Fq(\()p Fp(\030)t(;)15 b(\021)s(;)g(\026;)g(\027)6 b Fq(\))27 b(=)2074 3011 y Fo(b)2076 3035 y Fp(h)2128 3049 y Fk(w)2185 3035 y Fq(\()p Fp(\030)t(;)15 b(\021)s(;)g(\026;)g(\027)6 b Fq(\))-83 3231 y(or,)31 b(in)e(formal)g(series)h(expansions,)770 3349 y Fo(X)746 3546 y Fk(p;q)r Fn(2)p Fb(Z)941 3435 y Fp(g)987 3398 y Fl(\()p Fk(w)r Fl(\))984 3458 y Fk(pq)1099 3435 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))15 b Fp( )1484 3398 y Fl(\()p Fk(w)r Fl(\))1481 3458 y Fk(pq)1597 3435 y Fq(\()p Fp(\030)t(\021)s(;)g(\026\027)6 b Fq(\))15 b Fp(\030)1964 3398 y Fk(p)2005 3435 y Fp(\026)2060 3398 y Fk(q)2123 3435 y Fq(=)2243 3349 y Fo(X)2219 3546 y Fk(p;q)r Fn(2)p Fb(Z)2414 3435 y Fp(h)2466 3398 y Fl(\()p Fk(w)r Fl(\))2466 3458 y Fk(pq)2578 3435 y Fq(\()p Fp(\030)t(\021)s(;)g (\026\027)6 b Fq(\))15 b Fp(\030)2945 3398 y Fk(p)2986 3435 y Fp(\026)3041 3398 y Fk(q)3079 3435 y Fp(;)-83 3723 y Fq(whose)30 b(formal)g(solution)f(is)g(giv)m(en)h(b)m(y)1193 3904 y Fo(b)1176 3928 y Fp( )1235 3942 y Fk(w)1292 3928 y Fq(\()p Fp(\030)t(;)15 b(\021)s(;)g(\026;)g(\027)6 b Fq(\))27 b(=)1828 3841 y Fo(X)1803 4039 y Fk(p;q)r Fn(2)p Fb(Z)1999 3928 y Fp( )2061 3890 y Fl(\()p Fk(w)r Fl(\))2058 3950 y Fk(pq)2173 3928 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))15 b Fp(\030)2540 3890 y Fk(p)2580 3928 y Fp(\026)2635 3890 y Fk(q)3771 3928 y Fq(\(36\))-83 4250 y(with)29 b(the)i(functions)e Fp( )735 4202 y Fl(\()p Fk(w)r Fl(\))732 4262 y Fk(pq)846 4250 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))32 b(coming)e(from)580 4525 y Fp( )642 4488 y Fl(\()p Fk(w)r Fl(\))639 4548 y Fk(pq)754 4525 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))26 b(=)1194 4464 y Fp(h)1246 4416 y Fl(\()p Fk(w)r Fl(\))1246 4476 y Fk(pq)1358 4464 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))p 1194 4504 473 4 v 1197 4613 a Fp(g)1243 4565 y Fl(\()p Fk(w)r Fl(\))1240 4625 y Fk(pq)1355 4613 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))1702 4525 y(=)2285 4464 y Fp(h)2337 4416 y Fl(\()p Fk(w)r Fl(\))2337 4476 y Fk(pq)2449 4464 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))p 1808 4504 1427 4 v 1808 4613 a(\000)1865 4565 y Fl(\()p Fk(w)r Fl(\))1865 4625 y Fk(pq)1976 4613 y Fq(\()p Fp(\025)p Fq(\))21 b(+)f Fp(p)14 b Fo(b)-50 b Fp(a)2320 4627 y Fl(1)2359 4613 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))21 b(+)f Fp(q)d Fo(b)-50 b Fp(a)2886 4627 y Fl(2)2926 4613 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))3245 4525 y Fp(;)501 b Fq(\(37\))-83 4793 y(for)32 b Fp(w)f Fq(=)d(1)p Fp(;)15 b Fq(2)p Fp(;)g(:)g(:)g(:)j (;)d Fq(4)33 b(and)e Fp(p;)15 b(q)32 b Fj(2)c Fh(Z)p Fq(.)41 b(With)32 b(this)f(notation)i(co)s(e\016cien)m(ts)g(with)e Fp(p)d Fq(=)g Fj(\006)p Fq(1)k(and)g Fp(q)f Fq(=)d(0)k(or)h Fp(p)28 b Fq(=)g(0)k(and)-83 4905 y Fp(q)c Fq(=)d Fj(\006)p Fq(1)30 b(are)h(those)g(b)s(elonging)e(to)i(the)f(pro)5 b(jection)31 b Fj(P)1810 4882 y Fo(b)1800 4905 y Fq(\011.)-83 5144 y Fs(2.3.2)105 b(Solution)36 b(of)f(a)877 5121 y Fo(b)854 5144 y Fp(N)30 b Fq(+)1067 5121 y Fo(b)1048 5144 y Fp(B)f Fq(=)c Fj(P)1334 5121 y Fo(b)1312 5144 y Fp(H)8 b Fs(-t)m(yp)s(e)35 b(equation)-83 5316 y Fq(As)44 b(it)f(has)h(b)s(een)f(done)g(for)h(equations)g(of)g(t)m(yp)s(e)g Fj(L)1775 5330 y Fk(N)1841 5316 y Fq(\()p Fj(R)1963 5293 y Fo(b)1953 5316 y Fq(\011)q(\))k(=)f Fj(R)2324 5293 y Fo(b)2303 5316 y Fp(H)k Fq(w)m(e)44 b(are)g(going)g(to)h(pro)m(v)m(e) f(that)h(equation)-59 5406 y Fo(b)-83 5429 y Fp(N)20 b Fq(+)110 5406 y Fo(b)91 5429 y Fp(B)30 b Fq(=)25 b Fj(P)378 5406 y Fo(b)356 5429 y Fp(H)33 b Fq(determines)24 b(uniquely)f(the)j(co)s(e\016cien)m(ts)g(of)2013 5406 y Fo(b)1989 5429 y Fp(N)36 b Fq(and)2289 5406 y Fo(b)2269 5429 y Fp(B)30 b Fq(pro)m(vided)24 b(they)i(are)g(of)f(t)m(yp)s(e)h (\(2)p Fp(a)p Fq(\))g(and)f(\(4)p Fp(a)p Fq(\),)-83 5542 y(resp)s(ectiv)m(ely)-8 b(,)30 b(and)g(that)825 5519 y Fo(b)804 5542 y Fp(H)37 b Fq(is)30 b(kno)m(wn.)40 b(Th)m(us,)30 b(writing)1370 5755 y Fj(P)1462 5733 y Fo(b)1440 5755 y Fp(H)j Fq(=)1645 5655 y Fo(\020)1699 5755 y Fp(\030)1742 5731 y Fo(b)1743 5755 y Fp(h)1795 5769 y Fl(1)1835 5755 y Fp(;)15 b(\021)1922 5731 y Fo(b)1923 5755 y Fp(h)1975 5769 y Fl(2)2015 5755 y Fp(;)g(\026)2108 5731 y Fo(b)2110 5755 y Fp(h)2162 5769 y Fl(3)2202 5755 y Fp(;)g(\027)2291 5731 y Fo(b)2293 5755 y Fp(h)2345 5769 y Fl(4)2385 5655 y Fo(\021)2455 5755 y Fp(;)1291 b Fq(\(38\))1879 6018 y(11)p eop %%Page: 12 12 12 11 bop -83 191 a Fq(where)172 167 y Fo(b)174 191 y Fp(h)226 205 y Fk(w)307 191 y Fq(are)25 b(functions)d(of)j Fp(\030)t(\021)i Fq(and)d Fp(\026\027)6 b Fq(,)25 b(for)f Fp(w)k Fq(=)d(1)p Fp(;)15 b Fq(2)p Fp(;)g(:)g(:)g(:)j(;)d Fq(4,)26 b(the)f(solution)e(of)h(this)f(equation)h(is)g(giv)m(en)g (explicitly)-83 304 y(b)m(y)916 434 y Fo(b)-50 b Fp(a)965 448 y Fl(1)1088 434 y Fq(=)1251 399 y Fl(1)p 1251 414 36 4 v 1251 466 a(2)1312 333 y Fo(\020)1364 410 y(b)1366 434 y Fp(h)1418 448 y Fl(1)1478 434 y Fj(\000)1567 410 y Fo(b)1569 434 y Fp(h)1621 448 y Fl(2)1661 333 y Fo(\021)1730 434 y Fp(;)2098 410 y Fo(b)2103 434 y Fp(b)2142 448 y Fl(1)2265 434 y Fq(=)2428 399 y Fl(1)p 2428 414 V 2428 466 a(2)2489 333 y Fo(\020)2542 410 y(b)2543 434 y Fp(h)2595 448 y Fl(1)2655 434 y Fq(+)2744 410 y Fo(b)2746 434 y Fp(h)2798 448 y Fl(2)2838 333 y Fo(\021)2907 434 y Fp(;)916 598 y Fo(b)g Fp(a)965 612 y Fl(2)1088 598 y Fq(=)1251 562 y Fl(1)p 1251 577 V 1251 629 a(2)1312 497 y Fo(\020)1364 574 y(b)1366 598 y Fp(h)1418 612 y Fl(3)1478 598 y Fj(\000)1567 574 y Fo(b)1569 598 y Fp(h)1621 612 y Fl(4)1661 497 y Fo(\021)1730 598 y Fp(;)2098 574 y Fo(b)2103 598 y Fp(b)2142 612 y Fl(2)2265 598 y Fq(=)2428 562 y Fl(1)p 2428 577 V 2428 629 a(2)2489 497 y Fo(\020)2542 574 y(b)2543 598 y Fp(h)2595 612 y Fl(3)2655 598 y Fq(+)2744 574 y Fo(b)2746 598 y Fp(h)2798 612 y Fl(4)2838 497 y Fo(\021)2907 598 y Fp(:)3771 516 y Fq(\(39\))-83 862 y Fd(2.4)112 b(The)38 b(recurren)m(t)f(sc)m(heme:)50 b(an)38 b(impro)m(v)m(emen)m(t)-83 1033 y Fq(One)28 b(of)g(the)h(features)g(of)f(this)g(pro)s(cedure)f(is) g(that)i(it)f(pro)m(vides)g(a)g(constructiv)m(e)h(\(and,)g(therefore,)g (implemen)m(table)-83 1146 y(on)38 b(a)g(computer\))h(w)m(a)m(y)g(to)g (determine)1332 1123 y Fo(b)1325 1146 y Fq(\010,)h Fp(N)48 b Fq(and)1780 1123 y Fo(b)1761 1146 y Fp(B)5 b Fq(.)64 b(T)-8 b(o)38 b(do)g(it)g(w)m(e)g(need)g(to)h(de\014ne)e(\(and)h(allo)s (cate)h(memory)-83 1259 y(for)d(them\))g(data)g(v)m(ectors)i(represen)m (ting)d(these)h(v)m(ector)i(\014elds.)56 b(Unfortunately)-8 b(,)37 b(the)f(sc)m(heme)h(ab)s(o)m(v)m(e)g(implies)c(to)-83 1372 y(handle)27 b(\(and)i(to)g(recompute\))h(the)f Fm(c)-5 b(omplete)30 b Fq(v)m(ectors)g(storing)2149 1349 y Fo(b)2142 1372 y Fq(\010)o(,)g Fp(N)38 b Fq(and)2568 1349 y Fo(b)2548 1372 y Fp(B)5 b Fq(,)29 b(at)h(an)m(y)f(step)g(of)g(the)g(pro)s(cess.) 39 b(This)-83 1485 y(mak)m(es)c(it)e(slo)m(w)h(and)f(not)i(m)m(uc)m(h)f (e\016cien)m(t.)52 b(In)33 b(this)g(sense)h(it)g(is)f(easy)h(to)h (re\014ne)e(it)h(b)m(y)g(pa)m(ying)g(atten)m(tion)h(on)e(the)-83 1598 y(order)d(of)g(the)h(solutions)e(of)h(equations)g(\(27\){\(28\).) 58 1711 y(Before)36 b(going)f(on)g(with)f(this)g(re\014nemen)m(t,)i (let)f(us)g(in)m(tro)s(duce)f(some)h(notation.)55 b(W)-8 b(e)36 b(will)d(denote)i Fp(G)f Fq(=)f Fj(O)3825 1729 y Fl([)p Fk(K)5 b Fl(])-83 1824 y Fq(if)35 b Fp(G)h Fq(is)f(a)h (homogeneous)h(p)s(olynomial)c(in)h(the)i(spatial)f(v)-5 b(ariables)35 b Fp(\030)t(;)15 b(\021)s(;)g(\026;)g(\027)42 b Fq(of)36 b(order)g(exactly)g Fp(K)7 b Fq(.)57 b(Besides,)37 b(w)m(e)-83 1937 y(will)31 b(write)h Fp(G)f Fq(=)f Fj(O)602 1951 y Fk(K)704 1937 y Fq(if)i Fp(G)i Fq(con)m(tains)f(only)g(terms)g (of)h(order)f(greater)h(or)f(equal)g(than)g Fp(K)40 b Fq(in)32 b(these)i(v)-5 b(ariables)32 b(and)-83 2050 y Fp(G)26 b Fq(=)f Fj(O)182 2064 y Fn(\024)p Fk(K)336 2050 y Fq(if)k(all)g(the)i(terms)f(in)f Fp(G)i Fq(are)g(of)f(order)g (less)g(or)g(equal)g(than)g Fp(K)7 b Fq(.)40 b(Th)m(us,)30 b(w)m(e)h(ha)m(v)m(e)-83 2237 y Fs(Lemma)i(7)45 b Fm(A)n(t)32 b(any)i(step)f Fp(K)f Fj(\025)25 b Fq(1)33 b Fm(of)g(the)g(pr)-5 b(o)g(c)g(ess)34 b Fq(\(25\){\(28\))p Fm(,)j(the)c(terms)898 2441 y Fq(\010)964 2404 y Fl(\()p Fk(K)5 b Fl(+1\))1197 2441 y Fj(\000)20 b Fq(\010)1354 2404 y Fl(\()p Fk(K)5 b Fl(\))1477 2441 y Fp(;)108 b(N)1693 2404 y Fl(\()p Fk(K)5 b Fl(+1\))1927 2441 y Fj(\000)20 b Fp(N)2101 2404 y Fl(\()p Fk(K)5 b Fl(\))2224 2441 y Fp(;)2376 2419 y Fo(b)2357 2441 y Fp(B)2431 2404 y Fl(\()p Fk(K)g Fl(+1\))2664 2441 y Fj(\000)2774 2419 y Fo(b)2755 2441 y Fp(B)2829 2404 y Fl(\()p Fk(K)g Fl(\))-83 2646 y Fm(ar)-5 b(e)33 b(al)5 b(l)34 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5305 y Fo(b)1746 5328 y Fp(B)1819 5265 y Fl(\(2)p Fk(J)6 b Fn(\000)p Fl(1\))2074 5328 y Fq(=)25 b(0)p Fp(;)198 b(J)34 b Fj(\025)25 b Fq(2)p Fp(:)1083 b Fq(\(43\))1879 6018 y(12)p eop %%Page: 13 13 13 12 bop -83 191 a Fd(2.5)112 b(Con)m(v)m(ergence)38 b(of)g(the)f(recurren)m(t)g(sc)m(heme)-83 363 y Fs(2.5.1)105 b(De\014nition)36 b(of)f(the)f(norm,)h(estimates)e(and)i(tec)m(hnical)g (lemmas)-83 534 y Fq(The)30 b(domains)f(w)m(e)i(consider)e(are)i(those) g(of)f(t)m(yp)s(e)p 719 665 117 4 v 719 738 a Fj(D)789 752 y Fk(\033)861 738 y Fq(=)25 b Fj(f)q Fp(z)k Fq(=)c(\()p Fp(z)1247 752 y Fl(1)1287 738 y Fp(;)15 b(z)1369 752 y Fl(2)1410 738 y Fp(;)g(:)g(:)g(:)h(;)f(z)1653 752 y Fk(n)1701 738 y Fq(\))25 b Fj(2)g Fh(C)1907 701 y Fk(n)2015 738 y Fq(:)56 b Fj(j)q Fp(z)2164 752 y Fk(j)2200 738 y Fj(j)26 b(\024)f Fp(\033)94 b(j)31 b Fq(=)25 b(1)p Fp(;)15 b Fq(2)p Fp(;)g(:)g(:)g(:)j(;)d(n)p Fj(g)g Fp(;)-83 943 y Fq(where)34 b Fp(r)g(>)e Fq(0)j(and)f Fj(j)23 b(\001)g(j)34 b 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4143 y Ff(2)3830 4108 y Fm(.)-66 4311 y Fq(\()p Fp(iii)p Fq(\))47 b Fm(L)-5 b(et)33 b Fp(g)j Fm(b)-5 b(e)33 b(an)g(analytic)h(function)f(on)p 1454 4238 117 4 v 33 w Fj(D)1524 4325 y Fk(\033)1603 4311 y Fm(satisfying)g(that)h Fj(j)q Fp(g)s Fq(\()p Fp(z)t Fq(\))p Fj(j)27 b(\025)e Fp(C)39 b Fj(8)p Fp(z)29 b Fj(2)p 2840 4238 V 24 w(D)2910 4325 y Fk(\033)2957 4311 y Fm(.)42 b(Then,)33 b(one)g(has)h(that)1785 4433 y Fo(\015)1785 4488 y(\015)1785 4542 y(\015)1785 4597 y(\015)1846 4503 y Fq(1)p 1845 4544 47 4 v 1845 4627 a Fp(g)1902 4433 y Fo(\015)1902 4488 y(\015)1902 4542 y(\015)1902 4597 y(\015)1952 4655 y Fl(1)p Fk(;\033)2079 4565 y Fj(\024)2198 4503 y Fq(1)p 2185 4544 72 4 v 2185 4627 a Fp(C)2267 4565 y(:)-51 4873 y Fq(\()p Fp(iv)s Fq(\))47 b Fm(If)36 b Fp(G)314 4892 y Fl([)p Fk(K)5 b Fl(])452 4873 y Fq(=)30 b Fj(O)625 4892 y Fl([)p Fk(K)5 b Fl(])768 4873 y Fm(and)37 b Fp(H)1024 4892 y Fl([)p Fk(L)p Fl(])1145 4873 y Fq(=)30 b Fj(O)1318 4892 y Fl([)p Fk(L)p Fl(])1445 4873 y Fm(ar)-5 b(e)37 b(homo)-5 b(gene)g(ous)37 b(p)-5 b(olynomials)39 b(of)d(or)-5 b(ders)37 b Fp(K)42 b Fm(and)36 b Fp(L)p Fm(,)g(r)-5 b(esp)g(e)g(ctively,)144 4986 y(with)34 b Fp(K)e Fj(6)p Fq(=)25 b Fp(L)p Fm(,)32 b(then)1148 5022 y Fo(\015)1148 5077 y(\015)1199 5099 y Fp(G)1270 5118 y Fl([)p Fk(K)5 b Fl(])1398 5099 y Fq(+)20 b Fp(H)1565 5118 y Fl([)p Fk(L)p Fl(])1656 5022 y Fo(\015)1656 5077 y(\015)1706 5139 y Fl(1)p Fk(;\033)1801 5148 y Ff(2)1865 5099 y Fq(=)1961 5022 y Fo(\015)1961 5077 y(\015)2012 5099 y Fp(G)2083 5118 y Fl([)p Fk(K)5 b Fl(])2191 5022 y Fo(\015)2191 5077 y(\015)2241 5139 y Fl(1)p Fk(;\033)2336 5148 y Ff(2)2396 5099 y Fq(+)2487 5022 y Fo(\015)2487 5077 y(\015)2537 5099 y Fp(H)2613 5118 y Fl([)p Fk(L)p Fl(])2704 5022 y Fo(\015)2704 5077 y(\015)2754 5139 y Fl(1)p Fk(;\033)2849 5148 y Ff(2)2904 5099 y Fp(:)-83 5314 y Fq(F)-8 b(rom)32 b(this)f(p)s(oin)m(t)g(up)f(to)j(the)f(end)f (of)g(this)g(section)h(w)m(e)g(will)d(pro)m(v)m(e)k(some)f(tec)m (hnical)f(results)g(whic)m(h)f(will)f(b)s(e)i(used)-83 5427 y(during)j(the)i(pro)s(of)g(of)g(the)g(con)m(v)m(ergence)j(of)d (the)g(recurren)m(t)g(sc)m(heme)h(in)m(tro)s(duced)e(in)g(Sections)g (2.3)j(and)d(2.4.)59 b(In)-83 5540 y(particular,)28 b(next)i(lemma)f (pro)m(vides)g(a)h(lo)m(w)m(er)f(b)s(ound)f(for)h Fj(j)p Fp(q)2011 5554 y Fl(1)2050 5540 y Fp(\025)2103 5554 y Fl(1)2163 5540 y Fq(+)20 b Fp(q)2295 5554 y Fl(2)2334 5540 y Fp(\025)2387 5554 y Fl(2)2426 5540 y Fj(j)30 b Fq(whic)m(h)e(w)m(orks)i(in)e(b)s(oth)h(cases,)i(when)d(the)-83 5653 y(equilibrium)34 b(p)s(oin)m(t)k(is)g(a)h Fm(sadd)5 b(le-c)-5 b(enter)40 b Fq(or)f(a)g Fm(sadd)5 b(le-fo)-5 b(cus)40 b Fq(\(whose)f(c)m(haracteristic)h(exp)s(onen)m(ts)f(are)g (giv)m(en)g(b)m(y)-83 5766 y Fj(f\006)p Fp(\025;)31 b Fq(i)p Fp(\013)p Fj(g)f Fq(and)g Fj(f\006)p Fp(\025)20 b Fj(\006)g Fq(i)p Fp(\013)p Fj(g)p Fq(,)31 b(resp)s(ectiv)m(ely\).) 