Content-Type: multipart/mixed; boundary="-------------0306030940691" This is a multi-part message in MIME format. ---------------0306030940691 Content-Type: text/plain; name="03-256.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-256.keywords" Herman Rings, Quasiconformal Surgery, Complex Standard Family, Semistandard Map,Analytic Linearization, Brjuno Numbers, Hyperbolic Geometry, Siegel Disks ---------------0306030940691 Content-Type: application/postscript; name="anells.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="anells.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86e Copyright 2001 Radical Eye Software %%Title: MAIN.dvi %%Pages: 33 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips MAIN %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.06.03:1555 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind 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FI(PSfrag)30 b(replacemen)m(ts)p -173 -348 811 4 v 638 -345 a /Unhide PSfrag 638 -345 a 591 -275 a { 591 -275 a 569 -293 a FC(\013)p 554 -279 75 4 v 554 -232 a FB(2)p FC(\031)591 -275 y } 0/Place PSfrag 591 -275 a 619 -137 a { 619 -137 a 600 -119 a FD(")619 -137 y } 1/Place PSfrag 619 -137 a 620 -27 a { 620 -27 a 602 -6 a FJ(1)620 -27 y } 2/Place PSfrag 620 -27 a 620 86 a { 620 86 a 602 107 a FJ(0)620 86 y } 3/Place PSfrag 620 86 a 588 194 a { 588 194 a 598 167 a Fj(1)p 597 179 31 3 v 597 220 a(5)588 194 y } 4/Place PSfrag 588 194 a 587 307 a { 587 307 a 597 280 a Fj(1)p 597 292 31 3 v 597 333 a(4)587 307 y } 5/Place PSfrag 587 307 a 587 420 a { 587 420 a 597 393 a Fj(1)p 597 405 31 3 v 597 446 a(3)587 420 y } 6/Place PSfrag 587 420 a 587 533 a { 587 533 a 597 506 a Fj(2)p 597 518 31 3 v 597 559 a(5)587 533 y } 7/Place PSfrag 587 533 a 613 629 a { 613 629 a 598 618 a Fj(1)p 598 630 31 3 v 598 672 a(2)613 629 y } 8/Place PSfrag 613 629 a 587 758 a { 587 758 a 597 731 a Fj(3)p 597 743 31 3 v 597 785 a(5)587 758 y } 9/Place PSfrag 587 758 a 587 871 a { 587 871 a 597 844 a Fj(2)p 597 856 31 3 v 597 897 a(3)587 871 y } 10/Place PSfrag 587 871 a 598 997 a { 598 997 a Fl(\015)598 997 y } 11/Place PSfrag 598 997 a 613 1081 a { 613 1081 a 598 1070 a Fj(3)p 598 1082 31 3 v 598 1123 a(4)613 1081 y } 12/Place PSfrag 613 1081 a 587 1210 a { 587 1210 a 597 1183 a Fj(4)p 597 1195 31 3 v 597 1236 a(5)587 1210 y } 13/Place PSfrag 587 1210 a 620 1328 a { 620 1328 a 602 1349 a FJ(1)620 1328 y } 14/Place PSfrag 620 1328 a 621 1439 a { 621 1439 a 604 1462 a Fl(\022)621 1439 y } 15/Place PSfrag 621 1439 a 585 1548 a { 585 1548 a 562 1566 a Fl(T)603 1575 y Fj(0)585 1548 y } 16/Place PSfrag 585 1548 a 591 1643 a { 591 1643 a -47 w Fl(T)595 1647 y Fz(1)p 595 1655 29 3 v 595 1688 a(3)591 1643 y } 17/Place PSfrag 591 1643 a 591 1756 a { 591 1756 a -47 w Fl(T)595 1760 y Fz(2)p 595 1768 29 3 v 595 1801 a(3)591 1756 y } 18/Place PSfrag 591 1756 a 576 1869 a { 576 1869 a -32 w Fl(T)595 1873 y Fz(1)p 595 1881 29 3 v 595 1914 a(2)576 1869 y } 19/Place PSfrag 576 1869 a 585 1999 a { 585 1999 a 562 2017 a Fl(T)603 2026 y Fj(1)585 1999 y } 20/Place PSfrag 585 1999 a 600 2111 a { 600 2111 a 562 2128 a Fl(T)603 2140 y Fk(\022)600 2111 y } 21/Place PSfrag 600 2111 a 71 2322 a FI(Figure)h(1:)43 b FF(Rational)28 b(Arnold)g(tongues)f(in) i(the)g(parameter)e(space)h(of)g(the)h(standard)f(family)g(up)h(to)f (denominator)g(5)71 2435 y(\(note)k(that)g(these)f(are)g(sets)g(with)h (in)n(terior\).)49 