1879 6018 y(13)p eop %%Page: 14 14 14 13 bop -83 191 a Fs(Lemma)33 b(9)45 b Fm(L)-5 b(et)33 b(us)g(de\014ne)1380 304 y Fp(!)1437 318 y Fn(1)1537 304 y Fq(=)25 b Fp(!)1690 318 y Fn(1)1764 304 y Fq(\(\003\))h(:=)g(min) 13 b Fj(f)q Fp(\025;)30 b(\013)p Fj(g)1303 b Fq(\(46\))-83 470 y Fm(wher)-5 b(e)34 b(we)e(assume)i Fp(\025;)15 b(\013)26 b(>)f Fq(0)p Fm(.)42 b(Then,)33 b(we)g(have)g(that)1287 718 y Fj(j)p Fp(q)1353 732 y Fl(1)1392 718 y Fp(\025)1445 732 y Fl(1)1504 718 y Fq(+)20 b Fp(q)1636 732 y Fl(2)1675 718 y Fp(\025)1728 732 y Fl(2)1768 718 y Fj(j)25 b(\025)1914 590 y Fo(\022)1981 607 y(q)p 2072 607 278 4 v 111 x Fp(q)2116 686 y Fl(2)2113 744 y(1)2176 718 y Fq(+)20 b Fp(q)2311 686 y Fl(2)2308 744 y(2)2350 590 y Fo(\023)2432 718 y Fp(!)2489 732 y Fn(1)-83 965 y Fm(for)33 b(any)g Fp(q)277 979 y Fl(1)316 965 y Fp(;)15 b(q)397 979 y Fl(2)462 965 y Fj(2)25 b Fh(Z)p Fm(.)-83 1152 y Fi(Pro)s(of.)81 b Fq(W)-8 b(e)32 b(pro)s(ceed)e(separately)-8 b(.)41 b(Th)m(us,)53 1338 y Fj(\017)46 b Fm(Sadd)5 b(le-c)-5 b(enter)34 b(c)-5 b(ase)p Fq(:)41 b(as)31 b(it)f(has)g(b)s(een)g(men)m(tioned)f(ab)s(o)m (v)m(e,)j(w)m(e)f(ha)m(v)m(e)h Fp(\025)2639 1352 y Fl(1)2703 1338 y Fq(=)25 b Fp(\025)31 b Fq(and)e Fp(\025)3112 1352 y Fl(2)3177 1338 y Fq(=)c(i)p Fp(\013)30 b Fq(so)516 1586 y Fj(j)p Fp(q)582 1600 y Fl(1)621 1586 y Fp(\025)674 1600 y Fl(1)734 1586 y Fq(+)20 b Fp(q)866 1600 y Fl(2)905 1586 y Fp(\025)958 1600 y Fl(2)997 1586 y Fj(j)26 b Fq(=)1144 1475 y Fo(q)p 1234 1475 469 4 v 1234 1586 a Fp(q)1278 1555 y Fl(2)1275 1612 y(1)1318 1586 y Fp(\025)1371 1560 y Fl(2)1430 1586 y Fq(+)20 b Fp(q)1565 1555 y Fl(2)1562 1612 y(2)1605 1586 y Fp(\013)1663 1560 y Fl(2)1728 1586 y Fj(\025)1824 1458 y Fo(\022)1891 1475 y(q)p 1982 1475 278 4 v 111 x Fp(q)2026 1555 y Fl(2)2023 1612 y(1)2085 1586 y Fq(+)g Fp(q)2220 1555 y Fl(2)2217 1612 y(2)2259 1458 y Fo(\023)2341 1586 y Fq(min)14 b Fj(f)p Fp(\025;)31 b(\013)p Fj(g)26 b Fq(=)2887 1458 y Fo(\022)2954 1475 y(q)p 3045 1475 V 111 x Fp(q)3089 1555 y Fl(2)3086 1612 y(1)3148 1586 y Fq(+)20 b Fp(q)3283 1555 y Fl(2)3280 1612 y(2)3323 1458 y Fo(\023)3405 1586 y Fp(!)3462 1600 y Fn(1)3536 1586 y Fp(:)53 1871 y Fj(\017)46 b Fm(Sadd)5 b(le-fo)-5 b(cus)34 b(c)-5 b(ase)p Fq(:)41 b(no)m(w)31 b(w)m(e)f(ha)m(v)m(e)i Fp(\025)1481 1885 y Fl(1)1546 1871 y Fq(=)25 b Fp(\025)20 b Fq(+)g(i)p Fp(\013)30 b Fq(and)g Fp(\025)2150 1885 y Fl(2)2214 1871 y Fq(=)25 b Fp(\025)c Fj(\000)f Fq(i)p Fp(\013)p Fq(.)40 b(Then,)583 2074 y Fj(j)p Fp(q)649 2088 y Fl(1)688 2074 y Fp(\025)741 2088 y Fl(1)801 2074 y Fq(+)20 b Fp(q)933 2088 y Fl(2)972 2074 y Fp(\025)1025 2088 y Fl(2)1064 2074 y Fj(j)26 b Fq(=)f Fj(j)p Fq(\()p Fp(q)1312 2088 y Fl(1)1371 2074 y Fq(+)20 b Fp(q)1503 2088 y Fl(2)1542 2074 y Fq(\))p Fp(\025)h Fq(+)f(\()p Fp(q)1818 2088 y Fl(1)1877 2074 y Fj(\000)g Fp(q)2009 2088 y Fl(2)2048 2074 y Fq(\)i)p Fp(\013)p Fj(j)26 b Fq(=)2314 1991 y Fo(p)p 2405 1991 1065 4 v 83 x Fq(\()p Fp(q)2481 2088 y Fl(1)2540 2074 y Fq(+)20 b Fp(q)2672 2088 y Fl(2)2711 2074 y Fq(\))2746 2047 y Fl(2)2786 2074 y Fp(\025)2839 2047 y Fl(2)2899 2074 y Fq(+)g(\()p Fp(q)3066 2088 y Fl(1)3125 2074 y Fj(\000)g Fp(q)3257 2088 y Fl(2)3296 2074 y Fq(\))3331 2047 y Fl(2)3371 2074 y Fp(\013)3429 2047 y Fl(2)3469 2074 y Fp(:)144 2277 y Fq(If)30 b Fp(q)276 2291 y Fl(1)315 2277 y Fp(q)356 2291 y Fl(2)420 2277 y Fp(>)25 b Fq(0,)31 b(using)e(that)i Fj(j)p Fp(q)1118 2291 y Fl(1)1158 2277 y Fj(j)20 b Fq(+)g Fj(j)p Fp(q)1360 2291 y Fl(2)1399 2277 y Fj(j)26 b(\025)1546 2198 y Fo(p)p 1637 2198 278 4 v 79 x Fp(q)1681 2245 y Fl(2)1678 2303 y(1)1740 2277 y Fq(+)20 b Fp(q)1875 2245 y Fl(2)1872 2303 y(2)1914 2277 y Fq(,)31 b(one)f(obtains)g(that)720 2450 y Fo(p)p 811 2450 1065 4 v 83 x Fq(\()p Fp(q)887 2547 y Fl(1)947 2533 y Fq(+)20 b Fp(q)1079 2547 y Fl(2)1118 2533 y Fq(\))1153 2507 y Fl(2)1193 2533 y Fp(\025)1246 2507 y Fl(2)1305 2533 y Fq(+)g(\()p Fp(q)1472 2547 y Fl(1)1532 2533 y Fj(\000)g Fp(q)1664 2547 y Fl(2)1703 2533 y Fq(\))1738 2507 y Fl(2)1777 2533 y Fp(\013)1835 2507 y Fl(2)1901 2533 y Fj(\025)1997 2450 y Fo(p)p 2087 2450 474 4 v 2087 2533 a Fq(\()p Fp(q)2163 2547 y Fl(1)2223 2533 y Fq(+)g Fp(q)2355 2547 y Fl(2)2394 2533 y Fq(\))2429 2507 y Fl(2)2469 2533 y Fp(\025)2522 2507 y Fl(2)2587 2533 y Fj(\025)2682 2405 y Fo(\022)2749 2422 y(q)p 2840 2422 278 4 v 111 x Fp(q)2884 2502 y Fl(2)2881 2559 y(1)2944 2533 y Fq(+)g Fp(q)3079 2502 y Fl(2)3076 2559 y(2)3118 2405 y Fo(\023)3200 2533 y Fp(!)3257 2547 y Fn(1)3331 2533 y Fp(:)144 2781 y Fq(On)30 b(the)g(other)h(hand,)e(if)h Fp(q)1067 2795 y Fl(1)1106 2781 y Fp(q)1147 2795 y Fl(2)1211 2781 y Fp(<)25 b Fq(0)31 b(then)718 2946 y Fo(p)p 809 2946 1065 4 v 82 x Fq(\()p Fp(q)885 3042 y Fl(1)944 3028 y Fq(+)20 b Fp(q)1076 3042 y Fl(2)1115 3028 y Fq(\))1150 3002 y Fl(2)1190 3028 y Fp(\025)1243 3002 y Fl(2)1303 3028 y Fq(+)f(\()p Fp(q)1469 3042 y Fl(1)1529 3028 y Fj(\000)h Fp(q)1661 3042 y Fl(2)1700 3028 y Fq(\))1735 3002 y Fl(2)1775 3028 y Fp(\013)1833 3002 y Fl(2)1898 3028 y Fj(\025)1994 2946 y Fo(p)p 2085 2946 480 4 v 82 x Fq(\()p Fp(q)2161 3042 y Fl(1)2220 3028 y Fj(\000)g Fp(q)2352 3042 y Fl(2)2391 3028 y Fq(\))2426 3002 y Fl(2)2466 3028 y Fp(\013)2524 3002 y Fl(2)2589 3028 y Fj(\025)2685 2900 y Fo(\022)2752 2918 y(q)p 2843 2918 278 4 v 110 x Fp(q)2887 2997 y Fl(2)2884 3054 y(1)2947 3028 y Fq(+)f Fp(q)3081 2997 y Fl(2)3078 3054 y(2)3121 2900 y Fo(\023)3203 3028 y Fp(!)3260 3042 y Fn(1)3334 3028 y Fp(:)3865 3291 y Fa(2)-83 3664 y Fs(Remark)34 b(10)46 b Fm(In)28 b(fact,)h Fp(!)838 3678 y Fn(1)941 3664 y Fm(c)-5 b(onstitutes)29 b(a)f(lower)h(b)-5 b(ound)29 b(for)g(the)f(values)g Fp(!)2568 3679 y Fk(k)2638 3664 y Fm(intr)-5 b(o)g(duc)g(e)g(d)31 b(by)d(Bruno)g(in)g Fq(condition)-83 3777 y Fp(!)35 b Fm(\(se)-5 b(e)33 b(Se)-5 b(ction)33 b Fq(1)p Fm(\).)43 b(Mor)-5 b(e)g(over,)33 b(notic)-5 b(e)34 b(that,)f(in)g(the)g Fq(saddle-cen)m(ter)g Fm(c)-5 b(ase,)33 b(one)g(has)g(that)1650 3980 y Fp(\032)p Fq(\(\003)1795 3942 y Fn(\000)p Fl(1)1889 3980 y Fq(\))26 b(=)f Fp(!)2106 3942 y Fn(\000)p Fl(1)2103 4002 y Fn(1)-83 4183 y Fm(wher)-5 b(e)36 b Fp(\032)p Fq(\()p Fp(M)10 b Fq(\))35 b Fm(is)g(the)g Fq(sp)s(ectral)d(radius)h Fm(of)i(the)g(matrix)h Fp(M)10 b Fm(,)35 b(de\014ne)-5 b(d)36 b(as)f(the)h(maximum)g(of)f(the)g(mo)-5 b(dulus)36 b(of)f(their)-83 4295 y(eigenvalues.)-83 4507 y Fq(No)m(w,)f(w)m(e)f (presen)m(t)g(a)f(basic)g(result)g(whic)m(h)f(pro)m(vides)h(estimates)h (for)f(the)h(v)m(ector)h(\014elds)d Fj(R)3093 4484 y Fo(b)3083 4507 y Fq(\011,)3236 4484 y Fo(b)3212 4507 y Fp(N)42 b Fq(and)3525 4484 y Fo(b)3506 4507 y Fp(B)37 b Fq(that)c(are)-83 4619 y(solution)c(of)h(the)h(equations)1225 4727 y Fo(b)1201 4750 y Fp(N)f Fq(+)1414 4727 y Fo(b)1395 4750 y Fp(B)g Fq(=)25 b Fj(P)1682 4727 y Fo(b)1660 4750 y Fp(H)8 b(;)197 b Fj(L)2029 4764 y Fk(N)2111 4649 y Fo(\020)2165 4750 y Fj(R)2252 4727 y Fo(b)2242 4750 y Fq(\011)2313 4649 y Fo(\021)2393 4750 y Fq(=)25 b Fj(R)2587 4727 y Fo(b)2566 4750 y Fp(H)1129 b Fq(\(47\))-83 4934 y(and)30 b(whose)g(formal)g(approac)m(h)g(has)g(b)s(een)g(deriv)m(ed)f (in)g(Sections)i(2.3.2)h(and)d(2.3.1,)k(resp)s(ectiv)m(ely)-8 b(.)-83 5145 y Fs(Prop)s(osition)36 b(11)46 b Fm(L)-5 b(et)45 b(us)g(c)-5 b(onsider)46 b(a)g(ve)-5 b(ctor)45 b(\014eld)1904 5122 y Fo(b)1883 5145 y Fp(H)51 b Fm(analytic)46 b(on)p 2509 5072 117 4 v 46 w Fj(D)2579 5159 y Fk(\033)2671 5145 y Fm(and)f(let)h Fj(R)3087 5122 y Fo(b)3077 5145 y Fq(\011)e Fm(and)3405 5122 y Fo(b)3381 5145 y Fp(N)10 b Fm(,)3559 5122 y Fo(b)3540 5145 y Fp(B)49 b Fm(\(of)c(the)-83 5258 y(form)28 b Fq(\(2)p Fp(a)p Fq(\))d(and)e(\(4)p Fp(a)p Fq(\))p Fm(,)29 b(r)-5 b(esp)g(e)g(ctively\))29 b(b)-5 b(e)26 b(the)i(solutions)g(of)f(e)-5 b(quations)28 b Fq(\(47\))p Fm(,)h(\(formal)5 b(ly\))29 b(derive)-5 b(d)28 b(in)e(Se)-5 b(ctions)28 b Fq(2.3.2)-83 5371 y Fm(and)34 b Fq(2.3.1)p Fm(.)43 b(Then,)33 b(the)g(fol)5 b(lowing)34 b(estimates)f(hold.)-3 5557 y Fq(\()p Fp(i)p Fq(\))46 b Fm(First,)34 b(we)e(have)1454 5588 y Fo(\015)1454 5643 y(\015)1454 5697 y(\015)1529 5670 y(b)1505 5693 y Fp(N)1588 5588 y Fo(\015)1588 5643 y(\015)1588 5697 y(\015)1638 5756 y Fl(1)p Fk(;\033)1755 5693 y Fp(;)1828 5588 y Fo(\015)1828 5643 y(\015)1828 5697 y(\015)1898 5670 y(b)1879 5693 y Fp(B)1952 5588 y Fo(\015)1952 5643 y(\015)1952 5697 y(\015)2003 5756 y Fl(1)p Fk(;\033)2130 5693 y Fj(\024)2226 5588 y Fo(\015)2226 5643 y(\015)2226 5697 y(\015)2276 5693 y Fj(P)2368 5670 y Fo(b)2346 5693 y Fp(H)2430 5588 y Fo(\015)2430 5643 y(\015)2430 5697 y(\015)2480 5756 y Fl(1)p Fk(;\033)2597 5693 y Fp(:)1879 6018 y Fq(14)p eop %%Page: 15 15 15 14 bop -35 191 a Fq(\()p Fp(ii)p Fq(\))47 b Fm(Mor)-5 b(e)g(over,)1294 334 y Fo(\015)1294 388 y(\015)1294 443 y(\015)1344 438 y Fj(R)1431 415 y Fo(b)1421 438 y Fq(\011)1492 334 y Fo(\015)1492 388 y(\015)1492 443 y(\015)1542 501 y Fl(1)p Fk(;\033)1669 438 y Fj(\024)2093 222 y Fo(\015)2093 277 y(\015)2093 331 y(\015)2143 327 y Fj(R)2242 304 y Fo(b)2220 327 y Fp(H)2303 222 y Fo(\015)2303 277 y(\015)2303 331 y(\015)2354 390 y Fl(1)p Fk(;\033)p 1775 417 999 4 v 1775 560 a Fp(!)1832 574 y Fn(1)1922 432 y Fo(\022)1989 560 y Fq(1)20 b Fj(\000)2219 524 y Fl(4)p 2155 539 164 4 v 2155 591 a Fk(\033)15 b(!)2254 599 y Fe(1)2344 455 y Fo(\015)2344 510 y(\015)2344 565 y(\015)2394 560 y Fj(R)2493 537 y Fo(b)2471 560 y Fp(H)2554 455 y Fo(\015)2554 510 y(\015)2554 565 y(\015)2605 623 y Fl(1)p Fk(;\033)2706 432 y Fo(\023)144 760 y Fm(pr)-5 b(ovide)g(d)35 b(we)e(assume)g(that)h (the)f(fol)5 b(lowing)34 b(b)-5 b(ound)33 b(is)g(satis\014e)-5 b(d:)1681 853 y Fo(\015)1681 907 y(\015)1681 962 y(\015)1731 957 y Fj(R)1829 934 y Fo(b)1808 957 y Fp(H)1891 853 y Fo(\015)1891 907 y(\015)1891 962 y(\015)1942 1021 y Fl(1)p Fk(;\033)2069 957 y Fp(<)2175 896 y(\033)s(!)2287 910 y Fn(1)p 2175 936 187 4 v 2245 1020 a Fq(4)2371 957 y Fp(:)1375 b Fq(\(48\))-83 1205 y Fi(Pro)s(of.)-3 1394 y Fq(\()p Fp(i)p Fq(\))46 b(F)-8 b(rom)31 b(\(38\))h(and)e(\(39\))i(it) e(follo)m(ws)f(that)i Fj(k)1599 1371 y Fo(b)1575 1394 y Fp(N)11 b Fj(k)1704 1408 y Fl(1)p Fk(;\033)1836 1394 y Fq(and)30 b Fj(k)2077 1371 y Fo(b)2058 1394 y Fp(B)5 b Fj(k)2177 1408 y Fl(1)p Fk(;\033)2309 1394 y Fq(are)31 b(b)s(oth)f(b)s(ounded)e(b)m(y)i Fj(kP)3311 1371 y Fo(b)3288 1394 y Fp(H)8 b Fj(k)3417 1408 y Fl(1)p Fk(;\033)3519 1394 y Fq(.)