b(The)31 b(curv)n(es)g(corresp)r(ond)f(to)h(the)h (irrational)e(tongues)h(for)g FD(\015)j FF(=)3668 2354 y Fy(p)p 3722 2354 34 3 v 3722 2402 a FB(5)p Fy(\000)p FB(1)p 3668 2416 173 4 v 3738 2463 a(2)71 2547 y FF(and)27 b FD(\022)f FF(=)407 2508 y Fz(5)389 2478 y Fw(p)p 459 2478 42 4 v 459 2547 a FF(2)18 b Fw(\000)g FF(1.)71 2821 y FI(analytically)29 b(linearizable)g(is)h(equiv)-5 b(alen)m(t)31 b(to)h(the)f(existence)g(of)h(a)f Fr(Herman)j(ring)2968 2798 y Fn(e)2955 2821 y Fm(U)3017 2835 y Fl(")3085 2821 y FI(for)3242 2798 y Fn(e)3225 2821 y Fm(F)3283 2839 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")3472 2821 y FI(,)e(whic)m(h)d(is)71 2953 y(giv)m(en)j(b)m(y)451 2930 y Fn(e)438 2953 y Fm(U)500 2967 y Fl(")565 2953 y FI(:=)40 b Fn(e)-63 b Fm(')748 2967 y Fl(")785 2953 y FI(\()842 2930 y Fn(e)820 2953 y Fm(A)888 2967 y Fl(")925 2953 y FI(\).)46 b(In)1160 2930 y Fn(e)1147 2953 y Fm(U)1209 2967 y Fl(")1246 2953 y FI(,)32 b(ev)m(ery)h(orbit)e(under)2047 2930 y Fn(e)2029 2953 y Fm(F)2087 2971 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")2308 2953 y FI(lies)g(on)h(an)g(in)m(v)-5 b(arian)m(t)31 b(closed)h(curv)m(e)h(whic)m(h)71 3066 y(has)40 b(rotation)h(n)m(um)m(b)s(er)e Fm(\022)s FI(.)71 b(Since)51 b Fn(e)-63 b Fm(')1391 3080 y Fl(")1469 3066 y FI(is)40 b(unique)e(mo)s(dulus)g(the)j(comp)s(osition)e(with)g(an)m (y)i(rotation,)j(the)71 3178 y(constan)m(t)455 3155 y Fn(e)438 3178 y Fm(R)507 3192 y Fl(")574 3178 y FI(is)29 b(univ)m(o)s(cally)f(de\014ned)h(and)h(w)m(e)h(will)d(call)h(it)h Fr(the)j(size)38 b FI(of)30 b(the)h(Herman)f(ring.)212 3294 y(The)e(main)g(goal)h(of)g(this)e(pap)s(er)h(is)f(to)j(giv)m(e)f (an)f(asymptotic)h(estimate)h(for)e(the)h(size)3161 3271 y Fn(e)3144 3294 y Fm(R)3213 3308 y Fl(")3278 3294 y FI(of)g(the)f(Herman)71 3407 y(ring)271 3384 y Fn(e)258 3407 y Fm(U)320 3421 y Fl(")387 3407 y FI(as)j Fm(")25 b Fo(!)h FI(0.)212 3522 y(The)42 b(sharp)s(est)f(results)f(concerning)i (the)g(size)g(of)g(Herman)g(rings)f(for)h(univ)-5 b(alen)m(t)40 b(maps)i(on)g(a)g(giv)m(en)71 3635 y(ann)m(ulus)f(are)j(due)e(to)i(Y)-8 b(o)s(ccoz)45 b(\(see)g(Theorems)d(2.1)j(and)d(2.3\),)48 b(who)43 b(giv)m(es)h(an)f(estimate)h(that)f(can)h(b)s(e)71 3748 y(applied)26 b(to)i Fr(any)37 b FI(analytic)28 b(map)f Fm(F)41 b FI(that)29 b(lea)m(v)m(es)g(the)f(unit)f(circle)g Ft(C)2423 3762 y FJ(1)2490 3748 y FI(in)m(v)-5 b(arian)m(t)27 b(and)h(has)f(rotation)i(n)m(um)m(b)s(er)71 3861 y Fm(\022)40 b FI(\(i.e.,)g(a)e(lift)e(on)h Fq(R)47 b FI(of)37 b Fm(F)1009 3880 y Fp(j)p Fi(C)1088 3889 y Fj(1)1165 3861 y FI(has)g(rotation)h(n)m (um)m(b)s(er)e Fm(\022)s FI(\),)j(and)e(whic)m(h)g(dep)s(ends)f(only)g (on)i Fm(\022)i FI(and)d(on)g(the)71 3974 y(size)31 b(of)g(the)g (domain)f(where)g(the)h(map)g(is)f(univ)-5 b(alen)m(t.)41 b(In)30 b(Section)h(3)g(w)m(e)g(will)e(see)i(that)h(this)d(general)i (result)71 4087 y(applied)d(to)j(the)g(complex)f(standard)f(family)g (leads)h(to)1716 4319 y Fn(e)1699 4342 y Fm(R)1768 4356 y Fl(")1830 4342 y Fo(\025)25 b Fm(K)2019 4280 y(\033)s FI(\()p Fm(")p FI(\))p 2019 4321 169 4 v 2082 4404 a Fm(")2197 4342 y(;)71 4580 y FI(where)36 b Fm(K)41 b FI(=)35 b(exp\()p Fo(\000)p FI(\010\()p Fm(\022)s FI(\))23 b Fo(\000)h FI(2)p Fm(\031)s(C)1274 4594 y FJ(0)1314 4580 y FI(\),)38 b(\010)e(is)f(the)i Fr(Brjuno)h(function)43 b FI([MMY)q(],)38 b Fm(C)2869 4594 y FJ(0)2945 4580 y FI(is)d(a)i(univ)m(ersal)d(constan)m(t)71 4692 