-35 1573 y(\()p Fp(ii)p Fq(\))47 b(The)30 b(second)h(equation)f(in)f(\(47\))j(w)m(as)e(solv)m(ed)h(in)e(Section)h (2.3.1.)42 b(F)-8 b(rom)31 b(there)g(it)f(follo)m(ws)f(that)1081 1672 y Fo(\015)1081 1726 y(\015)1081 1781 y(\015)1130 1752 y(b)1132 1776 y Fp(h)1184 1790 y Fk(w)1241 1672 y Fo(\015)1241 1726 y(\015)1241 1781 y(\015)1291 1839 y Fl(1)p Fk(;\033)1418 1776 y Fq(=)1657 1690 y Fo(X)1514 1887 y Fk(j)t Fl(+)p Fk(k)r Fl(+)p Fk(`)p Fl(+)p Fk(m)p Fn(\025)p Fl(2)1946 1776 y Fj(j)q Fp(h)2024 1791 y Fk(j)t(k)r(`m)2190 1776 y Fj(j)i Fp(\033)2301 1739 y Fk(j)t Fl(+)p Fk(k)r Fl(+)p Fk(`)p Fl(+)p Fk(m)2657 1776 y Fq(=)1043 b(\(49\))1272 1961 y Fo(X)1247 2158 y Fk(p;q)r Fn(2)p Fb(Z)1549 1961 y Fo(X)1443 2162 y Fl(\()p Fk(k)r(;m)p Fl(\))p Fn(2)p Fk(Q)1721 2170 y Fc(pq)1802 1943 y Fo(\014)1802 1997 y(\014)1802 2052 y(\014)1833 2047 y Fp(h)1885 1999 y Fl(\()p Fk(w)r Fl(\))1885 2077 y Fk(p)p Fl(+)p Fk(k)r(;k)r(;q)r Fl(+)p Fk(m;m)2329 1943 y Fo(\014)2329 1997 y(\014)2329 2052 y(\014)2390 2047 y Fp(\033)2445 2010 y Fk(p)p Fl(+)p Fk(q)r Fl(+2\()p Fk(k)r Fl(+)p Fk(m)p Fl(\))2900 2047 y Fq(=)1272 2241 y Fo(X)1247 2438 y Fk(p;q)r Fn(2)p Fb(Z)1443 2222 y Fo(\015)1443 2277 y(\015)1443 2332 y(\015)1493 2327 y Fp(h)1545 2289 y Fl(\()p Fk(w)r Fl(\))1545 2350 y Fk(pq)1657 2327 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))15 b Fp(\030)2024 2289 y Fk(p)2065 2327 y Fp(\026)2120 2289 y Fk(q)2158 2222 y Fo(\015)2158 2277 y(\015)2158 2332 y(\015)2208 2390 y Fl(1)p Fk(;\033)2325 2327 y Fp(:)144 2625 y Fq(F)-8 b(rom)37 b(that)g(section)f(w)m(e)h(also)f(kno)m(w)g (that)h(the)g(solution)d Fj(R)2274 2602 y Fo(b)2264 2625 y Fq(\011)h(=)g(\()2527 2601 y Fo(b)2511 2625 y Fp( )2570 2639 y Fl(1)2610 2625 y Fp(;)2666 2601 y Fo(b)2650 2625 y Fp( )2709 2639 y Fl(2)2749 2625 y Fp(;)15 b(:)g(:)g(:)i(;)2967 2601 y Fo(b)2951 2625 y Fp( )3010 2639 y Fl(4)3050 2625 y Fq(\))36 b(is)f(giv)m(en,)j(in)d(terms)h(of)144 2754 y(formal)30 b(p)s(o)m(w)m(er)g(series)g(b)m(y)g(\(32\))i(where)e Fp( )1582 2706 y Fl(\()p Fk(w)r Fl(\))1579 2766 y Fk(pq)1694 2754 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))31 b(are)g(obtained)f(from)1000 3013 y Fp( )1062 2976 y Fl(\()p Fk(w)r Fl(\))1059 3036 y Fk(pq)1174 3013 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))26 b(=)2092 2952 y Fp(h)2144 2904 y Fl(\()p Fk(w)r Fl(\))2144 2964 y Fk(pq)2255 2952 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))p 1614 2992 1427 4 v 1614 3101 a(\000)1671 3053 y Fl(\()p Fk(w)r Fl(\))1671 3113 y Fk(pq)1783 3101 y Fq(\()p Fp(\025)p Fq(\))21 b(+)f Fp(p)14 b Fo(b)-50 b Fp(a)2127 3115 y Fl(1)2166 3101 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))21 b(+)f Fp(q)d Fo(b)-50 b Fp(a)2693 3115 y Fl(2)2732 3101 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))3051 3013 y Fp(;)695 b Fq(\(50\))144 3299 y(for)40 b Fp(w)k Fq(=)d(1)p Fp(;)15 b Fq(2)p Fp(;)g(:)g(:)g(:)j (;)d Fq(4,)43 b Fp(p;)15 b(q)45 b Fj(2)c Fh(Z)35 b Fq(and)40 b(the)g(co)s(e\016cien)m(ts)h(\000)2214 3251 y Fl(\()p Fk(w)r Fl(\))2214 3311 y Fk(pq)2325 3299 y Fq(\()p Fp(\025)p Fq(\))g(as)f(de\014ned)f(in)g(\(35\).)71 b(Notice)41 b(that)f(the)144 3430 y(functions)f Fp( )608 3382 y Fl(\()p Fk(w)r Fl(\))605 3442 y Fk(pq)760 3430 y Fq(in)g(\(50\))j(are)f (rational)e(functions)g(of)h Fp(\030)t(\021)s Fq(,)k Fp(\026\027)6 b Fq(.)69 b(Therefore,)43 b(expression)c(\(32\))j(is)d (not)i(an)144 3561 y(standard)30 b(represen)m(tation)g(in)f(p)s(o)m(w)m (er)i(series,)f(that)h(is,)e(form)m(ula)h(\(34\))i(do)s(es)e(not)g (apply)f(to)j Fp( )3388 3513 y Fl(\()p Fk(w)r Fl(\))3385 3572 y Fk(pq)3500 3561 y Fq(.)144 3716 y(T)-8 b(o)30 b(estimate)g(the)g(1-norm)f(of)h Fj(R)1298 3693 y Fo(b)1288 3716 y Fq(\011)f(on)p 1513 3643 117 4 v 29 w Fj(D)1583 3730 y Fk(\033)1659 3716 y Fq(w)m(e)h(ha)m(v)m(e)h(to)f(b)s(ound)d (their)h(comp)s(onen)m(ts.)41 b(T)-8 b(aking)29 b(in)m(to)g(accoun)m(t) 144 3828 y(Lemma)i(8\()p Fp(i)p Fq(\))h(w)m(e)e(ha)m(v)m(e)599 3927 y Fo(\015)599 3981 y(\015)599 4036 y(\015)666 4007 y(b)650 4031 y Fp( )709 4045 y Fk(w)766 3927 y Fo(\015)766 3981 y(\015)766 4036 y(\015)816 4094 y Fl(1)p Fk(;\033)943 4031 y Fj(\024)1064 3945 y Fo(X)1039 4142 y Fk(p;q)r Fn(2)p Fb(Z)1235 3927 y Fo(\015)1235 3981 y(\015)1235 4036 y(\015)1285 4031 y Fp( )1347 3994 y Fl(\()p Fk(w)r Fl(\))1344 4054 y Fk(pq)1459 4031 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))15 b Fp(\030)1826 3994 y Fk(p)1867 4031 y Fp(\026)1922 3994 y Fk(q)1960 3927 y Fo(\015)1960 3981 y(\015)1960 4036 y(\015)2010 4094 y Fl(1)p Fk(;\033)2137 4031 y Fq(=)790 4270 y Fo(X)765 4467 y Fk(p;q)r Fn(2)p Fb(Z)961 4197 y Fo(\015)961 4252 y(\015)961 4307 y(\015)961 4361 y(\015)961 4416 y(\015)1410 4295 y Fp(h)1462 4247 y Fl(\()p Fk(w)r Fl(\))1462 4307 y Fk(pq)1574 4295 y Fq(\()p Fp(\030)t(\021)s(;)g(\026\027)6 b Fq(\))p Fp(\030)1926 4262 y Fk(p)1967 4295 y Fp(\026)2022 4262 y Fk(q)p 1021 4336 1427 4 v 1021 4444 a Fq(\000)1078 4396 y Fl(\()p Fk(w)r Fl(\))1078 4456 y Fk(pq)1190 4444 y Fq(\()p Fp(\025)p Fq(\))21 b(+)e Fp(p)14 b Fo(b)-50 b Fp(a)1533 4458 y Fl(1)1573 4444 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))21 b(+)f Fp(q)d Fo(b)-50 b Fp(a)2100 4458 y Fl(2)2139 4444 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))2458 4197 y Fo(\015)2458 4252 y(\015)2458 4307 y(\015)2458 4361 y(\015)2458 4416 y(\015)2509 4474 y Fl(1)p Fk(;\033)2636 4357 y Fj(\024)1064 b Fq(\(51\))790 4595 y Fo(X)765 4793 y Fk(p;q)r Fn(2)p Fb(Z)961 4577 y Fo(\015)961 4632 y(\015)961 4686 y(\015)1011 4682 y Fp(h)1063 4644 y Fl(\()p Fk(w)r Fl(\))1063 4704 y Fk(pq)1175 4682 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))p Fp(\030)1527 4644 y Fk(p)1568 4682 y Fp(\026)1623 4644 y Fk(q)1661 4577 y Fo(\015)1661 4632 y(\015)1661 4686 y(\015)1711 4745 y Fl(1)p Fk(;\033)1828 4523 y Fo(\015)1828 4577 y(\015)1828 4632 y(\015)1828 4686 y(\015)1828 4741 y(\015)2579 4620 y Fq(1)p 1888 4661 V 1888 4770 a(\000)1945 4722 y Fl(\()p Fk(w)r Fl(\))1945 4781 y Fk(pq)2057 4770 y Fq(\()p Fp(\025)p Fq(\))21 b(+)f Fp(p)14 b Fo(b)-50 b Fp(a)2401 4784 y Fl(1)2440 4770 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))21 b(+)f Fp(q)d Fo(b)-50 b Fp(a)2967 4784 y Fl(2)3007 4770 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))3325 4523 y Fo(\015)3325 4577 y(\015)3325 4632 y(\015)3325 4686 y(\015)3325 4741 y(\015)3376 4800 y Fl(1)p Fk(;\033)144 4958 y Fq(Next)31 b(lemma)f(giv)m(es)h(an)f(upp)s(er)f(b)s(ound)f(for)i (the)h(second)f(norm)g(app)s(earing)f(in)g(this)g(form)m(ula)h(\(51\).) 144 5196 y Fs(Lemma)j(12)46 b Fm(Consider)38 b Fq(\000)1113 5148 y Fl(\()p Fk(w)r Fl(\))1113 5208 y Fk(pq)1224 5196 y Fq(\()p Fp(\025)p Fq(\))g Fm(as)f(de\014ne)-5 b(d)38 b(in)f Fq(\(35\))h Fm(and)f Fo(b)-50 b Fp(a)2359 5210 y Fl(1)2398 5196 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))p Fm(,)38 b Fo(b)-50 b Fp(a)2821 5210 y Fl(2)2860 5196 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))38 b Fm(c)-5 b(oming)38 b(fr)-5 b(om)38 b Fq(\(2)p Fp(a)p Fq(\))p Fm(.)144 5309 y(Then,)33 b(for)g(any)h Fp(p;)15 b(q)28 b Fj(2)d Fh(Z)j Fm(and)33 b Fq(\()p Fp(\030)t(;)15 b(\021)s(;)g(\026;)g(\027)6 b Fq(\))27 b Fj(2)p 1740 5236 117 4 v 25 w(D)1810 5323 y Fk(\033)1857 5309 y Fm(,)33 b(we)f(have)h(that)723 5434 y Fo(\014)723 5489 y(\014)723 5543 y(\014)754 5539 y Fq(\000)811 5501 y Fl(\()p Fk(w)r Fl(\))811 5561 y Fk(pq)922 5539 y Fq(\()p Fp(\025)p Fq(\))21 b(+)f Fp(p)14 b Fo(b)-50 b Fp(a)1266 5553 y Fl(1)1305 5539 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))21 b(+)f Fp(q)d Fo(b)-50 b Fp(a)1832 5553 y Fl(2)1872 5539 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))2180 5434 y Fo(\014)2180 5489 y(\014)2180 5543 y(\014)2236 5539 y Fj(\025)25 b Fp(!)2389 5553 y Fn(1)2479 5411 y Fo(\022)2546 5539 y Fq(1)20 b Fj(\000)2783 5477 y Fq(4)p 2712 5518 187 4 v 2712 5601 a Fp(\033)s(!)2824 5615 y Fn(1)2924 5434 y Fo(\015)2924 5489 y(\015)2924 5543 y(\015)2974 5539 y Fj(R)3073 5516 y Fo(b)3051 5539 y Fp(H)3134 5434 y Fo(\015)3134 5489 y(\015)3134 5543 y(\015)3185 5602 y Fl(1)p Fk(;\033)3287 5411 y Fo(\023)144 5769 y Fm(pr)-5 b(ovide)g(d)35 b(estimate)f Fq(\(48\))g Fm(is)e(satis\014e)-5 b(d.)1879 6018 y Fq(15)p eop %%Page: 16 16 16 15 bop 175 191 a Fq(Pro)s(of)30 b(\(lemma\))g(W)-8 b(e)32 b(will)27 b(distinguish)g(t)m(w)m(o)32 b(cases:)247 379 y Fs({)45 b Fq(If)30 b Fj(j)p Fp(p)p Fj(j)21 b Fq(+)f Fj(j)p Fp(q)s Fj(j)25 b(\025)g Fq(1)31 b(it)f(follo)m(ws)f(that)1056 501 y Fo(\014)1056 555 y(\014)1056 610 y(\014)1087 605 y Fq(\000)1144 568 y Fl(\()p Fk(w)r Fl(\))1144 628 y Fk(pq)1255 605 y Fq(\()p Fp(\025)p Fq(\))21 b(+)f Fp(p)14 b Fo(b)-50 b Fp(a)1599 619 y Fl(1)1658 605 y Fq(+)20 b Fp(q)d Fo(b)-50 b Fp(a)1856 619 y Fl(2)1896 501 y Fo(\014)1896 555 y(\014)1896 610 y(\014)1951 605 y Fj(\025)2047 501 y Fo(\014)2047 555 y(\014)2047 610 y(\014)2093 501 y(\014)2093 555 y(\014)2093 610 y(\014)2123 605 y Fq(\000)2180 568 y Fl(\()p Fk(w)r Fl(\))2180 628 y Fk(pq)2291 605 y Fq(\()p Fp(\025)p Fq(\))2414 501 y Fo(\014)2414 555 y(\014)2414 610 y(\014)2466 605 y Fj(\000)19 b(k)q Fp(p)14 b Fo(b)-50 b Fp(a)2711 619 y Fl(1)2770 605 y Fq(+)20 b Fp(q)d Fo(b)-50 b Fp(a)2968 619 y Fl(2)3008 605 y Fj(k)3053 632 y Fn(1)p Fk(;\033)3190 501 y Fo(\014)3190 555 y(\014)3190 610 y(\014)3771 605 y Fq(\(52\))344 862 y(F)-8 b(rom)31 b(the)g (de\014nition)d(of)i(\000)1304 814 y Fl(\()p Fk(w)r Fl(\))1304 874 y Fk(pq)1446 862 y Fq(in)f(\(35\))j(and)e(applying)e(Lemma)i(9)h (it)f(turns)f(out)i(that)1735 986 y Fo(\014)1735 1041 y(\014)1735 1095 y(\014)1765 1091 y Fq(\000)1822 1053 y Fl(\()p Fk(w)r Fl(\))1822 1113 y Fk(pq)1933 1091 y Fq(\()p Fp(\025)p Fq(\))2056 986 y Fo(\014)2056 1041 y(\014)2056 1095 y(\014)2113 1091 y Fj(\025)25 b Fp(M)2297 1105 y Fk(pq)2386 1091 y Fp(!)2443 1105 y Fn(1)2517 1091 y Fp(;)344 1313 y Fq(where)30 b(w)m(e)h(de\014ne)1144 1544 y Fp(M)1232 1558 y Fk(pq)1331 1544 y Fq(:=)25 b(min)1619 1416 y Fo(\032)1687 1426 y(q)p 1778 1426 558 4 v 118 x Fq(\()q Fj(j)p Fp(p)p Fj(j)20 b(\000)g Fq(1\))2102 1502 y Fl(2)2161 1544 y Fq(+)g Fp(q)2296 1518 y Fl(2)2336 1544 y Fp(;)2376 1426 y Fo(q)p 2467 1426 V 118 x Fp(p)2513 1518 y Fl(2)2572 1544 y Fq(+)g(\()p Fj(j)p Fp(q)s Fj(j)h(\000)f Fq(1\))2985 1502 y Fl(2)3024 1416 y Fo(\033)3108 1544 y Fp(:)344 1828 y Fq(W)-8 b(e)35 b(recall)e(that)h(the)g(terms)g Fp(h)1419 1780 y Fl(\()p Fk(w)r Fl(\))1419 1840 y Fk(pq)1530 1828 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))35 b(with)d Fj(j)p Fp(p)p Fj(j)f Fq(=)f(1)35 b(and)e Fp(q)g Fq(=)d(0)k(or)g Fp(p)c Fq(=)h(0)j(and)f Fj(j)p Fp(q)s Fj(j)e Fq(=)f(1)k(v)-5 b(anish)344 1950 y(since)30 b(they)h(b)s(elong)e (to)i(the)g(pro)5 b(jection)30 b Fj(P)1854 1927 y Fo(b)1832 1950 y Fp(H)38 b Fq(so,)31 b(in)e(particular,)g(this)g(implies)e(that) 1962 2154 y Fp(M)2050 2168 y Fk(pq)2149 2154 y Fj(\025)e Fq(1)p Fp(:)1456 b Fq(\(53\))344 2358 y(Moreo)m(v)m(er,)33 b(it)d(is)f(clear)i(that)1831 2471 y Fj(j)p Fp(p)p Fj(j)p Fp(;)g Fj(j)p Fp(q)s Fj(j)41 b(\024)25 b Fq(2)p Fp(M)2347 2485 y Fk(pq)2421 2471 y Fp(:)1325 b Fq(\(54\))344 2638 y(Coming)30 b(bac)m(k)h(to)g(equation)f(\(52\))i(w)m(e)f(ha)m(v)m(e)g (that)698 2760 y Fo(\014)698 2815 y(\014)698 2869 y(\014)743 2760 y(\014)743 2815 y(\014)743 2869 y(\014)773 2865 y Fq(\000)830 2827 y Fl(\()p Fk(w)r Fl(\))830 2887 y Fk(pq)942 2865 y Fq(\()p Fp(\025)p Fq(\))1065 2760 y Fo(\014)1065 2815 y(\014)1065 2869 y(\014)1116 2865 y Fj(\000)20 b(k)p Fp(p)14 b Fo(b)-50 b Fp(a)1361 2879 y Fl(1)1421 2865 y Fq(+)20 b Fp(q)d Fo(b)-50 b Fp(a)1619 2879 y Fl(2)1658 2865 y Fj(k)1704 2892 y Fn(1)p Fk(;\033)1841 2760 y Fo(\014)1841 2815 y(\014)1841 2869 y(\014)1896 2865 y Fj(\025)864 3089 y Fp(M)952 3103 y Fk(pq)1041 3089 y Fp(!)1098 3103 y Fn(1)1187 2957 y Fo(\014)1187 3012 y(\014)1187 3066 y(\014)1187 3121 y(\014)1217 3089 y Fq(1)21 b Fj(\000)1515 3027 y Fq(1)p 1384 3068 309 4 v 1384 3151 a Fp(M)1472 3165 y Fk(pq)1561 3151 y Fp(!)1618 3165 y Fn(1)1717 3089 y Fj(k)q Fp(p)14 b Fo(b)-50 b Fp(a)1872 3103 y Fl(1)1931 3089 y Fq(+)20 b Fp(q)d Fo(b)-50 b Fp(a)2129 3103 y Fl(2)2169 3089 y Fj(k)2214 3116 y Fn(1)p Fk(;\033)2351 2957 y Fo(\014)2351 3012 y(\014)2351 3066 y(\014)2351 3121 y(\014)2407 3089 y Fj(\025)25 b Fp(!)2560 3103 y Fn(1)2649 2961 y Fo(\022)2716 3089 y Fq(1)c Fj(\000)2961 3027 y Fq(4)p 2883 3068 202 4 v 2883 3151 a Fp(\033)d(!)3010 3165 y Fn(1)3109 2984 y Fo(\015)3109 3039 y(\015)3109 3093 y(\015)3160 3089 y Fj(R)3258 3066 y Fo(b)3237 3089 y Fp(H)3320 2984 y Fo(\015)3320 3039 y(\015)3320 3093 y(\015)3370 3152 y Fl(1)p Fk(;\033)3472 2961 y Fo(\023)3554 3089 y Fp(;)344 3346 y Fq(where)24 b(it)h(has)f(b)s(een)g(tak)m(en)i (in)m(to)f(accoun)m(t)h(assumption)d(\(48\))j(and,)g(from)e(\(54\),)k (\(39\))e(and)e(Lemma)h(8\()p Fp(i)p Fq(\),)344 3458 y(the)31 b(estimate)809 3617 y(1)p 678 3657 309 4 v 678 3741 a Fp(M)766 3755 y Fk(pq)855 3741 y Fp(!)912 3755 y Fn(1)1011 3678 y Fj(k)p Fp(p)14 b Fo(b)-50 b Fp(a)1165 3692 y Fl(1)1225 3678 y Fq(+)20 b Fp(q)d 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4073 y Fo(\020)1125 4174 y Fj(k)p Fp(\033)f Fo(b)-49 b Fp(a)1289 4188 y Fl(1)1328 4174 y Fj(k)1374 4201 y Fn(1)p Fk(;\033)1531 4174 y Fq(+)20 b Fj(k)p Fp(\033)d Fo(b)-49 b Fp(a)1786 4188 y Fl(2)1825 4174 y Fj(k)1870 4201 y Fn(1)p Fk(;\033)2008 4073 y Fo(\021)2087 4174 y Fj(\024)2271 4113 y Fq(4)p 2193 4153 V 2193 4237 a Fp(\033)18 b(!)2320 4251 y Fn(1)2435 4070 y Fo(\015)2435 4124 y(\015)2435 4179 y(\015)2485 4174 y Fj(R)2584 4151 y Fo(b)2562 4174 y Fp(H)2645 4070 y Fo(\015)2645 4124 y(\015)2645 4179 y(\015)2696 4238 y Fn(1)p Fk(;\033)2858 4174 y Fj(\024)3042 4113 y Fq(4)p 2964 4153 V 2964 4237 a Fp(\033)g(!)3091 4251 y Fn(1)3206 4070 y Fo(\015)3206 4124 y(\015)3206 4179 y(\015)3257 4174 y Fj(R)3355 4151 y Fo(b)3334 4174 y Fp(H)3417 4070 y Fo(\015)3417 4124 y(\015)3417 4179 y(\015)3467 4238 y Fl(1)p Fk(;\033)3584 4174 y Fp(:)247 4430 y Fs({)45 b Fq(If)30 b Fp(p)25 b Fq(=)g Fp(q)j Fq(=)d(0)31 b(one)g(has)f(that)1328 4547 y Fo(\014)1328 4602 y(\014)1328 4656 y(\014)1358 4652 y Fq(\000)1415 4614 y Fl(\()p Fk(w)r Fl(\))1415 4674 y Fk(pq)1526 4652 y Fq(\()p Fp(\025)p Fq(\))21 b(+)f Fp(p)14 b Fo(b)-50 b Fp(a)1870 4666 y Fl(1)1930 4652 y Fq(+)20 b Fp(q)d Fo(b)-50 b Fp(a)2128 4666 y Fl(2)2167 4547 y Fo(\014)2167 4602 y(\014)2167 4656 y(\014)2223 4652 y Fq(=)2319 4547 y Fo(\014)2319 4602 y(\014)2319 4656 y(\014)2349 4652 y Fq(\000)2406 4604 y Fl(\()p Fk(w)r Fl(\))2406 4678 y(00)2517 4652 y Fq(\()p Fp(\025)p Fq(\))2640 4547 y Fo(\014)2640 4602 y(\014)2640 4656 y(\014)2697 4652 y Fj(\025)25 b Fp(!)2850 4666 y Fn(1)2924 4652 y Fp(;)822 b Fq(\(55\))344 4879 y(and,)30 b(in)f(particular,)g(assuming)g (again)i(\(48\),)1237 5028 y Fo(\014)1237 5083 y(\014)1237 5137 y(\014)1267 5133 y Fq(\000)1324 5085 y Fl(\()p Fk(w)r Fl(\))1324 5159 y(00)1436 5133 y Fq(\()p Fp(\025)p Fq(\))1559 5028 y Fo(\014)1559 5083 y(\014)1559 5137 y(\014)1615 5133 y Fj(\025)25 b Fp(!)1768 5147 y Fn(1)1867 5133 y Fj(\025)g Fp(!)2020 5147 y Fn(1)2110 5005 y Fo(\022)2177 5133 y Fq(1)20 b Fj(\000)2421 5071 y Fq(4)p 2343 5112 V 2343 5195 a Fp(\033)f(!)2471 5209 y Fn(1)2570 5028 y Fo(\015)2570 5083 y(\015)2570 5137 y(\015)2620 5133 y Fj(R)2719 5110 y Fo(b)2697 5133 y Fp(H)2780 5028 y Fo(\015)2780 5083 y(\015)2780 5137 y(\015)2831 5196 y Fl(1)p Fk(;\033)2933 5005 y Fo(\023)3015 5133 y Fp(:)344 5382 y Fq(This)29 b(concludes)g(the)i(pro)s(of)f(of)g(this)f(lemma.) 1879 6018 y(16)p eop %%Page: 17 17 17 16 bop 3865 191 a Fa(2)144 504 y Fq(Since)25 b(w)m(e)h(are)g (assuming)f(that)h(\(48\))h(holds,)f(w)m(e)g(can)g(apply)e(this)h (lemma)g(together)i(with)e(lemma)g(8\()p Fp(iii)p Fq(\))j(and,)144 617 y(therefore,)j(it)f(follo)m(ws)g(that)631 734 y Fo(\015)631 789 y(\015)631 843 y(\015)631 898 y(\015)631 952 y(\015)1383 832 y Fq(1)p 692 872 1427 4 v 692 981 a(\000)749 933 y Fl(\()p Fk(w)r Fl(\))749 993 y Fk(pq)860 981 y Fq(\()p Fp(\025)p Fq(\))21 b(+)f Fp(p)14 b Fo(b)-50 b Fp(a)1204 995 y Fl(1)1243 981 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))22 b(+)d Fp(q)e Fo(b)-50 b Fp(a)1770 995 y Fl(2)1810 981 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))2129 734 y Fo(\015)2129 789 y(\015)2129 843 y(\015)2129 898 y(\015)2129 952 y(\015)2179 1011 y Fl(1)p Fk(;\033)2306 893 y Fj(\024)2889 832 y Fq(1)p 2412 872 999 4 v 2412 1015 a Fp(!)2469 1029 y Fn(1)2559 887 y Fo(\022)2626 1015 y Fq(1)20 b Fj(\000)2856 979 y Fl(4)p 2792 994 164 4 v 2792 1046 a Fk(\033)15 b(!)2891 1054 y Fe(1)2981 911 y Fo(\015)2981 965 y(\015)2981 1020 y(\015)3031 1015 y Fj(R)3129 992 y Fo(b)3108 1015 y Fp(H)3191 911 y Fo(\015)3191 965 y(\015)3191 1020 y(\015)3242 1078 y Fl(1)p Fk(;\033)3343 887 y Fo(\023)3420 893 y Fp(:)144 1269 y Fq(Th)m(us,)30 b(estimate)h(\(51\))h(join)m(tly)d(with)g(\(49\))j(giv)m(es)660 1407 y Fo(\015)660 1462 y(\015)660 1516 y(\015)727 1488 y(b)710 1512 y Fp( )769 1526 y Fk(w)826 1407 y Fo(\015)826 1462 y(\015)826 1516 y(\015)877 1575 y Fl(1)p Fk(;\033)1061 1512 y Fj(\024)1701 1450 y Fq(1)p 1225 1491 999 4 v 1225 1633 a Fp(!)1282 1647 y Fn(1)1372 1505 y Fo(\022)1439 1633 y Fq(1)20 b Fj(\000)1669 1598 y Fl(4)p 1605 1613 164 4 v 1605 1665 a Fk(\033)15 b(!)1704 1673 y Fe(1)1794 1529 y Fo(\015)1794 1583 y(\015)1794 1638 y(\015)1844 1633 y Fj(R)1942 1611 y Fo(b)1921 1633 y Fp(H)2004 1529 y Fo(\015)2004 1583 y(\015)2004 1638 y(\015)2055 1697 y Fl(1)p Fk(;\033)2156 1505 y Fo(\023)2273 1425 y(X)2248 1623 y Fk(p;q)r Fn(2)p Fb(Z)2444 1407 y Fo(\015)2444 1462 y(\015)2444 1516 y(\015)2494 1512 y Fp(h)2546 1474 y Fl(\()p Fk(w)r Fl(\))2546 1534 y Fk(pq)2658 1512 y Fq(\()p Fp(\030)t(\021)s(;)g(\026\027)6 b Fq(\))p Fp(\030)3010 1474 y Fk(p)3051 1512 y Fp(\026)3106 1474 y Fk(q)3144 1407 y Fo(\015)3144 1462 y(\015)3144 1516 y(\015)3194 1575 y Fl(1)p Fk(;\033)3321 1512 y Fq(=)1568 1753 y Fo(\015)1568 1808 y(\015)1568 1862 y(\015)1617 1834 y(b)1619 1858 y Fp(h)1671 1872 y Fk(w)1728 1753 y Fo(\015)1728 1808 y(\015)1728 1862 y(\015)1778 1921 y Fl(1)p Fk(;\033)p 1225 1948 999 4 v 1225 2091 a Fp(!)1282 2105 y Fn(1)1372 1963 y Fo(\022)1439 2091 y Fq(1)20 b Fj(\000)1669 2055 y Fl(4)p 1605 2070 164 4 v 1605 2122 a Fk(\033)15 b(!)1704 2130 y Fe(1)1794 1987 y Fo(\015)1794 2041 y(\015)1794 2096 y(\015)1844 2091 y Fj(R)1942 2068 y Fo(b)1921 2091 y Fp(H)2004 1987 y Fo(\015)2004 2041 y(\015)2004 2096 y(\015)2055 2154 y Fl(1)p Fk(;\033)2156 1963 y Fo(\023)2233 1969 y Fp(;)144 2345 y Fq(for)30 b Fp(w)e Fq(=)d(1)p Fp(;)15 b Fq(2)p Fp(;)g(:)g(:)g(:)j(;)d Fq(4.)42 b(Finally)-8 b(,)29 b(using)f(\(45\),)33 b(it)d(turns)f(out)h(that)1281 2579 y Fo(\015)1281 2634 y(\015)1281 2688 y(\015)1332 2684 y Fj(R)1419 2661 y Fo(b)1409 2684 y Fq(\011)1479 2579 y Fo(\015)1479 2634 y(\015)1479 2688 y(\015)1530 2747 y Fl(1)p Fk(;\033)1657 2684 y Fj(\024)2080 2468 y Fo(\015)2080 2522 y(\015)2080 2577 y(\015)2131 2572 y Fj(R)2229 2549 y Fo(b)2208 2572 y Fp(H)2291 2468 y Fo(\015)2291 2522 y(\015)2291 2577 y(\015)2341 2635 y Fl(1)p Fk(;\033)p 1763 2663 999 4 v 1763 2806 a Fp(!)1820 2820 y Fn(1)1909 2677 y Fo(\022)1976 2806 y Fq(1)21 b Fj(\000)2207 2770 y Fl(4)p 2143 2785 164 4 v 2143 2837 a Fk(\033)14 b(!)2241 2845 y Fe(1)2331 2701 y Fo(\015)2331 2756 y(\015)2331 2810 y(\015)2382 2806 y Fj(R)2480 2783 y Fo(b)2459 2806 y Fp(H)2542 2701 y Fo(\015)2542 2756 y(\015)2542 2810 y(\015)2592 2869 y Fl(1)p Fk(;\033)2694 2677 y Fo(\023)2771 2684 y Fp(:)3865 3069 y Fa(2)-83 3472 y Fs(2.5.2)105 b(Pro)s(of)36 b(of)f(the)g(con)m(v)m(ergence)-83 3644 y Fq(T)-8 b(o)30 b(ease)g(the)g(reading)e(of)h(this)g(pro)s(of,)g (let)g(us)f(recall)h(brie\015y)f(the)h(problem)f(w)m(e)h(are)h(dealing) e(with.)39 b(Let)30 b(consider)e(a)-83 3757 y(system)1487 3847 y(_)1451 3870 y Fp(X)33 b Fq(=)25 b Fp(F)13 b Fq(\()p Fp(X)7 b Fq(\))26 b(=)f(\003)c(+)2192 3847 y Fo(b)2174 3870 y Fp(F)13 b Fq(\()p Fp(X)7 b Fq(\))1374 b(\(56\))-83 4036 y(where)28 b Fp(F)42 b Fq(is)27 b(analytic)h(on)h(a)g(domain)p 1231 3963 128 4 v 27 w Fj(D)1301 4050 y Fk(R)1388 4036 y Fq(and)f(ha)m(ving)g(at)h Fp(X)k Fq(=)25 b(0)k(a)g(saddle-fo)s(cus)e (or)i(saddle-cen)m(ter)f(equilibrium)-83 4149 y(p)s(oin)m(t)23 b(with)g(c)m(haracteristic)i(exp)s(onen)m(ts)f Fj(f\006)p Fp(\025)1492 4163 y Fl(1)1532 4149 y Fp(;)15 b Fj(\006)p Fp(\025)1696 4163 y Fl(2)1735 4149 y Fj(g)25 b Fq(equal)e(to)i Fj(f)q(\006)p Fp(\025)20 b Fj(\006)g Fq(i)p Fp(\013)p Fj(g)k Fq(and)f Fj(f)q(\006)p Fp(\025;)30 b Fj(\006)p Fq(i)p Fp(\013)p Fj(g)p Fq(,)c(resp)s(ectiv)m(ely)-8 b(.)38 b(As)24 b(it)-83 4262 y(has)j(b)s(een)g(seen)h(at)g(the)g(b)s (eginning)d(of)j(Section)f(2.1,)i(w)m(e)g(can)e(assume)h(the)g(matrix)e (\003)i(to)g(b)s(e)f(written)g(in)f(\(complex\))-83 4375 y(diagonal)32 b(form.)46 b(This)30 b(allo)m(ws)i(us)g(to)h(deal)f(with) f(b)s(oth)g(cases)j(using)d(a)h(uni\014ed)e(approac)m(h.)47 b(W)-8 b(e)34 b(also)e(recall)g(that,)-83 4488 y(again)g(in)g(Section)g (2.1,)i(w)m(e)f(in)m(tro)s(duced)e(the)i(notation)f(\003)h(to)g(denote) g(b)s(oth)f(the)g(matrix)g(\003)h(and)e(the)i(v)m(ector)h(\014eld)-83 4601 y(\003id)o(.)41 b(W)-8 b(e)31 b(will)d(only)h(use)h(explicitly)e (the)j(second)f(expression)f(in)h(cases)h(of)f(p)s(ossible)e (misunderstanding.)58 4714 y(Our)35 b(aim)h(is)g(the)h(follo)m(wing:)51 b(w)m(e)37 b(are)g(lo)s(oking)e(for)i(an)f(analytic)g(transformation)g Fp(X)43 b Fq(=)35 b(\010\()p Fp(\037)p Fq(\))h(=)f Fp(\037)24 b Fq(+)3747 4691 y Fo(b)3739 4714 y Fq(\010\()p Fp(\037)p Fq(\))-83 4827 y(and)j(analytic)g(v)m(ector)i(\014elds)d Fp(N)37 b Fq(and)1237 4804 y Fo(b)1218 4827 y Fp(B)32 b Fq(\(that)c(w)m(e)g(can)g(assume)f(to)h(b)s(e)f(of)h(the)f(form)g (\(2)p Fp(a)p Fq(\))i(and)e(\(4)p Fp(a)p Fq(\),)i(resp)s(ectiv)m(ely\)) -83 4940 y(suc)m(h)h(that)h(the)g(equalit)m(y)1540 5053 y Fp(D)s Fq(\010)15 b Fp(N)30 b Fq(+)1912 5030 y Fo(b)1893 5053 y Fp(B)f Fq(=)c Fp(F)j Fq(\()q(\010\))1461 b(\(57\))-83 5220 y(is)40 b(satis\014ed.)73 b(W)-8 b(e)43 b(sa)m(y)f(in)e(that)i (case)h(that)f(\010)f(leads)f(system)i(\(56\))h(in)m(to)e(\011)p Fi(NF)p Fq(.)h(T)-8 b(o)42 b(get)g(suc)m(h)f(transformation)-83 5332 y(and)33 b(v)m(ector)h(\014elds)e(w)m(e)i(ha)m(v)m(e)g(dev)m(elop) s(ed)f(in)f(Sections)h(2.3)h(and)f(2.4)h(the)g(follo)m(wing)d Fm(r)-5 b(e)g(curr)g(ent)37 b(scheme)d Fq(to)g(whose)-83 5445 y(con)m(v)m(ergence)f(pro)s(of)c(is)h(dev)m(oted)h(this)e (section.)41 b(Setting)30 b(the)h(follo)m(wing)e(condition)g(on)2990 5422 y Fo(b)2982 5445 y Fq(\010,)1761 5650 y Fj(P)1839 5627 y Fo(b)1831 5650 y Fq(\010)c Fj(\021)g Fq(0)p Fp(;)1683 b Fq(\(58\))1879 6018 y(17)p eop %%Page: 18 18 18 17 bop -83 191 a Fq(w)m(e)31 b(tak)m(e)h(initial)27 b(v)-5 b(alues)1267 304 y(\010)1333 266 y Fl(\(1\))1452 304 y Fq(=)25 b(id)o Fp(;)106 b(N)1838 266 y Fl(\(1\))1958 304 y Fq(=)25 b(\003)p Fp(;)2267 281 y Fo(b)2248 304 y Fp(B)2322 266 y Fl(\(1\))2441 304 y Fq(=)g(0)1189 b(\(59\))-83 457 y(and)30 b(obtain,)g(recurren)m(tly)-8 b(,)734 634 y(\010)800 596 y Fl(\()p Fk(K)5 b Fl(+1\))1039 634 y Fq(=)25 b(id)19 b(+)g Fj(R)1406 611 y Fo(b)1398 634 y Fq(\010)1464 596 y Fl(\()p Fk(K)5 b Fl(+1\))1678 634 y Fp(;)106 b(N)1892 596 y Fl(\()p Fk(K)5 b Fl(+1\))2131 634 y Fq(=)25 b(\003)20 b(+)2425 611 y Fo(b)2401 634 y Fp(N)2484 596 y Fl(\()p Fk(K)5 b Fl(+1\))2697 634 y Fp(;)2848 611 y Fo(b)2828 634 y Fp(B)2902 596 y Fl(\()p Fk(K)g Fl(+1\))3771 634 y Fq(\(60\))-83 810 y(with)656 900 y Fo(b)648 923 y Fq(\010)714 886 y Fl(\()p Fk(K)g Fl(+1\))953 923 y Fq(=)25 b Fj(O)1121 937 y Fn(\024)p Fk(K)5 b Fl(+1)1334 923 y Fp(;)1580 900 y Fo(b)1557 923 y Fp(N)1640 886 y Fl(\()p Fk(K)g Fl(+1\))1878 923 y Fq(=)25 b Fj(O)2046 937 y Fn(\024)p Fk(K)5 b Fl(+1)2260 923 y Fp(;)2501 900 y Fo(b)2482 923 y Fp(B)2556 886 y Fl(\()p Fk(K)g Fl(+1\))2794 923 y Fq(=)25 b Fj(O)2962 937 y Fn(\024)p Fk(K)5 b Fl(+1)3176 923 y Fp(;)-83 1076 y Fq(from)30 b(equations)804 1177 y Fo(n)865 1278 y Fj(L)928 1304 y Fk(N)991 1284 y Ff(\()p Fc(K)s Ff(\))1117 1177 y Fo(\020)1171 1278 y Fj(R)1256 1255 y Fo(b)1248 1278 y Fq(\010)1314 1216 y Fl(\()p Fk(K)5 b Fl(+1\))1527 1177 y Fo(\021o)1642 1341 y Fn(\024)p Fk(K)g Fl(+1)1938 1278 y Fq(=)2092 1177 y Fo(n)2153 1278 y Fj(R)2245 1177 y Fo(\020)2317 1255 y(b)2299 1278 y Fp(F)2386 1177 y Fo(\020)2440 1278 y Fq(\010)2506 1241 y Fl(\()p Fk(K)g Fl(\))2629 1177 y Fo(\021\021o)2798 1341 y Fn(\024)p Fk(K)g Fl(+1)3771 1278 y Fq(\(61\))1185 1526 y Fo(b)1161 1549 y Fp(N)1244 1512 y Fl(\()p Fk(K)g Fl(+1\))1478 1549 y Fq(+)1588 1526 y Fo(b)1568 1549 y Fp(B)1642 1512 y Fl(\()p Fk(K)g Fl(+1\))1938 1549 y Fq(=)2092 1448 y Fo(n)2153 1549 y Fj(P)2239 1448 y Fo(\020)2311 1526 y(b)2293 1549 y Fp(F)2379 1448 y Fo(\020)2433 1549 y Fq(\010)2499 1512 y Fl(\()p Fk(K)g Fl(\))2622 1448 y Fo(\021)q(\021o)2792 1612 y Fn(\024)p Fk(K)g Fl(+1)3020 1549 y Fp(:)726 b Fq(\(62\))-83 1764 y(Let)32 b(us)f(start)h(with)f(the)h(pro)s(of.)44 b(First,)31 b(let)h(us)f(consider)f(a)i(p)s(ositiv)m(e)f(constan)m(t)i(0)28 b Fp(<)f(\015)33 b(<)27 b Fq(1)32 b(\(in)e(order)i(to)g(simplify)-83 1877 y(the)i(estimates,)h(w)m(e)f(can)g(assume)g Fp(\015)i Fj(\025)30 b Fq(1)p Fp(=)p Fq(2,)36 b(whic)m(h)d(is)f(not)i(restrictiv) m(e\).)51 b(As)34 b(it)f(is)g(commonly)g(done)h(in)e(Normal)-83 1990 y(F)-8 b(orm)33 b(Theory)-8 b(,)34 b(w)m(e)f(can)g(scale)g(our)f (system)h(b)m(y)f(means)h(of)g(a)g(c)m(hange)h Fp(X)i Fq(=)29 b Fp(\013Z)7 b Fq(,)33 b(where)g Fp(\013)c(>)g Fq(0)k(is)f(a)h(constan)m(t)h(to)-83 2103 y(determine.)40 b(Th)m(us)29 b(w)m(e)i(ha)m(v)m(e)h(a)e(new)g(system)1352 2256 y(_)1323 2279 y Fp(Z)h Fq(=)25 b Fp(F)1570 2293 y Fk(\013)1620 2279 y Fq(\()p Fp(Z)7 b Fq(\))26 b(:=)f(\003)20 b(+)g Fp(\013)2138 2242 y Fn(\000)p Fl(1)2251 2256 y Fo(b)2233 2279 y Fp(F)13 b Fq(\()p Fp(\013Z)7 b Fq(\))p Fp(;)1245 b Fq(\(63\))-83 2456 y(with)33 b Fp(F)186 2470 y Fk(\013)270 2456 y Fq(analytic)g(on)p 749 2383 109 4 v 34 w Fj(D)819 2470 y Fk(r)857 2456 y Fq(,)j(where)d Fp(r)h Fq(:=)e Fp(\013)1445 2423 y Fn(\000)p Fl(1)1539 2456 y Fp(R)q Fq(.)52 b(Let)35 b(us)e(consider)g(a)h(p)s(ositiv)m(e)f (constan)m(t)j(0)c Fp(<)f(\015)36 b(<)c Fq(1.)52 b(In)33 b(order)-83 2569 y(to)c(simplify)c(the)k(estimates,)g(w)m(e)g(can)g (assume)f Fp(\015)j Fj(\025)24 b Fq(1)p Fp(=)p Fq(2,)31 b(whic)m(h)c(is)g(not)i(restrictiv)m(e.)40 b(Then,)28 b(since)3386 2546 y Fo(b)3368 2569 y Fp(F)3426 2583 y Fk(\013)3504 2569 y Fq(starts)h(with)-83 2682 y(terms)37 b(of)h(order)f(at)h(least)g(2,)i(w)m(e)e(can)g(c)m(ho)s(ose)h Fp(\013)f Fq(big)e(enough)i(\(so)g Fp(r)i Fq(small)c(enough\))h(in)g (suc)m(h)g(a)h(w)m(a)m(y)g(that)g(the)-83 2795 y(follo)m(wing)29 b(estimate)i(holds)1410 2848 y Fo(\015)1410 2902 y(\015)1410 2957 y(\015)1478 2929 y(b)1461 2952 y Fp(F)1532 2848 y Fo(\015)1532 2902 y(\015)1532 2957 y(\015)1582 3016 y Fl(1)p Fk(;r)1700 2952 y Fj(\024)1796 2824 y Fo(\022)1873 2891 y Fq(\(1)21 b Fj(\000)f Fp(\015)5 b Fq(\))p Fp(!)2209 2905 y Fn(1)p 1873 2931 411 4 v 2056 3015 a Fq(8)2294 2824 y Fo(\023)2376 2952 y Fp(r)m(:)1332 b Fq(\(64\))-83 3168 y(Calling)24 b(again)i Fp(Z)33 b Fq(and)26 b Fp(F)790 3182 y Fk(\013)866 3168 y Fq(as)h Fp(X)33 b Fq(and)26 b Fp(F)13 b Fq(,)27 b(resp)s(ectiv)m(ely)-8 b(,)27 b(w)m(e)g(can)g (assume)f(our)g(system)g(\(56\))i(to)f(b)s(e)f(analytic)f(on)p 3825 3095 109 4 v 27 w Fj(D)3895 3182 y Fk(r)-83 3280 y Fq(and)j(satisfying)g(\(64\).)42 b(W)-8 b(e)30 b(are)f(going)g(to)g (pro)m(v)m(e)h(that)f(the)g(limit)e(v)m(ector)k(\014elds)c(\010,)i Fp(N)39 b Fq(and)3117 3257 y Fo(b)3098 3280 y Fp(B)33 b Fq(obtained)28 b(from)h(this)-83 3393 y(recurren)m(t)h(sc)m(heme)h (satisfy)f(\(57\))i(and)e(are)h(analytic)f(on)p 1897 3320 149 4 v 30 w Fj(D)1967 3407 y Fk(\015)t(r)2075 3393 y Fq(\(and)h(therefore,)g(rev)m(ersing)e(the)i(scaling,)f(on)p 3686 3320 169 4 v 30 w Fj(D)3756 3407 y Fk(\015)t(R)3854 3393 y Fq(\).)58 3506 y(W)-8 b(e)28 b(will)d(itemize)i(the)g(pro)s(of)f (in)g(sev)m(eral)i(parts:)38 b(the)28 b(\014rst)e(one)h(will)e(pro)m (vide)h(some)h(estimates)h(on)f(the)g(appro)m(x-)-83 3619 y(imations)i(pro)m(vided)g(b)m(y)h(the)h(recurren)m(t)f(sc)m (heme;)i(in)d(the)h(second)h(one,)g(their)e(con)m(v)m(ergence)k(will)27 b(b)s(e)j(deriv)m(ed.)