y(and)c Fm(\033)s FI(\()p Fm(")p 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b(shall)d(return)g(to)j(this)e(problem)f(in)g(a)i(momen)m(t,)71 5147 y(but)30 b(\014rst,)f(let)i(us)e(consider)h(what)g(is)f(kno)m(wn)h (as)h(the)f Fr(c)-5 b(omplex)35 b(semistandar)-5 b(d)35 b(map)j FI(of)30 b(parameter)h Fm(e)3607 5114 y FJ(i)p Fl(!)1675 5359 y Fm(G)p FI(\()p Fm(z)t FI(\))26 b(=)f Fm(z)t FI(e)2070 5321 y FJ(i)p Fl(!)2141 5359 y FI(e)2181 5321 y Fl(z)2221 5359 y Fm(:)71 5571 y FI(Observ)m(e)32 b(that)h Fm(z)f FI(=)c(0)k(is)g(a)g(\014xed)f(p)s(oin)m(t)h(of)g Fm(G)g FI(with)f(deriv)-5 b(ativ)m(e)32 b(e)2379 5538 y FJ(i)p Fl(!)2449 5571 y FI(.)46 b(Since)31 b Fm(!)g FI(=)d(2)p Fm(\031)s(\022)s FI(,)33 b(and)e Fm(\022)j FI(is)e(a)g(Brjuno)71 5684 y(n)m(um)m(b)s(er,)e(it)g(is)g(kno)m(wn)h ([Br1,)g(Br2)q(])g(that)g Fm(G)h FI(has)e(a)h Fr(Sie)-5 b(gel)33 b(disk)41 b FI(around)30 b(the)h(origin,)f(whic)m(h)g(w)m(e)h (denote)g(b)m(y)71 5796 y Fm(U)10 b FI(.)47 b(This)31 b(means)i(that)g(if)f(w)m(e)h(call)f Fq(D)1356 5810 y Fl(r)1432 5796 y FI(the)h(op)s(en)f(disk)f(of)i(cen)m(ter)h(0)f(and)f (radius)f Fm(r)s FI(,)i(there)g(exists)f(a)h(unique)p eop %%Page: 5 5 5 4 bop 71 274 a Fs(N.)31 b(F)-8 b(agella,)31 b(T.)f(M.)h(Seara)g(and)f (J.)g(Villan)m(uev)-5 b(a)2071 b FI(5)71 573 y(maximal)29 b(n)m(um)m(b)s(er)g Fm(R)842 587 y FJ(0)907 573 y Fm(>)c FI(0)31 b(and)e(a)i(unique)e(conformal)h(isomorphism)1109 770 y Fm(')25 b FI(:)h Fq(D)1304 784 y Fl(R)1357 793 y Fj(0)1427 770 y Fo(\000)-16 b(!)25 b Fm(U;)197 b(')p FI(\(0\))27 b(=)e(0)p Fm(;)198 b(')2506 733 y Fp(0)2530 770 y FI(\(0\))26 b(=)f(1)922 b(\(5\))71 968 y(that)36 b(conjugates)g Fm(G)f FI(to)h(the)g(rotation)f Fo(R)1539 982 y Fl(!)1590 968 y FI(,)h(i.e.,)h Fm(G)24 b Fo(\016)f Fm(')34 b FI(=)f Fm(')24 b Fo(\016)g(R)2418 982 y Fl(!)2468 968 y FI(.)55 b(The)35 b(n)m(um)m(b)s(er)e Fm(R)3144 982 y FJ(0)3219 968 y FI(is)h(kno)m(wn)h(as)g(the)71 1081 y Fr(c)-5 b(onformal)37 b(r)-5 b(adius)42 b FI(of)33 b(the)h(Siegel)e(disk.)47 b(Standard)32 b(argumen)m(ts)h(sho)m(w)g (that)h Fm(R)2876 1095 y FJ(0)2948 1081 y FI(is)e(alw)m(a)m(ys)i (\014nite.)48 b(Lo)m(w)m(er)71 1194 y(b)s(ounds)39 b(for)j Fm(R)618 1208 y FJ(0)698 1194 y FI(\(as)h(a)f(function)e(of)i Fm(!)s FI(\))f(could)g(b)s(e)g(obtained)g(applying)e(Y)-8 b(o)s(ccoz's)44 b(results)c([Y2)q(])i(to)g(the)71 1306 y(semistandard)29 b(map.)212 1419 y(W)-8 b(e)27 b(no)m(w)e(return)g(to) h(the)f(problem)f(of)i(estimating)f(the)h(size)f(of)h(the)f(Herman)h (ring)3073 1396 y Fn(e)3060 1419 y Fm(U)3122 1433 y Fl(")3159 1419 y FI(.)39 b(Our)24 b(main)g(result)71 1532 y(is)29 b(the)i(follo)m(wing)e(theorem.)71 1714 y Ft(Theorem)i(A.)40 b Fr(L)-5 b(et)31 b Fm(\022)i Fr(b)-5 b(e)31 b(a)g(Brjuno)g(numb)-5 b(er)31 b(and)h(c)-5 b(onsider)32 b(the)f(standar)-5 b(d)34 b(map)2973 1691 y Fn(e)2955 1714 y Fm(F)3013 1728 y Fl(\013;")3115 1714 y FI(\()p Fm(u)p FI(\))26 b(=)f Fm(u)p FI(e)3451 1681 y FJ(i)p Fl(\013)3520 1714 y FI(e)3571 1651 y Fk(")p 3571 1663 31 3 v 3571 1704 a Fj(2)3611 1678 y FJ(\()p Fl(u)p Fp(\000)3747 1651 y Fj(1)p 3744 1663 37 3 v 3744 1704 a Fk(u)3791 1678 y FJ(\))3823 1714 y Fr(,)71 1838 y(with)38 b Fm(\013)e FI(=)e Fm(\013)p FI(\()p Fm(")p FI(\))39 b Fr(such)f(that)1102 1815 y Fn(e)1084 1838 y Fm(F)1142 1856 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")1369 1838 y Fr(r)-5 b(estricte)g(d)40 b(to)e Ft(C)1962 1852 y FJ(1)2039 1838 y Fr(has)g(r)-5 