-3 3784 y(\()p Fp(i)p Fq(\))46 b(Consider)36 b(system)i(\(56\))i(ha)m(ving)d Fp(F)51 b Fq(analytic)38 b(on)g(a)g(domain)p 2351 3711 109 4 v 37 w Fj(D)2421 3798 y Fk(r)2497 3784 y Fq(and)f(satisfying)f(the)j (assumption)d(\(64\).)144 3897 y(Apply)29 b(on)m(to)j(it)d(the)i (recurren)m(t)f(sc)m(heme)h(\(58\){\(62\))j(and)c(consider)f(the)i (sequences)980 3995 y Fo(\032)1048 4019 y(\015)1048 4073 y(\015)1048 4128 y(\015)1098 4123 y Fq(\010)1164 4086 y Fl(\()p Fk(K)5 b Fl(\))1287 4019 y Fo(\015)1287 4073 y(\015)1287 4128 y(\015)1338 4187 y Fl(1)p Fk(;s)1430 3995 y Fo(\033)1498 4214 y Fk(K)1581 4123 y Fp(;)1713 3995 y Fo(\032)1781 4019 y(\015)1781 4073 y(\015)1781 4128 y(\015)1831 4123 y Fp(N)1914 4086 y Fl(\()p Fk(K)g Fl(\))2037 4019 y Fo(\015)2037 4073 y(\015)2037 4128 y(\015)2088 4187 y Fl(1)p Fk(;s)2180 3995 y Fo(\033)2248 4214 y Fk(K)2331 4123 y Fp(;)2463 3995 y Fo(\032)2531 4019 y(\015)2531 4073 y(\015)2531 4128 y(\015)2600 4100 y(b)2581 4123 y Fp(B)2655 4086 y Fl(\()p Fk(K)g Fl(\))2778 4019 y Fo(\015)2778 4073 y(\015)2778 4128 y(\015)2829 4187 y Fl(1)p Fk(;s)2920 3995 y Fo(\033)2988 4214 y Fk(K)3072 4123 y Fp(;)674 b Fq(\(65\))144 4349 y(de\014ned)29 b(for)i Fp(K)g Fj(\025)25 b Fq(1)31 b(and)f(b)s(eing)f Fp(s)c Fq(=)g Fp(\015)5 b(r)s Fq(.)40 b(Then,)30 b(the)g(follo)m(wing)f(prop)s (erties)g(are)i(satis\014ed.)180 4528 y(\()p Fp(a)p Fq(\))46 b(They)30 b(increase)g(monotonically)-8 b(,)30 b(that)h(is,)959 4622 y Fo(\015)959 4677 y(\015)959 4731 y(\015)1010 4727 y Fq(\010)1076 4689 y Fl(\()p Fk(K)5 b Fl(+1\))1289 4622 y Fo(\015)1289 4677 y(\015)1289 4731 y(\015)1340 4790 y Fl(1)p Fk(;s)1514 4727 y Fj(\025)1668 4622 y Fo(\015)1668 4677 y(\015)1668 4731 y(\015)1718 4727 y Fq(\010)1784 4689 y Fl(\()p Fk(K)g Fl(\))1907 4622 y Fo(\015)1907 4677 y(\015)1907 4731 y(\015)1958 4790 y Fl(1)p Fk(;s)2065 4727 y Fp(;)942 4836 y Fo(\015)942 4891 y(\015)942 4945 y(\015)993 4941 y Fp(N)1076 4903 y Fl(\()p Fk(K)g Fl(+1\))1289 4836 y Fo(\015)1289 4891 y(\015)1289 4945 y(\015)1340 5004 y Fl(1)p Fk(;s)1514 4941 y Fj(\025)1668 4836 y Fo(\015)1668 4891 y(\015)1668 4945 y(\015)1718 4941 y Fp(N)1801 4903 y Fl(\()p Fk(K)g Fl(\))1925 4836 y Fo(\015)1925 4891 y(\015)1925 4945 y(\015)1975 5004 y Fl(1)p Fk(;s)2082 4941 y Fp(;)2304 4836 y Fo(\015)2304 4891 y(\015)2304 4945 y(\015)2374 4918 y(b)2355 4941 y Fp(B)2429 4903 y Fl(\()p Fk(K)g Fl(+1\))2642 4836 y Fo(\015)2642 4891 y(\015)2642 4945 y(\015)2692 5004 y Fl(1)p Fk(;s)2809 4941 y Fj(\025)2905 4836 y Fo(\015)2905 4891 y(\015)2905 4945 y(\015)2975 4918 y(b)2956 4941 y Fp(B)3030 4903 y Fl(\()p Fk(K)g Fl(\))3152 4836 y Fo(\015)3152 4891 y(\015)3152 4945 y(\015)3203 5004 y Fl(1)p Fk(;s)3310 4941 y Fp(:)189 5169 y Fq(\()p Fp(b)p Fq(\))46 b(All)29 b(these)i(sequences)g(are)f(uniformly)e(upp)s(er-b)s(ounded.)37 b(Precisely)-8 b(,)30 b(for)g(all)f Fp(K)j Fj(\025)25 b Fq(1)31 b(w)m(e)g(ha)m(v)m(e)g(that)1865 5264 y Fo(\015)1865 5318 y(\015)1865 5373 y(\015)1916 5368 y Fq(\010)1982 5331 y Fl(\()p Fk(K)5 b Fl(\))2105 5264 y Fo(\015)2105 5318 y(\015)2105 5373 y(\015)2155 5432 y Fl(1)p Fk(;s)2272 5368 y Fj(\024)25 b Fp(r)1362 b Fq(\(66\))344 5580 y(and)30 b(that)1493 5588 y Fo(\015)1493 5643 y(\015)1493 5697 y(\015)1544 5693 y Fp(N)1627 5655 y Fl(\()p Fk(K)5 b Fl(\))1750 5588 y Fo(\015)1750 5643 y(\015)1750 5697 y(\015)1800 5756 y Fl(1)p Fk(;s)1907 5693 y Fp(;)1978 5588 y Fo(\015)1978 5643 y(\015)1978 5697 y(\015)2047 5670 y(b)2028 5693 y Fp(B)2102 5655 y Fl(\()p Fk(K)g Fl(\))2225 5588 y Fo(\015)2225 5643 y(\015)2225 5697 y(\015)2276 5756 y Fl(1)p Fk(;s)2393 5693 y Fj(\024)25 b(k)p Fp(F)13 b Fj(k)2651 5720 y Fl(1)p Fk(;r)2759 5693 y Fp(:)987 b Fq(\(67\))1879 6018 y(18)p eop %%Page: 19 19 19 18 bop 144 191 a Fq(Let)31 b(us)f(pro)m(v)m(e)h(these)g(assertions.) 180 377 y(\()p Fp(a)p Fq(\))46 b(F)-8 b(rom)33 b(Lemma)g(7,)g (expressions)e(\(42\),)j(\(43\))g(and)e(taking)g(in)m(to)h(accoun)m(t)g (Lemma)g(8\()p Fp(iv)s Fq(\),)i(it)c(turns)g(out)344 490 y(that)579 585 y Fo(\015)579 640 y(\015)579 694 y(\015)630 690 y Fq(\010)696 652 y Fl(\()p Fk(K)5 b Fl(+1\))909 585 y Fo(\015)909 640 y(\015)909 694 y(\015)959 753 y Fl(1)p Fk(;s)1076 690 y Fq(=)1172 585 y Fo(\015)1172 640 y(\015)1172 694 y(\015)1223 690 y Fq(\010)1289 652 y Fl(\()p Fk(K)g Fl(\))1432 690 y Fq(+)20 b Fj(R)p Fq(\001)1684 667 y Fo(b)1676 690 y Fq(\010)1741 628 y Fl(\()p Fk(K)5 b Fl(\))1864 585 y Fo(\015)1864 640 y(\015)1864 694 y(\015)1915 753 y Fl(1)p Fk(;s)2032 690 y Fq(=)2128 585 y Fo(\015)2128 640 y(\015)2128 694 y(\015)2178 690 y Fq(\010)2244 652 y Fl(\()p Fk(K)g Fl(\))2367 585 y Fo(\015)2367 640 y(\015)2367 694 y(\015)2418 753 y Fl(1)p Fk(;s)2529 690 y Fq(+)2620 585 y Fo(\015)2620 640 y(\015)2620 694 y(\015)2671 690 y Fj(R)p Fq(\001\010)2889 652 y Fl(\()p Fk(K)g Fl(\))3013 585 y Fo(\015)3013 640 y(\015)3013 694 y(\015)3063 753 y Fl(1)p Fk(;s)3180 690 y Fj(\025)3276 585 y Fo(\015)3276 640 y(\015)3276 694 y(\015)3326 690 y Fq(\010)3392 652 y Fl(\()p Fk(K)g Fl(\))3515 585 y Fo(\015)3515 640 y(\015)3515 694 y(\015)3566 753 y Fl(1)p Fk(;s)3673 690 y Fp(:)344 966 y Fq(The)30 b(result)f(for)924 888 y Fo(\015)924 943 y(\015)974 966 y Fp(N)1057 933 y Fl(\()p Fk(K)5 b Fl(+1\))1270 888 y Fo(\015)1270 943 y(\015)1321 1002 y Fl(1)p Fk(;s)1443 966 y Fq(and)1620 861 y Fo(\015)1620 916 y(\015)1620 970 y(\015)1689 943 y(b)1670 966 y Fp(B)1744 933 y Fl(\()p Fk(K)g Fl(+1\))1957 861 y Fo(\015)1957 916 y(\015)1957 970 y(\015)2008 1029 y Fl(1)p Fk(;s)2130 966 y Fq(can)30 b(b)s(e)g(deriv)m(ed)f(analogously)-8 b(.)189 1150 y(\()p Fp(b)p Fq(\))46 b(T)-8 b(o)31 b(see)g(it)f(w)m(e)h (pro)s(ceed)f(inductiv)m(ely)-8 b(.)39 b(Th)m(us,)29 b(for)i Fp(K)h Fq(=)24 b(1)31 b(equation)f(\(61\))i(b)s(ecomes)1304 1279 y Fo(n)1365 1380 y Fj(L)1428 1406 y Fk(N)1491 1386 y Ff(\(1\))1592 1279 y Fo(\020)1647 1380 y Fj(R)1732 1357 y Fo(b)1724 1380 y Fq(\010)1789 1318 y Fl(\(2\))1884 1279 y Fo(\021o)1999 1444 y Fn(\024)p Fl(2)2118 1380 y Fq(=)2214 1279 y Fo(n)2275 1380 y Fj(R)2367 1279 y Fo(\020)2439 1357 y(b)2421 1380 y Fp(F)13 b Fq(\(\010)2593 1343 y Fl(\(1\))2688 1380 y Fq(\))2723 1279 y Fo(\021o)2838 1444 y Fn(\024)p Fl(2)2947 1380 y Fp(:)344 1639 y Fq(Ha)m(ving)40 b(in)f(mind)f(that)i Fp(N)1311 1606 y Fl(\(1\))1447 1639 y Fq(=)h(\003)f(\(so)1842 1616 y Fo(b)1818 1639 y Fp(N)1901 1606 y Fl(\(1\))2037 1639 y Fq(=)h(0\),)i(\010)2363 1606 y Fl(\(1\))2498 1639 y Fq(=)e(id)e(and)g(the)h(de\014nition)e(\(21\))j (of)f(the)344 1751 y(op)s(erator)31 b Fj(L)p Fq(,)f(this)f(equation)i (is)e(equiv)-5 b(alen)m(t)30 b(to)1492 1977 y Fp(D)1585 1876 y Fo(\020)1639 1977 y Fj(R)1724 1954 y Fo(b)1716 1977 y Fq(\010)1782 1914 y Fl(\(2\))1876 1876 y Fo(\021)1960 1977 y Fq(\003)21 b Fj(\000)f Fq(\003)15 b Fj(R)2298 1954 y Fo(b)2290 1977 y Fq(\010)2356 1914 y Fl(\(2\))2475 1977 y Fq(=)25 b Fj(R)p Fp(F)2706 1995 y Fl([2])344 2194 y Fq(and)30 b(to)1783 2206 y Fo(h)1826 2307 y Fq(\003)p Fp(;)15 b Fj(R)2014 2284 y Fo(b)2006 2307 y Fq(\010)2073 2245 y Fl(\(2\))2167 2206 y Fo(i)2235 2307 y Fq(=)25 b Fp(F)2389 2325 y Fl([2])2468 2307 y Fp(;)344 2494 y Fq(where)38 b([)p Fp(H)r(;)15 b(G)p Fq(])41 b(=)d(\()p Fp(D)s(G)p Fq(\))p Fp(H)c Fj(\000)26 b Fq(\()p Fp(D)s(H)7 b Fq(\))p Fp(G)39 b Fq(stands)f(for)h(the)f(Lie)g(brac)m(k)m(et)j(of)e (the)f(v)m(ector)j(\014elds)c Fp(H)45 b Fq(and)344 2623 y Fp(G)p Fq(.)53 b(No)m(w,)37 b(from)d(Prop)s(osition)e(11\()p Fp(ii)p Fq(\),)38 b(taking)c(in)m(to)h(accoun)m(t)g(that)f Fo(b)-50 b Fp(a)2789 2575 y Fl(\(1\))2789 2649 y(1)2916 2623 y Fq(=)30 b Fo(b)-50 b Fp(a)3066 2575 y Fl(\(1\))3066 2649 y(2)3193 2623 y Fq(=)31 b(0)k(\(the)g(functions)344 2746 y(app)s(earing)29 b(in)896 2723 y Fo(b)872 2746 y Fp(N)955 2713 y Fl(\(1\))1050 2746 y Fq(\))h(and)g(using)f(estimate)i (\(55\))h(it)e(follo)m(ws)f(that)1688 2924 y Fo(\015)1688 2979 y(\015)1688 3033 y(\015)1739 3029 y Fj(R)1824 3006 y Fo(b)1816 3029 y Fq(\010)1881 2966 y Fl(\(2\))1976 2924 y Fo(\015)1976 2979 y(\015)1976 3033 y(\015)2026 3092 y Fl(1)p Fk(;s)2143 3029 y Fj(\024)2249 2864 y Fo(\015)2249 2918 y(\015)2300 2941 y Fp(F)2358 2960 y Fl([2])2437 2864 y Fo(\015)2437 2918 y(\015)2487 2980 y Fl(1)p Fk(;s)p 2249 3008 330 4 v 2348 3091 a Fp(!)2405 3105 y Fn(1)344 3263 y Fq(and,)h(in)f(particular,)1708 3401 y Fo(\015)1708 3455 y(\015)1708 3510 y(\015)1758 3505 y Fj(R)1843 3482 y Fo(b)1835 3505 y Fq(\010)1901 3443 y Fl(\(2\))1996 3401 y Fo(\015)1996 3455 y(\015)1996 3510 y(\015)2046 3569 y Fl(1)p Fk(;s)2163 3505 y Fj(\024)2269 3289 y Fo(\015)2269 3344 y(\015)2269 3398 y(\015)2337 3371 y(b)2319 3394 y Fp(F)2390 3289 y Fo(\015)2390 3344 y(\015)2390 3398 y(\015)2441 3457 y Fl(1)p Fk(;r)p 2269 3485 265 4 v 2336 3568 a Fp(!)2393 3582 y Fn(1)2544 3505 y Fp(:)1202 b Fq(\(68\))344 3730 y(Th)m(us,)33 b(applying)f(Lemma)h(8\()p Fp(iv)s Fq(\),)j(the)d(assumption)f Fj(P)2280 3707 y Fo(b)2272 3730 y Fq(\010)e(=)g(0)k(and)f(the)g(estimate)h(\(64\),)i (one)d(obtains)344 3843 y(that)957 3963 y Fo(\015)957 4018 y(\015)957 4072 y(\015)1007 4068 y Fq(\010)1073 4030 y Fl(\(2\))1167 3963 y Fo(\015)1167 4018 y(\015)1167 4072 y(\015)1218 4131 y Fl(1)p Fk(;s)1335 4068 y Fj(\024)25 b Fp(s)19 b Fq(+)1594 3852 y Fo(\015)1594 3906 y(\015)1594 3961 y(\015)1663 3933 y(b)1645 3956 y Fp(F)1716 3852 y Fo(\015)1716 3906 y(\015)1716 3961 y(\015)1766 4019 y Fl(1)p Fk(;r)p 1594 4047 V 1661 4130 a Fp(!)1718 4144 y Fn(1)1894 4068 y Fj(\024)25 b Fp(\015)5 b(r)23 b Fq(+)2207 4006 y(1)e Fj(\000)f Fp(\015)p 2207 4047 209 4 v 2289 4130 a Fq(8)2441 4068 y Fp(r)28 b Fj(\024)d Fp(\015)5 b(r)22 b Fq(+)e(\(1)h Fj(\000)f Fp(\015)5 b Fq(\))p Fp(r)29 b Fq(=)c Fp(r)m(:)344 4293 y Fq(Concerning)k(v)m(ector)j(\014elds)d Fp(N)1413 4260 y Fl(\(2\))1538 4293 y Fq(and)1734 4270 y Fo(b)1714 4293 y Fp(B)1788 4260 y Fl(\(2\))1913 4293 y Fq(w)m(e)h(ha)m(v)m(e)i(that)1386 4492 y Fp(N)1469 4455 y Fl(\(2\))1588 4492 y Fq(=)25 b Fp(N)1767 4455 y Fl(\(1\))1887 4492 y Fq(=)g(\003)p Fp(;)2287 4469 y Fo(b)2268 4492 y Fp(B)2342 4455 y Fl(\(2\))2461 4492 y Fq(=)2576 4469 y Fo(b)2557 4492 y Fp(B)2631 4455 y Fl(\(1\))2750 4492 y Fq(=)g(0)344 4692 y(and,)h(therefore,)i(estimate)e (\(67\))h(satis\014ed.)38 b(By)26 b(induction)e(h)m(yp)s(othesis,)h (let)g(us)g(no)m(w)h(assume)f(that)h(the)344 4805 y(b)s(ounds)63 b(\(66\),)34 b(\(67\))g(hold.)46 b(W)-8 b(e)33 b(are)g(going)g(to)g (sho)m(w)f(that)h(they)g(are)f(also)h(true)f(for)g Fp(K)c Fq(+)22 b(1.)47 b(In)31 b(fact,)344 4918 y(equation)f(\(61\))i(is)e(of) g(t)m(yp)s(e)h Fj(L)1370 4932 y Fk(N)1437 4918 y Fq(\()p Fj(R)1559 4895 y Fo(b)1549 4918 y Fq(\011\))26 b(=)e Fj(R)1875 4895 y Fo(b)1853 4918 y Fp(H)38 b Fq(pro)m(vided)29 b(w)m(e)i(tak)m(e)992 5148 y Fp(N)k Fq(=)25 b Fp(N)1279 5111 y Fl(\()p Fk(K)5 b Fl(\))1403 5148 y Fp(;)197 b Fj(R)1712 5125 y Fo(b)1702 5148 y Fq(\011)25 b(=)g Fj(R)1979 5125 y Fo(b)1971 5148 y Fq(\010)2036 5086 y Fl(\()p Fk(K)5 b Fl(+1\))2250 5148 y Fp(;)197 b Fj(R)2570 5125 y Fo(b)2549 5148 y Fp(H)32 b Fq(=)25 b Fj(R)2845 5047 y Fo(\020)2918 5125 y(b)2900 5148 y Fp(F)13 b Fq(\(\010)3072 5111 y Fl(\()p Fk(K)5 b Fl(\))3195 5148 y Fq(\))3230 5047 y Fo(\021)344 5366 y Fq(and)36 b(consider)g(just)g(terms)h(up)e(to)j (order)e Fp(K)31 b Fq(+)24 b(1.)61 b(Setting)36 b Fp(\033)j Fq(=)d Fp(s)g Fq(and)g(taking)h(in)m(to)f(accoun)m(t)j(esti-)344 5478 y(mate)32 b(\(64\),)g(the)e(induction)e(h)m(yp)s(othesis)h(and)h (Lemma)h(8\()p Fp(i;)15 b(ii)p Fq(\))32 b(it)e(follo)m(ws)g(that)964 5624 y Fo(\015)964 5678 y(\015)964 5733 y(\015)1015 5728 y Fj(R)1113 5705 y Fo(b)1092 5728 y Fp(H)1175 5624 y Fo(\015)1175 5678 y(\015)1175 5733 y(\015)1225 5791 y Fl(1)p Fk(;s)1342 5728 y Fq(=)1438 5624 y Fo(\015)1438 5678 y(\015)1438 5733 y(\015)1489 5728 y Fj(R)1581 5627 y Fo(\020)1653 5705 y(b)1635 5728 y Fp(F)13 b Fq(\(\010)1807 5691 y Fl(\()p Fk(K)5 b Fl(\))1930 5728 y Fq(\))1965 5627 y Fo(\021)2020 5624 y(\015)2020 5678 y(\015)2020 5733 y(\015)2070 5791 y Fl(1)p Fk(;s)2187 5728 y Fj(\024)2283 5624 y Fo(\015)2283 5678 y(\015)2283 5733 y(\015)2352 5705 y(b)2334 5728 y Fp(F)2405 5624 y Fo(\015)2405 5678 y(\015)2405 5733 y(\015)2455 5791 y Fl(1)p Fk(;r)2574 5728 y Fj(\024)2670 5600 y Fo(\022)2746 5667 y Fq(\(1)21 b Fj(\000)f Fp(\015)5 b Fq(\))p Fp(!)3082 5681 y Fn(1)p 2746 5707 411 4 v 2929 5791 a Fq(8)3167 5600 y Fo(\023)3249 5728 y Fp(r)m(:)1879 6018 y Fq(19)p eop %%Page: 20 20 20 19 bop 344 191 a Fq(Using)30 b(that)h(1)p Fp(=)p Fq(2)26 b Fj(\024)f Fp(\015)31 b(<)25 b Fq(1)30 b(and)g(that)h Fp(s)25 b Fq(=)g Fp(\015)5 b(r)s Fq(,)30 b(this)f(estimate)j(reads)1056 341 y Fo(\015)1056 395 y(\015)1056 450 y(\015)1106 445 y Fj(R)1204 422 y Fo(b)1183 445 y Fp(H)1266 341 y Fo(\015)1266 395 y(\015)1266 450 y(\015)1317 508 y Fl(1)p Fk(;s)1433 445 y Fj(\024)1529 317 y Fo(\022)1606 384 y Fq(\(1)21 b Fj(\000)f Fp(\015)5 b Fq(\))p Fp(!)1942 398 y Fn(1)p 1606 424 411 4 v 1789 507 a Fq(8)2027 317 y Fo(\023)2109 445 y Fp(r)28 b Fj(\024)2284 384 y Fp(\015)p 2284 424 53 4 v 2287 507 a Fq(8)2361 445 y Fp(!)2418 459 y Fn(1)2492 445 y Fp(r)g Fq(=)2667 384 y Fp(s)15 b(!)2782 398 y Fn(1)p 2667 424 190 4 v 2739 507 a Fq(8)2891 445 y Fp(<)2997 384 y(s)g(!)3112 398 y Fn(1)p 2997 424 V 3069 507 a Fq(4)3196 445 y Fp(;)344 699 y Fq(whic)m(h)29 b(is)h(assumption)f(\(48\).)42 b(Applying)28 b(Prop)s(osition)g(11\()p Fp(ii)p Fq(\))33 b(and)d(that)739 942 y(1)21 b Fj(\000)1004 880 y Fq(4)p 906 921 243 4 v 906 1004 a Fp(\015)5 b(r)18 b(!)1074 1018 y Fn(1)1173 837 y Fo(\015)1173 892 y(\015)1173 946 y(\015)1224 942 y Fj(R)1316 841 y Fo(\020)1388 919 y(b)1370 942 y Fp(F)13 b Fq(\(\010)1542 904 y Fl(\()p Fk(K)5 b Fl(\))1666 942 y Fq(\))1701 841 y Fo(\021)1755 837 y(\015)1755 892 y(\015)1755 946 y(\015)1806 1005 y Fl(1)p Fk(;\015)t(r)2022 942 y Fq(=)83 b(1)20 b Fj(\000)2440 880 y Fq(4)p 2342 921 V 2342 1004 a Fp(\015)5 b(r)18 b(!)2510 1018 y Fn(1)2609 837 y Fo(\015)2609 892 y(\015)2609 946 y(\015)2660 942 y Fj(R)2758 919 y Fo(b)2737 942 y Fp(H)2820 837 y Fo(\015)2820 892 y(\015)2820 946 y(\015)2870 1005 y Fl(1)p Fk(;s)2987 942 y Fj(\025)2176 1187 y Fq(1)i Fj(\000)2332 1059 y Fo(\022)2481 1125 y Fq(4)p 2409 1166 190 4 v 2409 1249 a Fp(s)15 b(!)2524 1263 y Fn(1)2608 1059 y Fo(\023)2690 1086 y(\020)2754 1125 y Fp(s)g(!)2869 1139 y Fn(1)p 2754 1166 V 2826 1249 a Fq(8)2953 1086 y Fo(\021)3033 1187 y Fq(=)25 b(1)c Fj(\000)3295 1125 y Fq(1)p 3295 1166 46 4 v 3295 1249 a(2)3376 1187 y(=)3482 1125 y(1)p 3482 1166 V 3482 1249 a(2)344 1436 y(w)m(e)31 b(obtain)938 1647 y Fo(\015)938 1702 y(\015)938 1756 y(\015)989 1752 y Fj(R)1074 1729 y Fo(b)1066 1752 y Fq(\010)1131 1689 y Fl(\()p Fk(K)5 b Fl(+1\))1345 1647 y Fo(\015)1345 1702 y(\015)1345 1756 y(\015)1395 1815 y Fl(1)p Fk(;\015)t(r)1612 1752 y Fj(\024)2109 1536 y Fo(\015)2109 1590 y(\015)2109 1645 y(\015)2159 1640 y Fj(R)2251 1539 y Fo(\020)2323 1617 y(b)2306 1640 y Fp(F)13 b Fq(\(\010)2478 1607 y Fl(\()p Fk(K)5 b Fl(\))2601 1640 y Fq(\))2636 1539 y Fo(\021)2691 1536 y(\015)2691 1590 y(\015)2691 1645 y(\015)2741 1703 y Fl(1)p Fk(;\015)t(r)p 1775 1731 1433 4 v 1775 1874 a Fp(!)1832 1888 y Fn(1)1922 1745 y Fo(\022)1989 1874 y Fq(1)20 b Fj(\000)2235 1838 y Fl(4)p 2155 1853 195 4 v 2155 1905 a Fk(\015)t(r)14 b(!)2285 1913 y Fe(1)2375 1769 y Fo(\015)2375 1824 y(\015)2375 1878 y(\015)2426 1874 y Fj(R)2518 1773 y Fo(\020)2590 1851 y(b)2572 1874 y Fp(F)f Fq(\(\010)2744 1847 y Fl(\()p Fk(K)5 b Fl(\))2867 1874 y Fq(\))2902 1773 y Fo(\021)2957 1769 y(\015)2957 1824 y(\015)2957 1878 y(\015)3007 1937 y Fl(1)p Fk(;\015)t(r)3141 1745 y Fo(\023)3243 1752 y Fj(\024)1785 2043 y Fl(\(1)p Fn(\000)p Fk(\015)t Fl(\))p Fk(!)2013 2051 y Fe(1)p 1785 2067 295 4 v 1915 2119 a Fl(8)2105 2088 y Fp(r)p 1775 2134 373 4 v 1851 2217 a(!)1908 2231 y Fn(1)1982 2217 y Fp(=)p Fq(2)2183 2155 y(=)2289 2093 y(\(1)21 b Fj(\000)f Fp(\015)5 b Fq(\))p Fp(r)p 2289 2134 323 4 v 2428 2217 a Fq(4)2622 2155 y Fp(:)344 2407 y Fq(Finally)-8 b(,)29 b(from)h(Lemma)h(8\()p Fp(iv)s Fq(\))h(one)e(obtains)g(that)906 2537 y Fo(\015)906 2592 y(\015)906 2646 y(\015)956 2642 y Fq(\010)1022 2604 y Fl(\()p Fk(K)5 b Fl(+1\))1235 2537 y Fo(\015)1235 2592 y(\015)1235 2646 y(\015)1286 2705 y Fl(1)p Fk(;s)1461 2642 y Fq(=)1614 2537 y Fo(\015)1614 2592 y(\015)1614 2646 y(\015)1665 2642 y Fq(\010)1731 2604 y Fl(\()p Fk(K)g Fl(+1\))1944 2537 y Fo(\015)1944 2592 y(\015)1944 2646 y(\015)1994 2705 y Fl(1)p Fk(;\015)t(r)2153 2642 y Fq(=)25 b Fj(k)p Fq(id)o Fj(k)2415 2669 y Fl(1)p Fk(;\015)t(r)2569 2642 y Fq(+)2660 2537 y Fo(\015)2660 2592 y(\015)2660 2646 y(\015)2710 2642 y Fj(R)2795 2619 y Fo(b)2787 2642 y Fq(\010)2853 2580 y Fl(\()p Fk(K)5 b Fl(+1\))3066 2537 y Fo(\015)3066 2592 y(\015)3066 2646 y(\015)3117 2705 y Fl(1)p Fk(;\015)t(r)3275 2642 y Fj(\024)1614 2881 y Fp(\015)g(r)23 b Fq(+)1831 2819 y(\(1)e Fj(\000)f Fp(\015)5 b Fq(\))p Fp(r)p 1831 2860 V 1970 2943 a Fq(4)2189 2881 y Fj(\024)25 b Fp(\015)5 b(r)23 b Fq(+)d(\(1)h Fj(\000)f Fp(\015)5 b Fq(\))p Fp(r)28 b Fq(=)d Fp(r)m(:)344 3132 y Fq(Concerning)34 b Fp(N)912 3099 y Fl(\()p Fk(K)5 b Fl(+1\))1160 3132 y Fq(and)1360 3109 y Fo(b)1341 3132 y Fp(B)1415 3099 y Fl(\()p Fk(K)g Fl(+1\))1628 3132 y Fq(,)36 b(ha)m(ving)f(in)e(mind)g(the)i(induction)e(h)m(yp)s(othesis)g Fj(k)p Fq(\010)3467 3099 y Fl(\()p Fk(K)5 b Fl(\))3591 3132 y Fj(k)3636 3146 y Fl(1)p Fk(;s)3760 3132 y Fj(\024)33 b Fp(r)s Fq(,)344 3244 y(equation)d(\(62\))i(and)e(section)h(2.3.2,)h (one)f(obtains)e(that)941 3367 y Fo(\015)941 3421 y(\015)941 3476 y(\015)1015 3448 y(b)992 3471 y Fp(N)1075 3434 y Fl(\()p Fk(K)5 b Fl(+1\))1288 3367 y Fo(\015)1288 3421 y(\015)1288 3476 y(\015)1338 3535 y Fl(1)p Fk(;s)1455 3471 y Fj(\024)1551 3367 y Fo(\015)1551 3421 y(\015)1551 3476 y(\015)1620 3448 y(b)1602 3471 y Fp(F)1673 3367 y Fo(\015)1673 3421 y(\015)1673 3476 y(\015)1723 3535 y Fl(1)p Fk(;r)1831 3471 y Fp(;)2054 3367 y Fo(\015)2054 3421 y(\015)2054 3476 y(\015)2123 3448 y(b)2104 3471 y Fp(B)2178 3434 y Fl(\()p Fk(K)g Fl(+1\))2391 3367 y Fo(\015)2391 3421 y(\015)2391 3476 y(\015)2442 3535 y Fl(1)p Fk(;s)2559 3471 y Fj(\024)2654 3367 y Fo(\015)2654 3421 y(\015)2654 3476 y(\015)2723 3448 y(b)2705 3471 y Fp(F)2776 3367 y Fo(\015)2776 3421 y(\015)2776 3476 y(\015)2827 3535 y Fl(1)p Fk(;r)2945 3471 y Fj(\024)25 b(k)p Fp(F)13 b Fj(k)3203 3498 y Fl(1)p Fk(;r)3311 3471 y Fp(:)344 3736 y Fq(Since)30 b Fp(N)665 3703 y Fl(\()p Fk(K)5 b Fl(+1\))903 3736 y Fq(=)25 b(\003)c(+)1197 3713 y Fo(b)1173 3736 y Fp(N)1256 3703 y Fl(\()p Fk(K)5 b Fl(+1\))1500 3736 y Fq(and)30 b Fp(F)38 b Fq(=)25 b(\003)c(+)2061 3713 y Fo(b)2043 3736 y Fp(F)44 b Fq(it)30 b(turns)f(out)h(that)1706 3835 y Fo(\015)1706 3890 y(\015)1706 3945 y(\015)1756 3940 y Fp(N)1839 3903 y Fl(\()p Fk(K)5 b Fl(+1\))2053 3835 y Fo(\015)2053 3890 y(\015)2053 3945 y(\015)2103 4003 y Fl(1)p Fk(;s)2220 3940 y Fj(\024)25 b(k)q Fp(F)13 b Fj(k)2478 3967 y Fl(1)p Fk(;r)344 4184 y Fq(whic)m(h)29 b(concludes)h(the)h(pro)s(of)e(of)i(\()p Fp(b)p Fq(\).)-35 4372 y(\()p Fp(ii)p Fq(\))47 b(A)m(t)32 b(\()p Fp(i)p Fq(\))f(it)f(has)g(b)s(een)g(pro)m(v)m(ed)g(that)h(the)g(sequences)980 4498 y Fo(\032)1048 4521 y(\015)1048 4576 y(\015)1048 4630 y(\015)1098 4626 y Fq(\010)1164 4588 y Fl(\()p Fk(K)5 b Fl(\))1287 4521 y Fo(\015)1287 4576 y(\015)1287 4630 y(\015)1338 4689 y Fl(1)p Fk(;s)1430 4498 y Fo(\033)1498 4716 y Fk(K)1581 4626 y Fp(;)1713 4498 y Fo(\032)1781 4521 y(\015)1781 4576 y(\015)1781 4630 y(\015)1831 4626 y Fp(N)1914 4588 y Fl(\()p Fk(K)g Fl(\))2037 4521 y Fo(\015)2037 4576 y(\015)2037 4630 y(\015)2088 4689 y Fl(1)p Fk(;s)2180 4498 y Fo(\033)2248 4716 y Fk(K)2331 4626 y Fp(;)2463 4498 y Fo(\032)2531 4521 y(\015)2531 4576 y(\015)2531 4630 y(\015)2600 4603 y(b)2581 4626 y Fp(B)2655 4588 y Fl(\()p Fk(K)g Fl(\))2778 4521 y Fo(\015)2778 4576 y(\015)2778 4630 y(\015)2829 4689 y Fl(1)p Fk(;s)2920 4498 y Fo(\033)2988 4716 y Fk(K)3072 4626 y Fp(;)144 4879 y Fq(increase)23 b(monotonically)f(and)h(are)h(uniformly)c(upp)s (er-b)s(ounded.)34 b(Applying)21 b(on)m(to)j(them)f(the)h (Ascoli-Arzel\022)-45 b(a)144 4992 y(theorem)31 b(it)f(follo)m(ws)f (that)i(they)g(admit)f(con)m(v)m(ergen)m(t)i(subsequences)952 5113 y Fo(\032)1020 5137 y(\015)1020 5191 y(\015)1020 5246 y(\015)1070 5241 y Fq(\010)1136 5204 y Fl(\()p Fk(K)1223 5215 y Fc(J)1266 5204 y Fl(\))1297 5137 y Fo(\015)1297 5191 y(\015)1297 5246 y(\015)1348 5304 y Fl(1)p Fk(;s)1440 5113 y Fo(\033)1508 5332 y Fk(J)1572 5241 y Fp(;)1703 5113 y Fo(\032)1771 5137 y(\015)1771 5191 y(\015)1771 5246 y(\015)1822 5241 y Fp(N)1905 5204 y Fl(\()p Fk(K)1992 5215 y Fc(J)2035 5204 y Fl(\))2066 5137 y Fo(\015)2066 5191 y(\015)2066 5246 y(\015)2117 5304 y Fl(1)p Fk(;s)2208 5113 y Fo(\033)2277 5332 y Fk(J)2341 5241 y Fp(;)2472 5113 y Fo(\032)2540 5137 y(\015)2540 5191 y(\015)2540 5246 y(\015)2610 5218 y(b)2591 5241 y Fp(B)2665 5204 y Fl(\()p Fk(K)2752 5215 y Fc(J)2794 5204 y Fl(\))2826 5137 y Fo(\015)2826 5191 y(\015)2826 5246 y(\015)2876 5304 y Fl(1)p Fk(;s)2968 5113 y Fo(\033)3036 5332 y Fk(J)3100 5241 y Fp(:)144 5494 y Fq(Therefore,)f(if)e(w)m(e)i(de\014ne)e(a)i(v)m (ector)h(\014eld)d(\010)h(giv)m(en)g(b)m(y)1591 5699 y(\010\()p Fp(\037)p Fq(\))25 b(:=)55 b(lim)1931 5759 y Fk(J)6 b Fn(!1)2132 5699 y Fq(\010)2198 5661 y Fl(\()p Fk(K)2285 5672 y Fc(J)2327 5661 y Fl(\))2359 5699 y Fq(\()p Fp(\037)p Fq(\))1879 6018 y(20)p eop %%Page: 21 21 21 20 bop 144 191 a Fq(for)30 b(an)m(y)h Fp(\037)25 b Fj(2)p 623 118 107 4 v 25 w(D)693 205 y Fk(s)730 191 y Fq(,)31 b(it)e(follo)m(ws)h(that)h(the)f(limit)1543 391 y Fj(k)q Fq(\010)p Fj(k)1700 418 y Fl(1)p Fk(;s)1817 391 y Fq(=)55 b(lim)1913 451 y Fk(J)6 b Fn(!1)2114 286 y Fo(\015)2114 341 y(\015)2114 395 y(\015)2164 391 y Fq(\010)2230 353 y Fl(\()p Fk(K)2317 364 y Fc(J)2360 353 y Fl(\))2391 286 y Fo(\015)2391 341 y(\015)2391 395 y(\015)2442 454 y Fl(1)p Fk(;s)144 608 y Fq(exists)37 b(and)f(is)g(\014nite.)59 b(F)-8 b(rom)37 b(W)-8 b(eierstrass)37 b(theorem)g(it)g(follo)m(ws)e(that)j(\010)e(is)g(an)h(analytic)f(v)m (ector)i(\014eld)e(on)p 144 648 V 144 721 a Fj(D)214 735 y Fk(s)280 721 y Fq(=)p 380 648 149 4 v 29 w Fj(D)450 735 y Fk(\015)t(r)528 721 y Fq(.)47 b(Moreo)m(v)m(er,)36 b(since)c(the)g(recurren)m(t)h(sc)m(heme)g(\(58\){\(62\))j(and)c(Lemma) h(7,)g(pro)m(vide)f(v)m(ector)i(\014elds)144 833 y(\010)210 800 y Fl(\()p Fk(K)5 b Fl(+1\))454 833 y Fq(of)30 b(the)h(form)1027 1016 y(\010)1093 978 y Fl(\()p Fk(K)5 b Fl(+1\))1332 1016 y Fq(=)25 b(\010)1494 978 y Fl(\()p Fk(K)5 b Fl(\))1637 1016 y Fq(+)20 b Fj(R)p Fq(\001)1889 993 y Fo(b)1881 1016 y Fq(\010)1946 953 y Fl(\()p Fk(K)5 b Fl(\))2069 1016 y Fp(;)197 b Fj(R)p Fq(\001)2452 993 y Fo(b)2444 1016 y Fq(\010)2510 953 y Fl(\()p Fk(K)5 b Fl(\))2658 1016 y Fq(=)25 b Fj(O)2826 1034 y Fl([)p Fk(K)5 b Fl(+1])3024 1016 y Fp(;)144 1211 y Fq(it)20 b(can)h(b)s(e)e(deriv)m(ed)h(that)g (the)h(subsequence)e Fj(fk)p Fq(\010)1796 1178 y Fl(\()p Fk(K)1883 1189 y Fc(J)1927 1178 y Fl(\))1958 1211 y Fj(k)2003 1225 y Fl(1)p Fk(;s)2096 1211 y Fj(g)2141 1225 y Fk(J)2210 1211 y Fq(is,)j(in)d(fact,)k(the)e(complete)f(sequence)h Fj(fk)p Fq(\010)3654 1178 y Fl(\()p Fk(K)5 b Fl(\))3778 1211 y Fj(k)3823 1225 y Fl(1)p Fk(;s)3915 1211 y Fj(g)3960 1225 y Fk(K)4029 1211 y Fq(.)144 1335 y(In)30 b(a)h(similar)c(w)m(a)m (y)32 b(one)e(obtains)g Fp(N)40 b Fq(and)1611 1312 y Fo(b)1592 1335 y Fp(B)5 b Fq(,)30 b(analytic)g(v)m(ector)i(\014elds)d (on)p 2699 1262 V 30 w Fj(D)2769 1349 y Fk(\015)t(r)2878 1335 y Fq(de\014ned)g(as)1302 1527 y Fp(N)35 b Fq(:=)25 b(lim)1562 1588 y Fk(K)1673 1527 y Fp(N)1756 1490 y Fl(\()p Fk(K)5 b Fl(\))1879 1527 y Fp(;)2211 1504 y Fo(b)2192 1527 y Fp(B)30 b Fq(:=)25 b(lim)2443 1588 y Fk(K)2572 1504 y Fo(b)2553 1527 y Fp(B)2627 1490 y Fl(\()p Fk(K)5 b Fl(\))2750 1527 y Fp(:)144 1743 y Fq(T)-8 b(ogether)32 b(with)d(\010,)h(they)h(satisfy)e(equation)i(\(57\))h(and,)e (therefore,)h(they)f(lead)g(system)h(\(56\))h(in)m(to)e(\011)p Fi(NF)p Fq(.)