b(otation)40 b(numb)-5 b(er)39 b Fm(\022)s Fr(.)56 b(L)-5 b(et)3195 1815 y Fn(e)3178 1838 y Fm(R)3247 1852 y Fl(")3322 1838 y Fr(b)g(e)37 b(the)h(size)g(of)71 1951 y(its)c(Herman)h(ring)f(and)h (let)g Fm(R)1124 1965 y FJ(0)1197 1951 y Fr(b)-5 b(e)34 b(the)h(c)-5 b(onformal)37 b(r)-5 b(adius)35 b(of)g(the)f(Sie)-5 b(gel)34 b(disk)h(of)f(the)h(semistandar)-5 b(d)37 b(map)71 2064 y Fm(G)p FI(\()p Fm(z)t FI(\))27 b(=)e Fm(z)t FI(e)467 2031 y FJ(i)p Fl(!)537 2064 y FI(e)577 2031 y Fl(z)617 2064 y Fr(,)33 b(wher)-5 b(e)34 b Fm(!)28 b FI(=)d(2)p Fm(\031)s(\022)s Fr(.)41 b(Then,)1485 2276 y Fn(e)1468 2299 y Fm(R)1537 2313 y Fl(")1599 2299 y FI(=)1705 2238 y(2)p 1705 2278 46 4 v 1707 2362 a Fm(")1760 2299 y FI(\()p Fm(R)1864 2313 y FJ(0)1924 2299 y FI(+)20 b Fo(O)s FI(\()p Fm(")15 b FI(log)i Fm(")p FI(\)\))p Fm(:)71 2517 y Ft(Remark)30 b(1.1.)40 b FI(W)-8 b(e)28 b(b)s(eliev)m(e)e(that)i(this)e(is)g(the)i (b)s(est)e(estimate)i(that)g(can)f(b)s(e)g(obtained)f(with)g(our)h (metho)s(ds.)71 2630 y(Ho)m(w)m(ev)m(er,)41 b(some)d(recen)m(t)g(dev)m (elopmen)m(ts)g(\(w)m(ork)f(in)f(progress\))i(seem)f(to)h(indicate)e (that)i(a)g(more)f(optimal)71 2743 y(estimate)i(could)f(b)s(e)843 2720 y Fn(e)826 2743 y Fm(R)895 2757 y Fl(")971 2743 y FI(=)1091 2707 y FJ(2)p 1091 2722 36 4 v 1092 2775 a Fl(")1136 2743 y FI(\()p Fm(R)1240 2757 y FJ(0)1306 2743 y FI(+)25 b Fo(O)1474 2757 y FJ(2)1514 2743 y FI(\()p Fm(")p FI(\)\))40 b(\(with)1969 2720 y Fn(e)1952 2743 y Fm(R)2021 2757 y Fl(")2096 2743 y FI(analytic)e(on)h Fm(")p FI(\).)66 b(As)39 b(a)g(v)-5 b(ague)40 b(indication,)e(this)71 2856 y(could)31 b(follo)m(w)h(from)g(kno)m(wing)f(that)i(the)f (complexi\014cation)g(of)g(the)g(Arnold)f(tongue)i Fm(T)3115 2871 y Fl(\022)3154 2856 y FI(,)g(with)e Fm(\022)j FI(a)f(Brjuno)71 2969 y(n)m(um)m(b)s(er,)25 b(can)g(b)s(e)f(parametrized)h (holomorphically)d(b)m(y)j(a)g(complex)g(parameter)g(strongly)g (related)f(with)g(the)71 3082 y(mo)s(dulus)k(of)i(the)h(ring.)212 3264 y(An)44 b(analogue)g(of)g(Theorem)g(A,)h(for)e Fr(Chirikov's)j (standar)-5 b(d)49 b(and)d(semistandar)-5 b(d)48 b(maps)53 b FI(of)44 b Fq(R)3588 3231 y FJ(2)3633 3264 y FI(,)k(w)m(as)71 3377 y(pro)m(v)m(ed)22 b(in)f([SV)o(])h(using)f(KAM)h(metho)s(ds)f(and) g(complex)h(matc)m(hing,)i(and)d(therefore)h(restricting)f(the)h (result)f(to)71 3490 y(Diophan)m(tine)k(rotation)h(n)m(um)m(b)s(ers.)38 b(In)25 b(the)h(presen)m(t)f(pap)s(er,)h(w)m(e)g(pro)m(v)m(e)h(Theorem) f(A)f(using)g(quasiconformal)71 3603 y(surgery)-8 b(,)36 b(inspired)c(b)m(y)i(a)i(qualitativ)m(e)e(construction)h(of)g(Gey)m(er) h(in)d([G)q(])i(whic)m(h)f(relates)h(the)g(standard)f(and)71 3716 y(the)h(semistandard)f(maps.)55 b(W)-8 b(e)36 b(shall)e(mo)s(dify) f(this)i(construction)g(b)m(y)g(in)m(tro)s(ducing)e(the)i(dep)s (endence)f(on)71 3829 y(the)28 b(parameter)g Fm(")g FI(and)f(b)m(y)g (making)g(most)h(of)g(its)f(ingredien)m(ts)f(completely)i(explicit.)38 b(These)27 b(additions)f(will)71 3941 y(giv)m(e)31 b(us)e(the)i(p)s (ossibilit)m(y)c(of)j(obtaining)f(quan)m(titativ)m(e)i(estimates)g (from)f(the)h(geometric)g(construction.)212 4054 y(From)41 b(all)e(the)i(partial)e(results)g(in)m(v)m(olv)m(ed)h(in)g(the)g(pro)s (of)g(of)h(Theorem)f(A)h(w)m(e)g(c)m(ho)s(ose)g(the)g(follo)m(wing)71 4167 y(one)36 b(to)h(b)s(e)f(remark)m(ed)g(here,)h(b)s(ecause)g(of)f (its)f(in)m(terest)i(in)d(itself)h(and)h(its)f(p)s(ossible)f(use)i(in)f (other)h(surgery)71 4280 y(constructions.)k(Under)28 b(certain)h(conditions,)f(this)f(result)h(giv)m(es)i(an)f(estimate)g (for)g(ho)m(w)g(close)g(the)g(Ahlfors-)71 4393 y(Bers')i(map)f(\(see)h (Theorem)f(2.12\))i(is)e(to)h(the)f(iden)m(tit)m(y)-8 b(.)212 4506 y(Throughout)23 b(the)h(pap)s(er,)h Fm(\016)1177 4520 y Fp(U)1256 4506 y FI(indicates)e(the)i(h)m(yp)s(erb)s(olic)c (distance)j(inside)e Fo(U)9 b FI(,)26 b(where)d Fo(U)33 b FI(is)23 b(a)i(h)m(yp)s(erb)s(olic)71 4619 y(set)31 b(\(see)g(Section)f(2.3\).)42 b(The)30 b(notation)h Fo(k)21 b(\001)f(k)31 b FI(denotes)g(the)f(in\014nit)m(y)e(norm)i(and)g Fq(D)48 b FI(=)25 b Fq(D)3127 4633 y FJ(1)3173 4619 y FI(.)71 4801 y Ft(Prop)s(osition)35 b(B.)41 b Fr(L)-5 b(et)32 b Fm(\026)g Fr(b)-5 b(e)31 b(a)h(Beltr)-5 b(ami)34 b(form)e(on)h Fq(C)55 b Fr(\(se)-5 b(e)32 b(Se)-5 b(ction)32 b(2.2\))h(and)g Fm(h)25 b FI(:)h Fq(C)49 b Fo(!)25 b Fq(C)55 b Fr(b)-5 b(e)32 b(the)g(unique)71 4914 y(quasic)-5 b(onformal)37 b(solution)e(of)f(the)h(Beltr)-5 b(ami)36 b(e)-5 b(quation)2055 4878 y Fl(@)t(h)p 2055 4893 82 4 v 2058 4945 a(@)p 2099 4908 36 3 v 4 w(z)2175 4914 y FI(=)27 b Fm(\026)2338 4878 y Fl(@)t(h)p 2338 4893 82 4 v 2341 4945 a(@)t(z)2464 4914 y Fr(\014xing)34 b(0)h(and)g(1)g (\(se)-5 b(e)34 b(The)-5 b(or)g(em)36 b(2.12\).)71 5027 y(Then:)241 5140 y FI(\(a\))47 b Fr(F)-7 b(or)34 b(any)f Fm(z)c Fo(2)c Fq(C)44 b Fo(n)21 b(f)p FI(0)p Fm(;)15 b FI(1)p Fo(g)p Fr(,)34 b(we)f(have)g Fm(\016)1722 5158 y Fh(C)10 b Fp(nf)p FJ(0)p Fl(;)p FJ(1)p Fp(g)1971 5140 y FI(\()p Fm(z)t(;)15 b(h)p FI(\()p Fm(z)t FI(\)\))28 b Fo(\024)d Fm(\016)2459 5154 y Fh(D)2511 5140 y FI(\(0)p Fm(;)15 b Fo(jj)p Fm(\026)p Fo(jj)p FI(\))p Fr(.)241 5289 y FI(\(b\))46 b Fr(Ther)-5 b(e)30 b(exists)f(a)g(universal)h(c)-5 b(onstant)30 b FI(0)c Fm(<)f(\032)g(<)g FI(1)k Fr(such)h(that)g(if)e Fo(k)p Fm(\026)p Fo(k)e(\024)f Fm(\032)p Fr(,)30 b(then)f(for)h(any)f Fm(z)h Fo(2)24 b Fq(D)3708 5256 y Fp(\003)3708 5311 y Fl(\032)3780 5289 y FI(=)287 5402 y Fq(D)346 5416 y Fl(\032)413 5402 y Fo(n)d(f)p FI(0)p Fo(g)33 b Fr(verifying)g Fo(k)p Fm(\026)p Fo(kj)15 b FI(log)i Fo(j)p Fm(z)t Fo(jj)27 b(\024)e Fm(\032)p Fr(,)32 b(one)h(has:)1497 5599 y Fo(j)p Fm(h)p FI(\()p Fm(z)t FI(\))22 b Fo(\000)e Fm(z)t Fo(j)26 b(\024)f Fm(C)7 b Fo(jj)p Fm(\026)p Fo(jjj)p Fm(z)t Fo(jj)15 b FI(log)j Fo(j)p Fm(z)t Fo(jj)p Fm(;)287 5796 y Fr(wher)-5 b(e)33 b Fm(C)f(>)25 b FI(0)33 b Fr(only)h(dep)-5 b(ends)34 b(on)f Fm(\032)p Fr(.)p eop %%Page: 6 6 6 5 bop 71 274 a FI(6)1046 b Fs(Asymptotic)30 b(Size)g(of)h(Herman)f (Rings)f(of)i(the)f(Complex)g(Standard)f(F)-8 b(amily)212 573 y FI(The)25 b(pap)s(er)f(is)g(organized)h(as)h(follo)m(ws.)38 b(Section)25 b(2)h(con)m(tains)f(basic)g(in)m(tro)s(ductions)e(to)j (some)g(of)f(the)g(to)s(ols)71 686 y(and)34 b(preliminary)d(results)j (that)h(will)d(b)s(e)i(used)g(during)f(the)i(pro)s(ofs)f(of)h(Theorem)f (A)h(and)f(Prop)s(osition)f(B.)71 799 y(The)e(exp)s(ert)h(reader)g(can) g(go)h(directly)e(to)h(Section)g(3,)h(where)f(the)g(problem)e(is)h (scaled)h(and)f(restated)i(more)71 912 y(precisely)-8 b(.)48 b(Section)33 b(4)h(and)f(Section)g(5)h(con)m(tain)f(the)h (actual)f(pro)s(ofs)g(of)g(Theorem)g(A)h(and)e(Prop)s(osition)g(B,)71 1024 y(resp)s(ectiv)m(ely)-8 b(.)71 1322 y Fu(2)135 b(Preliminaries)71 1529 y FI(In)31 b(this)f(section)h(w)m(e)h(state)h(the)f(basic)e (results)h(that)h(w)m(e)f(need)h(to)g(pro)m(v)m(e)g(Theorem)f(A)h(and)e (Prop)s(osition)g(B.)71 1642 y(In)38 