-83 2026 y Fr(3)135 b(Pro)t(of)45 b(of)g(Prop)t(ositions)h (H1)f(and)f(R1)-83 2232 y Fd(3.1)112 b(Pro)s(of)37 b(of)h(Prop)s (osition)e(H1)-83 2404 y Fq(It)30 b(is)g(clear)g(that)h(if)623 2381 y Fo(b)604 2404 y Fp(B)f Fj(\021)25 b Fq(0)30 b(then)g(\011)p Fi(NF)g Fq(is)g(just)f Fi(BNF)h Fq(so,)h(let)f(us)g(consider)f(the)h (con)m(v)m(erse)i(situation.)40 b(T)-8 b(o)30 b(\014x)g(ideas,)-83 2517 y(let)d(us)f(deal)g(with)f(a)i(4-dimensional)e(Hamiltonian)g (system)i(with)e(the)i(origin)e(b)s(eing)h(a)h Fm(sadd)5 b(le-c)-5 b(enter)27 b Fq(equilibrium)-83 2629 y(p)s(oin)m(t.)39 b(The)27 b Fm(sadd)5 b(le-fo)-5 b(cus)28 b Fq(case)h(can)f(b)s(e)e (done)i(in)e(a)i(similar)d(w)m(a)m(y)-8 b(.)41 b(Assume)27 b(moreo)m(v)m(er)i(that)f(the)f(cen)m(ter)i(v)-5 b(ariables)-83 2742 y(ha)m(v)m(e)38 b(b)s(een)f(complexi\014ed)e(\(b)s(ecoming)i (complex)g(conjugated\).)62 b(Applying)35 b(Moser's)j(Theorem)f([21)q (],)i(w)m(e)f(kno)m(w)-83 2855 y(the)31 b(existence)f(of)h(an)f (analytic)g(con)m(v)m(ergen)m(t)j(transformation)d(\011,)g(close)h(to)g (the)f(iden)m(tit)m(y)-8 b(,)31 b(leading)e(it)h(in)m(to)g Fi(BNF)p Fq(,)1426 2961 y Fo(8)1426 3043 y(>)1426 3070 y(>)1426 3097 y(<)1426 3261 y(>)1426 3288 y(>)1426 3316 y(:)1578 3011 y Fq(_)1559 3035 y Fp(\030)87 b Fq(=)160 b Fp(\030)19 b(a)2024 3049 y Fl(1)2064 3035 y Fq(\()p Fp(\030)t(\021)s(;)c(\026\027)6 b Fq(\))1571 3148 y(_)-42 b Fp(\021)87 b Fq(=)e Fj(\000)p Fp(\021)18 b(a)2024 3162 y Fl(1)2064 3148 y Fq(\()p Fp(\030)t(\021)s(;)d(\026\027)6 b Fq(\))1565 3261 y(_)-42 b Fp(\026)83 b Fq(=)149 b Fp(\026)15 b(a)2024 3275 y Fl(2)2064 3261 y Fq(\()p Fp(\030)t(\021)s(;)g(\026\027) 6 b Fq(\))1567 3374 y(_)-40 b Fp(\027)89 b Fq(=)83 b Fj(\000)p Fp(\027)20 b(a)2024 3388 y Fl(2)2064 3374 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))3771 3202 y(\(69\))-83 3570 y(with)28 b Fp(a)171 3584 y Fl(1)210 3570 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))26 b(=)f Fp(\025)18 b Fq(+)g Fj(\001)d(\001)g(\001)45 b Fq(and)28 b Fp(a)1173 3584 y Fl(2)1213 3570 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))26 b(=)f(i)p Fp(\013)18 b Fq(+)f Fj(\001)e(\001)g(\001)h Fq(.)40 b(It)30 b(is)e(clear)h(that)2620 3546 y(~)2619 3570 y Fp(h)2671 3584 y Fl(1)2711 3570 y Fq(\()p Fp(\030)t(\021)s Fq(\))d(=)f Fp(\030)t(\021)18 b(a)3150 3584 y Fl(1)3190 3570 y Fq(\()p Fp(\030)t(\021)s(;)d Fq(0\))27 b(=)e Fp(\025\030)t(\021)c Fq(+)d Fj(\001)d(\001)g(\001)-83 3683 y Fq(and)95 3659 y(~)94 3683 y Fp(h)146 3697 y 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b(the)i(initial)c(system)k(in)m(to)f(\011)p Fi(NF)p Fq(,)h(that)g(is,)e (suc)m(h)h(that)h(the)g(new)f(system)g(is)g(of)g(the)h(form)1370 4342 y(_)-46 b Fp(\037)25 b Fq(=)g Fp(N)10 b Fq(\()p Fp(\037)p Fq(\))21 b(+)f(\()p Fp(D)s Fq(\010\()p Fp(\037)p Fq(\)\))2190 4301 y Fn(\000)p Fl(1)2319 4320 y Fo(b)2300 4342 y Fp(B)t Fq(\()p Fp(\037)p Fq(\))1271 b(\(70\))-83 4525 y(where)30 b Fp(\037)25 b Fq(=)g(\()p Fp(\030)t(;)15 b(\021)s(;)g(\026;)g(\027)6 b Fq(\))31 b(denotes)g(no)m(w)f(the)g(\011) p Fi(NF)p Fq(-v)-5 b(ariables.)40 b(Since)29 b(\010)h(starts)g(with)f (the)h(iden)m(tit)m(y)g(and)f Fp(h)3594 4539 y Fl(1)3634 4525 y Fq(,)h Fp(h)3741 4539 y Fl(2)3811 4525 y Fq(are)-83 4638 y(indep)s(enden)m(t)d(\014rst)i(in)m(tegrals)h(of)f(the)h (original)e(system,)i(it)g(follo)m(ws)e(that)2483 4614 y(\025)2482 4638 y Fp(h)2534 4652 y Fl(1)2599 4638 y Fq(=)d Fp(h)2747 4652 y Fl(1)2805 4638 y Fj(\016)20 b Fq(\010)29 b(and)3142 4614 y(\025)3141 4638 y Fp(h)3193 4652 y Fl(2)3258 4638 y Fq(=)c Fp(h)3406 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Fo(1)2409 5470 y(C)2409 5524 y(C)2409 5579 y(C)2409 5637 y(A)1879 6018 y Fq(21)p eop %%Page: 22 22 22 21 bop -83 197 a Fq(with)37 b Fp(b)171 149 y Fl(\(1\))171 209 y Fk(r)r(s)303 197 y Fj(6)p Fq(=)h(0)g(or)g Fp(b)653 149 y Fl(\(2\))653 227 y Fk(r)687 208 y Fe(0)709 227 y Fk(s)742 208 y Fe(0)807 197 y Fj(6)p Fq(=)f(0)i(\(and)e Fp(r)28 b Fq(+)d Fp(s)38 b Fq(not)g(necessarily)f(equal)h(to)g Fp(r)2504 164 y Fn(0)2553 197 y Fq(+)25 b Fp(s)2692 164 y Fn(0)2714 197 y Fq(\).)64 b(Using)38 b(that)3307 173 y(\025)3306 197 y Fp(h)3358 211 y Fl(1)3436 197 y Fq(=)g Fp(\025\030)t(\021)29 b Fq(+)c Fj(\001)15 b(\001)g(\001)-83 324 y Fq(and)35 b(\()p Fp(D)s Fq(\010\))313 291 y Fn(\000)p Fl(1)441 324 y Fq(=)f(I)24 b Fj(\000)f Fq(\()p Fp(D)817 301 y Fo(b)810 324 y Fq(\010\))h(+)f Fj(\001)15 b(\001)g(\001)h Fq(,)38 b(the)d(term)h(of)g(t)m(yp)s(e)g(\()p Fp(\030)t(\021)s Fq(\))2078 291 y Fk(`)2112 324 y Fq(\()p Fp(\026\027)6 b Fq(\))2288 291 y Fk(m)2390 324 y Fq(of)36 b(minimal)c(order)k 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Fq(v)-5 b(anishes.)-83 1342 y Fd(3.2)112 b(Pro)s(of)37 b(of)h(Prop)s(osition)e(R1)-83 1513 y Fq(The)i(problem)g(of)h(the)g(con)m(v)m(ergence)i(of)e(the)g (\011)p Fi(NF)f Fq(\(and)h Fi(BNF)p Fq(\))g(around)f(an)g(equilibrium)c (is)k(certainly)g(a)h(lo)s(cal)-83 1626 y(problem.)71 b(In)41 b(the)g(rev)m(ersible)f(setting,)k(this)c(implies)e(that)k(b)s (oth)f(the)g(linearized)e(system)i(and)g(the)g(rev)m(ersing)-83 1739 y(in)m(v)m(olution)20 b(can)h(b)s(e)g(tak)m(en)i(in)d(suitable)g (w)m(a)m(y)-8 b(.)39 b(Namely)-8 b(,)24 b(w)m(e)d(ha)m(v)m(e)i(the)f (follo)m(wing)d(lemma)i(whose)h(pro)s(of)e(is)h(essen)m(tially)-83 1852 y(con)m(tained)31 b(in)e([24)q(].)-83 2056 y Fs(Lemma)k(13)46 b Fm(L)-5 b(et)44 b(us)g(c)-5 b(onsider)46 b(a)e(system)1542 2033 y Fq(_)1506 2056 y Fp(X)54 b Fq(=)46 b Fp(F)13 b Fq(\()p Fp(X)7 b Fq(\))p Fm(,)48 b(analytic)d(ar)-5 b(ound)46 b(the)f(origin,)i(a)e(sadd)5 b(le-c)-5 b(enter)45 b(or)-83 2169 y(a)40 b(sadd)5 b(le-fo)-5 b(cus)42 b(e)-5 b(quilibrium,)41 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4693 y(Once)32 b(w)m(e)g(ha)m(v)m(e)h(set)f(the)g(linear)e(framew)m (ork,)i(w)m(e)g(presen)m(t)g(a)g(prop)s(ert)m(y)f(whic)m(h)f(c)m (haracterizes)j(those)g(transforma-)-83 4806 y(tions)d(that)h(preserv)m (e)f(a)h(giv)m(en)f(linear)f(rev)m(ersibilit)m(y)-8 b(.)-83 4986 y Fs(Lemma)33 b(14)46 b Fm(L)-5 b(et)33 b Fq(\011)f Fm(b)-5 b(e)32 b(a)h(di\013e)-5 b(omorphism)36 b(satisfying)1578 5182 y Fg(R)p Fq(\011\()p Fg(R)p Fp(\037)p Fq(\))26 b(=)f(\011\()p Fp(\037)p Fq(\))p Fp(:)1500 b Fq(\(77\))-83 5378 y Fm(Then)33 b(the)g(tr)-5 b(ansformation)36 b Fp(X)d Fq(=)25 b(\011\()p Fp(\037)p Fq(\))33 b Fm(pr)-5 b(eserves)33 b(the)g Fg(R)p Fm(-r)-5 b(eversibility,)33 b(that)h(is,)f(the)g(new)f(system)1487 5573 y Fq(_)-46 b Fp(\037)25 b Fq(=)g Fp(G)p Fq(\()p Fp(\037)p Fq(\))h(:=)f(\()q(\011)2096 5536 y Fn(\003)2135 5573 y Fp(F)13 b Fq(\))i(\()p Fp(\037)p Fq(\))-83 5769 y Fm(is)33 b(also)g Fg(R)p Fm(-r)-5 b(eversible.)1879 6018 y Fq(22)p eop %%Page: 23 23 23 22 bop 58 191 a Fq(The)30 b(pro)s(of)g(of)g(Prop)s(osition)f(R1)h (is)g(based)g(on)g(the)g(follo)m(wing)f(t)m(w)m(o)j(p)s(oin)m(ts:)53 372 y Fj(\017)46 b Fq(Applying)40 b(Theorem)j(1,)j(there)d(exist)f(an)g (analytic)g(transformation)g Fp(X)53 b Fq(=)45 b(\010\()p Fp(\037)p Fq(\))e(and)f(analytic)g(v)m(ector)144 485 y(\014elds)33 b Fp(N)10 b Fq(\()p Fp(\037)p Fq(\),)671 462 y Fo(b)652 485 y Fp(B)5 b Fq(\()p Fp(\037)p Fq(\))35 b(leading)e(the)i(original)d(system)j(in)m(to)f(\011)p Fi(NF)p Fq(,)h(pro)m(vided)e(the)h(origin)f(is)g(a)i(saddle-cen)m(ter) 144 598 y(or)c(saddle-fo)s(cus)e(equilibrium)c(p)s(oin)m(t.)40 b(That)30 b(is,)g(satisfying)f(equalit)m(y)h(\(57\).)53 783 y Fj(\017)46 b Fq(The)30 b(v)m(ector)j(\014elds)c(obtained)h(from)g (the)h(recurren)m(t)g(sc)m(heme)g(satisfy:)41 b(\()p Fp(a)p Fq(\))32 b(the)f(transformation)f Fp(X)j Fq(=)25 b(\010\()p Fp(\037)p Fq(\))144 896 y(v)m(eri\014es)38 b(relation)g(\(77\),)k(so)c(it)g(preserv)m(es)g Fg(R)p Fq(-rev)m(ersibilit)m(y;)j(\()p Fp(b)p Fq(\))e Fp(N)48 b Fq(and)2753 873 y Fo(b)2734 896 y Fp(B)43 b Fq(are)38 b Fg(R)p Fq(-rev)m(ersible.)64 b(This)37 b(last)144 1009 y(prop)s(ert)m(y)30 b(will)e(imply)g(that)1159 986 y Fo(b)1140 1009 y Fp(B)35 b Fq(has)30 b(to)h(v)-5 b(anish)29 b(and,)h(therefore,)h(\011)p Fi(NF)f Fq(will)e(b)s(ecome)i Fi(BNF)p Fq(.)-83 1213 y Fs(Lemma)j(15)46 b Fm(L)-5 b(et)33 b(us)g(c)-5 b(onsider)35 b(an)f Fg(R)p Fm(-r)-5 b(eversible)33 b(system)h Fq(\(75\))p Fm(,)h(analytic)g(on)f(a)f(neighb)-5 b(orho)g(o)g(d)37 b(of)d(the)f(origin,)h(a)-83 1326 y(sadd)5 b(le-c)-5 b(enter)34 b(or)g(a)g(sadd)5 b(le-fo)-5 b(cus)35 b(e)-5 b(quilibrium)33 b(p)-5 b(oint.)45 b(L)-5 b(et)34 b(us)f(take)g Fq(\010)2453 1293 y Fl(\()p Fk(K)5 b Fl(\))2576 1326 y Fm(,)33 b Fp(N)2720 1293 y Fl(\()p Fk(K)5 b Fl(\))2877 1326 y Fm(and)3073 1303 y Fo(b)3054 1326 y Fp(B)3128 1293 y Fl(\()p Fk(K)g Fl(\))3251 1326 y Fm(,)33 b(the)h(ve)-5 b(ctor)34 b(\014elds)-83 1439 y(pr)-5 b(ovide)g(d)35 b(by)d(the)h Fq(\011)p Fi(NF)p Fm(-r)-5 b(e)g(curr)g(ent)34 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Fo(b)2223 2122 y Fp(B)2297 2085 y Fl(\()p Fk(K)5 b Fl(\))2436 2122 y Fq(\()p Fg(R)p Fp(\037)p Fq(\))26 b(=)f Fj(\000)2849 2099 y Fo(b)2831 2122 y Fp(B)2905 2085 y Fl(\()p Fk(K)5 b Fl(\))3027 2122 y Fq(\()p Fp(\037)p Fq(\))p Fp(:)-35 2321 y Fq(\()p Fp(ii)p Fq(\))47 b Fm(The)33 b(ve)-5 b(ctor)33 b(\014elds)h Fq(\010)p Fm(,)e Fp(N)42 b Fm(and)1268 2298 y Fo(b)1249 2321 y Fp(B)37 b Fm(pr)-5 b(ovide)g(d)35 b(by)d(The)-5 b(or)g(em)35 b(1)e(and)g(de\014ne)-5 b(d)34 b(as)880 2518 y Fq(\010)25 b(=)64 b(lim)1067 2578 y Fk(K)5 b Fn(!1)1287 2518 y Fq(\010)1353 2480 y Fl(\()p Fk(K)g Fl(\))1476 2518 y Fp(;)202 b(N)35 b Fq(=)64 b(lim)1907 2578 y Fk(K)5 b Fn(!1)2127 2518 y Fp(N)2210 2480 y Fl(\()p Fk(K)g Fl(\))2333 2518 y Fp(;)2579 2495 y Fo(b)2560 2518 y Fp(B)29 b Fq(=)65 b(lim)2754 2578 y Fk(K)5 b Fn(!1)2994 2495 y Fo(b)2975 2518 y Fp(B)3049 2480 y Fl(\()p Fk(K)g Fl(\))3172 2518 y Fp(;)144 2768 y Fm(verify)33 b(the)g(same)g(pr)-5 b(op)g(erties)35 b(as)e Fq(\010)1382 2735 y Fl(\()p Fk(K)5 b 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b(.)-83 3588 y(Namely)g(,)-20 3769 y(\()p Fp(a)p Fq(\))46 b(If)30 b(a)h(v)m(ector)h(\014eld)d Fp(H)37 b Fq(is)30 b Fg(R)p Fq(-rev)m(ersible)f(then)h(its)g(pro)5 b(jections)30 b Fj(P)7 b Fp(H)38 b Fq(and)30 b Fj(R)p Fp(H)37 b Fq(are)31 b(also)f Fg(R)p Fq(-rev)m(ersible.)-11 3954 y(\()p Fp(b)p Fq(\))46 b(Let)31 b Fj(R)394 3931 y Fo(b)384 3954 y Fq(\011)e(b)s(e)h(the)g(solution)e(of)i(an)g (equation)g(of)g(t)m(yp)s(e)g(\(29\).)42 b(Then,)30 b(if)e Fj(R)2685 3931 y Fo(b)2663 3954 y Fp(H)37 b Fq(is)29 b Fg(R)p Fq(-rev)m(ersible)h(it)f(follo)m(ws)g(that)144 4067 y Fj(R)231 4044 y Fo(b)221 4067 y Fq(\011)22 b(satis\014es)f (\(77\))j(and,)f(therefore,)h(the)e(transformation)g Fp(X)32 b Fq(=)25 b Fp(\037)s Fq(+)s Fj(R)2603 4044 y Fo(b)2593 4067 y Fq(\011\()p Fp(\037)p Fq(\))e(preserv)m(es)f(the)g Fg(R)p Fq(-rev)m(ersibilit)m(y)-8 b(.)144 4216 y(T)g(o)31 b(see)g(it,)f(let)h(us)e(denote)i Fj(R)1169 4193 y Fo(b)1159 4216 y Fq(\011)25 b(=)g(\()1403 4192 y Fo(b)1386 4216 y Fp( )1445 4230 y Fl(1)1485 4216 y Fp(;)1542 4192 y Fo(b)1525 4216 y Fp( )1584 4230 y Fl(2)1624 4216 y Fp(;)1681 4192 y Fo(b)1664 4216 y Fp( )1723 4230 y Fl(3)1763 4216 y Fp(;)1820 4192 y Fo(b)1803 4216 y Fp( )1862 4230 y Fl(4)1902 4216 y Fq(\),)31 b(where)1412 4404 y Fo(b)1396 4428 y Fp( )1455 4442 y Fk(w)1512 4428 y Fq(\()p Fp(\030)t(;)15 b(\021)s(;)g(\026;)g(\027)6 b Fq(\))27 b(=)2166 4342 y Fo(X)2023 4539 y Fk(j)t Fl(+)p Fk(k)r Fl(+)p Fk(`)p Fl(+)p Fk(m)p Fn(\025)p Fl(2)2455 4428 y Fp( )2517 4380 y Fl(\()p Fk(w)r Fl(\))2514 4458 y Fk(j)t(k)r(`m)3771 4428 y Fq(\(78\))144 4742 y(for)32 b Fp(w)f Fq(=)d(1)p Fp(;)15 b Fq(2)p Fp(;)g(:)g(:)g(:)i(;)e Fq(4.)47 b(It)32 b(is)f(not)i(di\016cult)d(to)j(c)m(hec)m(k)g(that)g(if)e Fj(R)2351 4719 y Fo(b)2341 4742 y Fq(\011)h(satis\014es)f(equation)h (\(29\),)i(whose)e(explicit)144 4855 y(solution)d(is)h(giv)m(en)g(in)f (section)i(2.3.1,)h(then)e(the)h(co)s(e\016cien)m(ts)g(in)e(\(78\))j(v) m(erify)d(that)1355 5067 y Fp( )1417 5019 y Fl(\(2\))1414 5096 y Fk(j)t(k)r(`m)1606 5067 y Fq(=)c Fp( )1764 5019 y Fl(\(1\))1761 5096 y Fk(k)r(j)t(m`)1927 5067 y Fp(;)198 b( )2212 5019 y Fl(\(4\))2209 5096 y Fk(j)t(k)r(`m)2400 5067 y Fq(=)25 b Fp( )2558 5019 y Fl(\(3\))2555 5096 y Fk(k)r(j)t(m`)144 5290 y Fq(and,)30 b(consequen)m(tly)-8 b(,)31 b Fj(R)989 5267 y Fo(b)979 5290 y Fq(\011)f(satis\014es)g (relation)g(\(77\).)