b(Section)h(2.1)h(w)m(e)g(review)e (previous)g(results)g(ab)s(out)h(the)g(linearization)e(of)i(analytic)g (circle)f(maps)h(and)71 1754 y(their)24 b(translation)h(to)h(maps)f(of) h(the)g(complex)f(plane)g(ha)m(ving)g(an)g(in)m(v)-5 b(arian)m(t)25 b(circle.)38 b(Section)51 b(2.2)27 b(is)d(dev)m(oted)71 1867 y(to)38 b(quasiconformal)e(mappings)f(and)i(Ahlfors-Bers')g (theorem.)62 b(Finally)-8 b(,)38 b(in)e(Section)h(2.3)h(w)m(e)g(giv)m (e)g(some)71 1980 y(de\014nitions)28 b(and)i(results)f(in)g(h)m(yp)s (erb)s(olic)e(geometry)-8 b(.)71 2235 y Fg(2.1)112 b(Analytic)36 b(linearization)71 2410 y FI(Let)31 b Fm(f)k FI(:)25 b Fq(T)423 2377 y FJ(1)490 2410 y Fo(!)g Fq(T)665 2377 y FJ(1)737 2410 y FI(b)s(e)k(an)i(orien)m(tation)f(preserving)f (homeomorphism)g(of)i(the)f(circle)g Fq(T)3127 2377 y FJ(1)3193 2410 y FI(=)c Fq(R)r Fm(=)q Fq(Z)p FI(,)32 b(with)d(the)71 2523 y(normalization)g Fm(f)10 b FI(\(0\))26 b Fo(2)e FI([0)p Fm(;)15 b FI(1\).)43 b(T)-8 b(o)31 b(suc)m(h)f(a)h (map)f(one)g(can)h(assign)f(a)g Fr(r)-5 b(otation)35 b(numb)-5 b(er)41 b FI(de\014ned)29 b(as)1498 2781 y Fm(\032)p FI(\()p Fm(f)10 b FI(\))26 b(=)53 b(lim)1792 2835 y Fl(n)p Fp(!1)2001 2719 y Fm(f)2056 2686 y Fl(n)2102 2719 y FI(\()p Fm(x)p FI(\))21 b Fo(\000)f Fm(x)p 2001 2760 387 4 v 2167 2843 a(n)2398 2781 y(;)71 3026 y FI(where)31 b Fm(x)g FI(is)f(an)m(y)i(p)s(oin)m(t)e(in)g Fq(T)1087 2993 y FJ(1)1128 3026 y FI(.)43 b(It)31 b(is)f(w)m(ell)h(kno)m(wn)f (\(see)i(e.)g(g.)g([dMvS]\))f(that)h Fm(f)41 b FI(b)s(eing)30 b(a)h(homeomorphism)71 3138 y(guaran)m(tees)f(that)g(this)e(limit)f (exists)i(and)f(is)g(indep)s(enden)m(t)f(of)j(the)f(p)s(oin)m(t)f Fm(x)p FI(.)40 b(With)29 b(this)f(de\014nition,)f Fm(\032)p FI(\()p Fm(f)10 b FI(\))29 b(is)71 3251 y(a)i(rational)e(n)m(um)m(b)s (er)g(if)g(and)h(only)f(if)h Fm(f)39 b FI(has)30 b(a)h(p)s(erio)s(dic)d (orbit.)39 b(If)30 b Fm(\032)p FI(\()p Fm(f)10 b FI(\))31 b(is)e(irrational,)g(all)g(orbits)g(are)i(dense)71 3364 y(in)e(the)h(circle.)40 b(W)-8 b(e)32 b(are)f(in)m(terested)f(in)f (maps)h(with)f(an)i(irrational)d(rotation)j(n)m(um)m(b)s(er.)212 3479 y(If)38 b(the)h(rotation)g(n)m(um)m(b)s(er)e(of)i Fm(f)48 b FI(is)37 b(an)i(irrational)d(n)m(um)m(b)s(er)i Fm(\022)i FI(and)e Fm(f)49 b Fo(2)38 b(C)2872 3446 y FJ(2)2912 3479 y FI(\()p Fq(T)3006 3446 y FJ(1)3047 3479 y FI(\),)j(Denjo)m(y's)f(theorem)71 3592 y(\(see)33 b([dMvS]\))f (ensures)g(that)h Fm(f)41 b FI(is)31 b(top)s(ologically)h(conjugate)h (to)g(the)f(rigid)f(rotation)h(of)h(angle)f Fm(\022)s FI(,)g Fo(T)3590 3607 y Fl(\022)3629 3592 y FI(\()p Fm(x)p FI(\))d(=)71 3705 y Fm(x)19 b FI(+)g Fm(\022)s FI(.)39 b(That)30 b(is,)f(there)h(exists)g(a)g(homeomorphism)e Fm(\021)g FI(:)e Fq(T)2108 3672 y FJ(1)2174 3705 y Fo(!)f Fq(T)2349 3672 y FJ(1)2420 3705 y FI(suc)m(h)30 b(that)g Fm(\021)22 b Fo(\016)e(T)3003 3720 y Fl(\022)3067 3705 y FI(=)25 b Fm(f)j Fo(\016)20 b Fm(\021)s FI(,)30 b(making)f(the)71 3818 y(follo)m(wing)g(diagram)g(comm)m(ute:)1691 3959 y Fq(T)1750 3926 y FJ(1)1924 3904 y Fp(T)1963 3916 y Fk(\022)1836 3959 y Fo(\000)-42 b(\000)-21 b(\000)h(\000)-43 b(!)46 b Fq(T)2190 3926 y FJ(1)1673 4120 y Fl(\021)1710 4031 y Fn(?)1710 4085 y(?)1710 4140 y(y)2150 4031 y(?)2150 4085 y(?)2150 4140 y(y)2211 4120 y Fl(\021)1691 4332 y Fq(T)1750 4299 y FJ(1)1940 4276 y Fl(f)1836 4332 y Fo(\000)-42 b(\000)-21 b(\000)h(\000)-43 b(!)46 b Fq(T)2190 4299 y FJ(1)71 4469 y FI(If)30 b(w)m(e)h(require)e