-11 5475 y(\()p Fp(c)p Fq(\))46 b(If)34 b Fp(H)41 b Fq(is)32 b Fg(R)p Fq(-rev)m(ersible)i(then)f Fp(H)1251 5489 y Fn(\024)p Fk(K)1408 5475 y Fq(\(constituted)h(b)m(y)g (its)f(terms)h(of)g(order)g(less)f(or)h(equal)f(than)h Fp(K)7 b Fq(\))34 b(is)f(also)144 5588 y Fg(R)p Fq(-rev)m(ersible,)d (for)g(an)m(y)h Fp(K)h Fj(\025)25 b Fq(1.)58 5769 y(W)-8 b(e)32 b(are)e(no)m(w)h(in)e(conditions)g(of)h(pro)m(ving)g(the)g (assertions)g(\()p Fp(i)p Fq(\),)i(\()p Fp(ii)p Fq(\))g(and)d(\()p Fp(iii)p Fq(\).)1879 6018 y(23)p eop %%Page: 24 24 24 23 bop -3 191 a Fq(\()p Fp(i)p Fq(\))46 b(F)-8 b(rom)39 b(its)f(form,)i(it)d(is)h(clear)g(that)1428 168 y Fo(b)1404 191 y Fp(N)1487 158 y Fl(\()p Fk(K)5 b Fl(\))1648 191 y Fq(and)38 b Fp(N)1916 158 y Fl(\()p Fk(K)5 b Fl(\))2077 191 y Fq(are)39 b Fg(R)p Fq(-rev)m(ersible)e(v)m(ector)j(\014elds,)f (for)f(an)m(y)h Fp(K)45 b Fj(\025)38 b Fq(1.)144 304 y(No)m(w,)33 b(w)m(e)f(are)g(going)g(to)h(pro)m(v)m(e)f(that)g(\010) 1533 271 y Fl(\()p Fk(K)5 b Fl(\))1688 304 y Fq(v)m(eri\014es)31 b(condition)f(\(77\))j(and)2784 281 y Fo(b)2765 304 y Fp(B)2839 271 y Fl(\()p Fk(K)5 b Fl(\))2993 304 y Fq(is)31 b Fg(R)p Fq(-rev)m(ersible)g(using)f(an)144 417 y(inductiv)m(e)f (argumen)m(t.)144 567 y(F)-8 b(or)31 b Fp(K)h Fq(=)25 b(2)31 b(\(the)g(case)g Fp(K)h Fq(=)25 b(1)31 b(is)f(trivial\))e(w)m(e) j(ha)m(v)m(e)h(that)1198 701 y Fo(n)1259 802 y Fj(L)1322 827 y Fk(N)1385 808 y Ff(\(1\))1486 701 y Fo(\020)1540 802 y Fj(R)1625 779 y Fo(b)1617 802 y Fq(\010)1683 739 y Fl(\(2\))1777 701 y Fo(\021)q(o)1892 865 y Fn(\024)p Fl(2)2012 802 y Fq(=)2108 701 y Fo(n)2169 802 y Fj(R)2261 701 y Fo(\020)2333 779 y(b)2315 802 y Fp(F)2401 701 y Fo(\020)2456 802 y Fq(\010)2522 764 y Fl(\(1\))2616 701 y Fo(\021\021o)2785 865 y Fn(\024)p Fl(2)144 1039 y Fq(or,)f (simplifying,)1658 1177 y Fj(L)1721 1191 y Fl(\003)1789 1076 y Fo(\020)1844 1177 y Fj(R)1929 1154 y Fo(b)1921 1177 y Fq(\010)1986 1115 y Fl(\(2\))2081 1076 y Fo(\021)2160 1177 y Fq(=)25 b Fp(F)2314 1195 y Fl([2])2393 1177 y Fp(:)144 1410 y Fq(Applying)k(prop)s(erties)g(\()p Fp(c)p Fq(\))i(and)f(\()p Fp(b)p Fq(\))i(ab)s(o)m(v)m(e)g(one)f(obtains)f (that)h Fj(R)2451 1387 y Fo(b)2443 1410 y Fq(\010)2509 1348 y Fl(\(2\))2634 1410 y Fq(preserv)m(es)f Fg(R)p Fq(-rev)m(ersibilit)m(y)-8 b(.)41 b(On)29 b(the)144 1523 y(other)39 b(hand,)673 1500 y Fo(b)654 1523 y Fp(B)728 1490 y Fl(\(2\))861 1523 y Fq(=)g(0)g(so)h(it)e(is)g(trivially)e(a)k Fg(R)p Fq(-rev)m(ersible)e(v)m(ector)i(\014eld.)65 b(Assume)39 b(no)m(w,)i(as)e Fm(induction)144 1636 y(hyp)-5 b(otheses)p Fq(,)33 b(that)e(for)f(a)h(giv)m(en)g Fp(K)h Fj(\025)25 b Fq(1,)247 1849 y Fs({)45 b Fq(\010)410 1816 y Fl(\()p Fk(K)5 b Fl(\))558 1849 y Fq(=)25 b(id)19 b(+)h Fj(R)926 1826 y Fo(b)918 1849 y Fq(\010)984 1787 y Fl(\()p Fk(K)5 b Fl(\))1137 1849 y Fq(satis\014es)30 b(\(77\))i(\(so)f(it)f(preserv)m (es)h Fg(R)p Fq(-rev)m(ersibilit)m(y\),)247 1995 y Fs({)363 1972 y Fo(b)344 1995 y Fp(B)418 1962 y Fl(\()p Fk(K)5 b Fl(\))571 1995 y Fq(is)30 b(a)g Fg(R)p Fq(-rev)m(ersible)g(v)m(ector) i(\014eld.)144 2183 y(Applying)j(prop)s(erties)g(\()p Fp(a)p Fq(\),)k(\()p Fp(c)p Fq(\))f(and)f(\()p Fp(b)p Fq(\))g(on)g(equation)g(\(61\))h(it)e(follo)m(ws)g(that)i Fj(R)3038 2160 y Fo(b)3030 2183 y Fq(\010)3096 2120 y Fl(\()p Fk(K)5 b Fl(+1\))3346 2183 y Fq(and,)38 b(therefore,)144 2326 y(\010)210 2293 y Fl(\()p Fk(K)5 b Fl(+1\))449 2326 y Fq(=)25 b(id)18 b(+)i Fj(R)816 2303 y Fo(b)808 2326 y Fq(\010)874 2264 y Fl(\()p Fk(K)5 b Fl(+1\))1118 2326 y Fq(preserv)m(e)30 b Fg(R)p Fq(-rev)m(ersibilit)m(y)-8 b(.)40 b(Moreo)m(v)m(er,)32 b(from)e(equation)h(\(62\))g(w)m(e)g(ha)m (v)m(e)1207 2530 y Fo(b)1188 2553 y Fp(B)1262 2515 y Fl(\()p Fk(K)5 b Fl(+1\))1500 2553 y Fq(=)1596 2452 y Fo(n)1657 2553 y Fj(P)1743 2452 y Fo(\020)1815 2530 y(b)1797 2553 y Fp(F)13 b Fq(\(\010)1969 2515 y Fl(\()p Fk(K)5 b Fl(\))2092 2553 y Fq(\))2127 2452 y Fo(\021)q(o)2243 2616 y Fn(\024)p Fk(K)g Fl(+1)2476 2553 y Fj(\000)2591 2530 y Fo(b)2567 2553 y Fp(N)2650 2515 y Fl(\()p Fk(K)g Fl(+1\))2864 2553 y Fp(:)144 2816 y Fq(Th)m(us,)27 b(since)g(\010)683 2783 y Fl(\()p Fk(K)5 b Fl(\))833 2816 y Fq(preserv)m(es)27 b Fg(R)p Fq(-rev)m(ersibilit)m(y)-8 b(,)1860 2793 y Fo(b)1837 2816 y Fp(N)1920 2783 y Fl(\()p Fk(K)5 b Fl(+1\))2160 2816 y Fq(is)26 b Fg(R)p Fq(-rev)m(ersible)h(and)f(taking)i(in)m(to)f (accoun)m(t)h(prop-)144 2929 y(erties)i(\()p Fp(a)p Fq(\),)i(\()p Fp(c)p Fq(\),)f(it)f(turns)f(out)i(that)1434 2906 y Fo(b)1415 2929 y Fp(B)1489 2896 y Fl(\()p Fk(K)5 b Fl(+1\))1732 2929 y Fq(is)30 b(also)g Fg(R)p Fq(-rev)m(ersible.)-35 3116 y(\()p Fp(ii)p Fq(\))47 b(It)35 b(follo)m(ws)f(from)h(\()p Fp(i)p Fq(\))h(letting)e Fp(K)42 b Fq(tend)35 b(to)h(in\014nit)m(y)c (and)j(applying)e(the)i(\(analytic\))g(con)m(v)m(ergence)i(of)f(\010) 3785 3083 y Fl(\()p Fk(K)5 b Fl(\))3907 3116 y Fq(,)144 3229 y Fp(N)227 3196 y Fl(\()p Fk(K)g Fl(\))381 3229 y Fq(and)577 3206 y Fo(b)558 3229 y Fp(B)632 3196 y Fl(\()p Fk(K)g Fl(\))754 3229 y Fq(.)-66 3417 y(\()p Fp(iii)p Fq(\))47 b(F)-8 b(rom)31 b(the)g Fg(R)p Fq(-rev)m(ersibilit)m(y)d(of) 1264 3394 y Fo(b)1245 3417 y Fp(B)t Fq(,)1623 3530 y Fg(R)1733 3507 y Fo(b)1713 3530 y Fp(B)20 b Fq(\()q Fg(R)p Fp(\037)p Fq(\))25 b(=)g Fj(\000)2216 3507 y Fo(b)2197 3530 y Fp(B)20 b Fq(\()p Fp(\037)p Fq(\))c Fp(;)144 3696 y Fq(it)30 b(turns)f(out)i(that)768 3877 y Fo(b)774 3901 y Fp(b)813 3915 y Fl(1)852 3901 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))26 b(=)f Fj(\000)1347 3877 y Fo(b)1353 3901 y Fp(b)1392 3915 y Fl(1)1432 3901 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))213 b(and)2305 3877 y Fo(b)2311 3901 y Fp(b)2350 3915 y Fl(2)2390 3901 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))26 b(=)f Fj(\000)2885 3877 y Fo(b)2891 3901 y Fp(b)2930 3915 y Fl(2)2969 3901 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))p Fp(;)144 4105 y Fq(so)250 4081 y Fo(b)256 4105 y Fp(b)295 4119 y Fl(1)334 4105 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))27 b(=)759 4081 y Fo(b)764 4105 y Fp(b)803 4119 y Fl(2)843 4105 y Fq(\()p Fp(\030)t(\021)s(;)15 b(\026\027)6 b Fq(\))26 b(=)f(0)31 b(and)f(the)g(lemma)g(is)g(pro)m(v)m(ed.)3865 4293 y Fa(2)58 4568 y Fq(F)-8 b(rom)39 b(this)f(lemma)g(the)h(pro)s(of) f(of)g(Prop)s(osition)f(R)8 b(1)39 b(follo)m(ws)f(straigh)m(tforw)m (ardly)-8 b(.)65 b(The)38 b(transformation)g(\010)-83 4681 y(preserv)m(es)30 b Fg(R)p Fq(-rev)m(ersibilit)m(y)-8 b(,)30 b(the)g(v)m(ector)i(\014eld)d Fp(N)41 b Fq(is)29 b Fg(R)p Fq(-rev)m(ersible)h(and)2465 4658 y Fo(b)2446 4681 y Fp(B)g Fq(=)25 b(0)31 b(so,)g(in)e(fact,)i(the)g(\011)p Fi(NF)f Fq(is)g(nothing)-83 4794 y(else)g(but)g(the)g Fi(BNF)p Fq(.)58 5143 y Fm(A)-5 b(cknow)5 b(le)-5 b(dgement.)40 b Fq(The)24 b(authors)g(wish)f(to)j(express)e(their)f(appreciation)h (to)i(A.)f(V)-8 b(anderbau)m(whede,)25 b(C.)g(Sim\023)-45 b(o)-83 5256 y(and)105 5233 y(\022)94 5256 y(A.)30 b(Jorba)h(for)f(v)m (ery)g(stim)m(ulating)f(discussions)f(and)i(fruitful)d(remarks.)1879 6018 y(24)p eop %%Page: 25 25 25 24 bop -83 191 a Fr(References)-38 394 y Fq([1])47 b(Arnol'd,)40 b(V.)f(I.)66 b(Chapitres)38 b(suppl)m(\023)-43 b(emen)m(taires)38 b(de)h(la)f(th)m(\023)-43 b(eorie)40 b(des)c(\023)-43 b(equations)40 b(di\013)m(\023)-43 b(eren)m(tielles)38 b(ordinaries.)112 484 y(\023)104 507 y(Editions)28 b(Mir,)i(Moscou,)i (1980.)-38 692 y([2])47 b(Bo)s(c)m(hner,)38 b(S.)56 b(Compact)37 b(groups)e(of)h(di\013eren)m(tiable)f(transformations.)56 b Fm(A)n(nn.)37 b(Math.)p Fq(,)h(second)e(series,)h(v)m(ol.)104 805 y(46,)31 b(issue)e(3,)j(372{381,)h(1945.)-38 991 y([3])47 b(Bro)s(er,)28 b(H.W.,)h(Huitema,)e(G.B.)h(and)f(Sevryuk,)g (M.B.)36 b(Quasi-p)s(erio)s(dic)23 b(motions)k(in)e(families)g(of)i (Dynamical)104 1104 y(Systems.)40 b Fm(L)-5 b(e)g(ctur)g(e)34 b(Notes)e(in)h(Mathematics)p Fq(,)f(1645,)g(Springer,)c(1997.)-38 1290 y([4])47 b(Bruno,)36 b(A.D.)55 b(Analytic)34 b(form)g(of)h (di\013eren)m(tial)f(equations)g(\(I\).)55 b Fm(T)-7 b(r)i(ans.)38 b(Mosc.)e(Math.)i(So)-5 b(c.)35 b Fq(25,)i(131{288,)104 1403 y(1971.)-38 1588 y([5])47 b(Bruno,)23 b(A.D.)28 b(The)22 b(analytic)f(form)h(of)g(di\013eren)m(tial)f(equations)g(\(I)s (I\).)27 b Fm(T)-7 b(r)i(ans.)27 b(Mosc.)d(Math.)i(So)-5 b(c.)22 b Fq(26,)j(199{239,)104 1701 y(1972.)-38 1887 y([6])47 b(Bruno,)37 b(A.D.)58 b(The)36 b(normal)f(form)h(of)g(a)g (Hamiltonian)f(system.)58 b Fm(R)n(ussian)38 b(Math.)g(Surveys)e Fq(43,)j(1,)f(25{66,)104 2000 y(1988.)-38 2186 y([7])47 b(Bruno,)29 b(A.D.)39 b(Lo)s(cal)28 b(metho)s(ds)g(in)g(nonlinear)f (di\013eren)m(tial)g(equations.)38 b Fm(Springer)32 b(Series)f(in)g (Soviet)h(Mathe-)104 2298 y(matics)p Fq(,)f(Springer,)e(1989.)-38 2484 y([8])47 b(DeLatte,)39 b(D.)57 b(On)35 b(normal)g(forms)g(in)f (Hamiltonian)g(dynamics,)i(a)g(new)g(approac)m(h)g(to)g(some)g(con)m(v) m(ergence)104 2597 y(questions.)k Fm(Er)-5 b(go)g(d.)33 b(Th.)g(&)g(Dynam.)g(Sys.)d Fq(15,)i(49{66,)h(1995.)-38 2783 y([9])47 b(Delshams,)32 b(D.,)h(Gelfreic)m(h,)f(V.,)h(Jorba,)1548 2760 y(\022)1537 2783 y(A.)f(and)f(Seara,)i(T.M.)46 b(Exp)s(onen)m (tially)30 b(small)g(splitting)g(of)i(separa-)104 2896 y(trices)e(under)f(fast)i(quasip)s(erio)s(dic)26 b(forcing.)40 b Fm(Commun.)34 b(Math.)f(Phys.)e Fq(189,)g(35{71,)i(1997.)-83 3081 y([10])47 b(Delshams,)40 b(A.,)h(Guillamon,)d(A.)g(and)g(L\023)-45 b(azaro)39 b(J.T.)64 b(A)39 b(pseudo-normal)d(form)i(for)g(planar)f(v)m (ector)j(\014elds.)104 3194 y Fm(Qualitative)33 b(The)-5 b(ory)34 b(of)f(Dynamic)-5 b(al)34 b(Systems)d Fq(3)g(\(1\),)h(51{82,)g (2002.)-83 3380 y([11])47 b(Dev)-5 b(aney)d(,)32 b(R.L.)40 b(Rev)m(ersible)30 b(di\013eomorphims)e(and)h(\015o)m(ws.)41 b Fm(T)-7 b(r)i(ans.)34 b(A)n(m.)e(Math.)h(So)-5 b(c.)p Fq(,)31 b(218,)h(89{113,)h(1976.)-83 3566 y([12])47 b(Gaeta,)25 b(G.)c(and)f(Cicogna)h(G.)26 b(Symmetry)20 b(and)g(p)s(erturbation)f (theory)i(in)e(nonlinear)g(dynamics.)24 b(V)-8 b(olume)20 b(M57)104 3679 y(of)30 b Fm(L)-5 b(e)g(ctur)g(e)34 b(Notes)f(in)f (Physics)p Fq(,)f(Springer,)d(Berlin,)h(1999.)-83 3864 y([13])47 b(Giorgilli,)39 b(A.)67 b(Unstable)38 b(equilibria)e(of)j (Hamiltonian)f(systems)67 b Fm(Discr)-5 b(ete)41 b(and)g(Cont.)h(Dyn.)e (Sys.)p Fq(,)i Fs(7)p Fq(,)g(4,)104 3977 y(855{871,)34 b(2001.)-83 4163 y([14])47 b(Henrici,)34 b(P)-8 b(.)53 b(Applied)32 b(and)i(computational)g(complex)h(analysis,)f(V)-8 b(ol.)35 b(I:)g(p)s(o)m(w)m(er)f(series,)h(in)m(tegration,)h(con-)104 4276 y(formal)29 b(mapping,)g(lo)s(cation)h(of)h(zeros.)41 b Fm(Wiley)33 b(Classics)g(Libr)-5 b(ary)p Fq(,)32 b(John)d(Wiley)h (and)g(sons,)g(1974.)-83 4462 y([15])47 b(Lam)m(b,)41 b(J.S.W.)e(and)f(Rob)s(erts,)i(J.A.G.)66 b(Time-rev)m(ersal)38 b(symmetry)g(in)f(dynamical)g(systems:)58 b(A)38 b(surv)m(ey)-8 b(.)104 4574 y Fm(Physic)j(a)33 b(D)p Fq(,)e(112,)g(1{39,)i(1998.)-83 4760 y([16])47 b(Ly)m(apuno)m(v,)26 b(A.M.)34 b(The)24 b(general)i(problem)d(of)j(the)f(stabilit)m(y)f(of)i(motion,)g(Mosco)m (w,)i(1892.)34 b(Probl)m(\022)-43 b(eme)26 b(g)m(\023)-43 b(en)m(\023)g(eral)104 4873 y(de)30 b(la)g(estabilit)m(\023)-43 b(e)30 b(du)g(mouv)m(emen)m(t.)41 b Fm(A)n(nn.)32 b(F)-7 b(ac.)33 b(Sci.)g(T)-7 b(oulouse)p Fq(,)31 b(9)g(\(2\),)g(202{474,)j (1907.)-83 5059 y([17])47 b(Lom)m(bardi,)31 b(E.)46 b(Oscillatory)31 b(in)m(tegrals)g(and)h(phenomena)f(b)s(ey)m(ond)h(all)f(algebraic)g (orders.)46 b(V)-8 b(olume)32 b(1741)h(of)104 5172 y Fm(L)-5 b(e)g(ctur)g(e)33 b(Notes)g(in)f(Mathematics)p Fq(,)g(Springer,)d(Berlin,)g(2000.)-83 5357 y([18])47 b(v)-5 b(an)21 b(der)f(Meer,)k(J.C.,)f(Sanders,)f(J.A.)g(and)e(V)-8 b(anderbau)m(whede,)23 b(A.)i(Hamiltonian)20 b(structure)g(of)i(the)f (rev)m(ersible)104 5470 y(nonsemisimple)31 b(1:1)36 b(resonance)54 b Fm(Dynamics,)38 b(Bifur)-5 b(c)g(ation)38 b(and)f(Symmetry)p Fq(,)h(221{240,)h(P)-8 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