Fm(\021)s FI(\(0\))d(=)f(0)31 b(then)f(the)h(conjugacy)g(is)e(unique.)212 4584 y(W)-8 b(e)45 b(restrict)f(from)f(no)m(w)h(on)g(to)h(the)f(case)h (where)e Fm(f)54 b FI(is)43 b(an)g(analytic)h(di\013eomorphism)d(of)j Fq(T)3590 4551 y FJ(1)3631 4584 y FI(,)k(and)71 4697 y(therefore)31 b(it)f(can)g(b)s(e)g(extended)g(to)h(a)g(complex)f (strip,)f(of)i(certain)f(width)f(\001)c Fm(>)g FI(0,)31 b(around)e Fq(T)3331 4664 y FJ(1)3372 4697 y FI(:)1320 4907 y Fo(A)1393 4921 y FJ(\001)1481 4907 y FI(=)c Fo(f)p Fm(z)30 b Fo(2)25 b Fq(C)17 b Fm(=)q Fq(Z)27 b FI(:)55 b Fo(j)p FI(Im\()p Fm(z)t FI(\))p Fo(j)27 b Fm(<)e FI(\001)p Fo(g)p Fm(:)1133 b FI(\(6\))71 5117 y(Abusing)26 b(notation,)j(w)m(e)f (shall)e(denote)j(this)d(extension)i(again)g(b)m(y)f Fm(f)10 b FI(.)40 b(If)27 b(the)h(conjugacy)h Fm(\021)i FI(is)c(also)g(analytic,)71 5230 y(the)h(map)g Fm(f)37 b FI(is)27 b(said)g(to)i(b)s(e)e Fr(analytic)-5 b(al)5 b(ly)33 b(line)-5 b(arizable)p FI(.)41 b(Then)27 b(again,)i Fm(\021)i FI(can)d(b)s(e)g(extended)g(to)h(a)f(neigh)m(b)s(our-)71 5343 y(ho)s(o)s(d)36 b(of)h(the)h(circle,)g(and)f(it)g(is)f(easy)i(to)g (c)m(hec)m(k)h(\(b)m(y)e(the)h(principle)33 b(of)38 b(analytic)f(con)m (tin)m(uation\))g(that)h(its)71 5456 y(extension)30 b(also)g (conjugates)i Fm(f)39 b FI(to)31 b Fo(T)1348 5471 y Fl(\022)1417 5456 y FI(wherev)m(er)g Fm(\021)i FI(is)d(de\014ned.)212 5571 y(W)-8 b(e)33 b(are)g(particularly)c(in)m(terested)j(in)f(the)h (case)h(where)e Fm(F)41 b FI(:)28 b Fo(U)37 b(\032)28 b Fq(C)51 b Fo(!)28 b Fq(C)56 b FI(is)31 b(an)h(analytic)f(map)h(ha)m (ving)71 5684 y(the)41 b(unit)e(circle)h Ft(C)764 5698 y FJ(1)843 5684 y FI(in)m(v)-5 b(arian)m(t,)43 b(and)d Fm(f)49 b FI(is)40 b(the)h(lift)e(to)i Fq(T)2147 5651 y FJ(1)2229 5684 y FI(of)g Fm(F)2401 5702 y Fp(j)p Fi(C)2480 5711 y Fj(1)2518 5684 y FI(.)72 b(In)39 b(this)h(case,)k(w)m(e)d(sa)m (y)h(that)f Fm(F)3733 5702 y Fp(j)p Fi(C)3812 5711 y Fj(1)71 5796 y FI(is)35 b(analytically)g(linearizable)f(if)h(there)i (exists)f(an)g(analytic)g(di\013eomorphism)d Fm(')j FI(:)f Ft(C)3095 5810 y FJ(1)3169 5796 y Fo(!)g Ft(C)3371 5810 y FJ(1)3410 5796 y FI(,)j(suc)m(h)e(that)p eop %%Page: 7 7 7 6 bop 71 274 a Fs(N.)31 b(F)-8 b(agella,)31 b(T.)f(M.)h(Seara)g(and)f (J.)g(Villan)m(uev)-5 b(a)2071 b FI(7)71 573 y Fm(')21 b Fo(\016)h(R)295 587 y Fl(!)373 573 y FI(=)k Fm(F)35 b Fo(\016)21 b Fm(')p FI(,)33 b(where)e Fo(R)1087 587 y Fl(!)1137 573 y FI(\()p Fm(u)p FI(\))d(=)f(e)1425 540 y FJ(i)p Fl(!)1495 573 y Fm(u)32 b FI(and)f Fm(!)f FI(=)d(2)p Fm(\031)s(\022)s FI(.)44 b(If)31 b(w)m(e)h(ask)g Fm(')p FI(\(1\))d(=)e(1,)32 b Fm(')g FI(is)f(univ)m(o)s(cally)f(de\014ned)71 686 y(and)g(the)g(relation)g(b)s(et)m(w)m(een)h Fm(\021)i FI(and)d Fm(')h FI(is)e(giv)m(en)i(b)m(y)1351 908 y Fm(')p FI(\(e)1485 871 y FJ(2)p Fl(\031)r FJ(i)p Fl(x)1628 908 y FI(\))26 b(=)f(e)1825 871 y FJ(2)p Fl(\031)r FJ(i)p Fl(\021)r FJ(\()p Fl(x)p FJ(\))2059 908 y Fm(;)198 b(x)25 b Fo(2)g Fq(T)2504 871 y FJ(1)2545 908 y Fm(:)71 1130 y FI(The)37 b(image)i(b)m(y)f Fm(')g FI(of)h(the)f(maximal)f(ann)m (ulus)f(where)i Fm(')g FI(can)h(b)s(e)e(analytically)g(con)m(tin)m(ued) h(is)f(called)g(the)71 1243 y Fr(Herman)42 b(ring)48 b FI(of)40 b Fm(F)13 b FI(.)70 b(If)40 b Fm(R)h FI(is)e(the)h(outer)g (radius)f(of)h(this)f(ann)m(ulus)f(\(in)h(the)h(understanding)e(that)i (this)71 1356 y(ann)m(ulus)28 b(is)i(symmetric)f(with)g(resp)s(ect)i (to)g(the)g(unit)d(circle,)i(and)g(then)g(it)g(is)g(of)g(the)h(form)f Fm(A)p FI(\(1)p Fm(=R)q(;)15 b(R)q FI(\)\),)32 b(the)71 1469 y(width)f(of)j(the)g(strip)e(of)h(analyticit)m(y)g(of)h Fm(f)42 b FI(around)33 b Fq(T)1940 1436 y FJ(1)2014 1469 y FI(is)2140 1433 y FJ(1)p 2119 1448 79 4 v 2119 1500 a(2)p Fl(\031)2222 1469 y FI(log)16 b Fm(R)q FI(.)50 b(The)33 b(quan)m(tit)m(y)3069 1433 y FJ(1)p 3065 1448 43 4 v 3065 1500 a Fl(\031)3133 1469 y FI(log)17 b Fm(R)34 b FI(is)f(called)f(the)71 1582 y Fr(mo)-5 b(dulus)34 b(of)f(the)g(ring)38 b FI(and)30 b(w)m(e)h(will)d(call)h Fm(R)i FI(the)g Fr(size)i(of)f(the)h(ring)p FI(.)212 1701 y(Arnold)d(sho)m(w)m(ed)i(in)f([Ar])h(that)g(if)f Fm(\022)i FI(is)e(a)h(Diophan)m(tine)f(n)m(um)m(b)s(er)g(and)g Fm(f)41 b FI(is)30 b(close)i(enough)g(to)g(the)g(rigid)71 1814 y(rotation)21 b Fo(T)460 1829 y Fl(\022)498 1814 y FI(,)i(then)e Fm(f)29 b FI(is)20 b(analytically)f(linearizable.)35 b(This)19 b(result)h(w)m(as)h(later)g(impro)m(v)m(ed)f(b)m(y)g(R)s (\177)-48 b(ussmann)19 b([Ru1,)71 1927 y(Ru2],)33 b(Herman)f([Her1)q(,) h(Her2)q(])f(and)g(Y)-8 b(o)s(ccoz)35 b([Y2)q(,)d(PM)q(].)47 b(The)32 b(sharp)s(est)f(results)g(are)i(due)f(to)h(Y)-8 b(o)s(ccoz)35 b(and)71 2040 y(w)m(e)c(state)i(them)e(b)s(elo)m(w.)42 b(In)30 b(the)h(statemen)m(t,)j(\010)26 b(:)g Fq(R)k Fo(n)21 b Fq(Q)41 b Fo(!)26 b Fq(R)2247 2007 y FJ(+)2333 2040 y Fo([)20 b(f1g)32 b FI(denotes)f(the)h Fr(Brjuno)h(function)p FI(,)f(a)71 2153 y(purely)26 b(arithmetic)g Fq(Z)p Fo(\000)p FI(p)s(erio)s(di)o(c)c(function.)38 b(Its)28 b(most)g(imp)s(ortan)m(t)e (prop)s(ert)m(y)h(is)g(that)h(\010\()p Fm(\013)p FI(\))g(is)f(\014nite) f(if)h(and)71 2266 y(only)i(if)h Fm(\013)g FI(is)g(a)h(Brjuno)e(n)m(um) m(b)s(er)g(\(see)j([MMY)q(])e(for)g(details)g(on)g(\010\).)212 2384 y(The)36 b Fr(L)-5 b(o)g(c)g(al)41 b(Conjugacy)e(The)-5 b(or)g(em)p FI(,)40 b(due)d(to)g(Y)-8 b(o)s(ccoz,)41 b(states)d(that)f(an)m(y)h(analytic)e(circle)g(map)g(with)g(a)71 2497 y(Brjuno)j(rotation)g(n)m(um)m(b)s(er)g(and)g(whic)m(h)f(is)g (univ)-5 b(alen)m(t)38 b(in)h(a)h(su\016cien)m(tly)e(large)h(strip)f (\(where)i(\\large")g(is)71 2610 y(de\014ned)32 b(only)h(in)f(terms)i (of)f(the)h(rotation)g(n)m(um)m(b)s(er\))f(is)f(analytically)g (linearizable.)48 b(Moreo)m(v)m(er,)37 b(it)c(giv)m(es)h(a)71 2723 y(lo)m(w)m(er)i(b)s(ound)d(for)j(the)g(linearization)d(domain)i (whic)m(h,)h(again,)h(only)e(dep)s(ends)e(on)j(the)g(initial)d(domain)h (of)71 2836 y(univ)-5 b(alency)28 b(and)i(the)h(rotation)f(n)m(um)m(b)s (er.)71 3048 y Ft(Theorem)43 b(2.1)i(\(Lo)s(cal)f(Conjugacy)h (Theorem\).)f Fr(L)-5 b(et)41 b Fm(\022)h Fr(b)-5 b(e)40 b(a)g(Brjuno)h(numb)-5 b(er)40 b(and)h FI(\001)d Fm(>)h FI(0)h Fr(such)71 3161 y(that)k FI(\001)f Fm(>)532 3125 y FJ(1)p 510 3140 79 4 v 510 3192 a(2)p Fl(\031)598 3161 y FI(\010\()p Fm(\022)s FI(\))28 b(+)f Fm(C)971 3175 y FJ(0)1010 3161 y Fr(,)45 b(wher)-5 b(e)44 b Fm(C)1415 3175 y FJ(0)1497 3161 y Fr(is)f(a)g(universal)g(c)-5 b(onstant.)74 b(L)-5 b(et)43 b Fm(f)53 b FI(:)44 b Fq(T)2916 3128 y FJ(1)3000 3161 y Fo(!)g Fq(T)3194 3128 y FJ(1)3277 3161 y Fr(b)-5 b(e)43 b(an)g(analytic)71 3274 y(cir)-5 b(cle)32 b(di\013e)-5 b(omorphism,)34 b(orientation)g(pr)-5 b(eserving)32 b(and)h(with)f(r)-5 b(otation)34 b(numb)-5 b(er)32 b Fm(\022)s Fr(.)40 b(We)32 b(assume)g(that)h Fm(f)40 b Fr(is)71 3387 y(holomorphic)h(and)e(univalent)f(in)g(the)g (strip)h Fo(A)1739 3401 y FJ(\001)1839 3387 y Fr(\(se)-5 b(e)38 b(\(6\)\).)59 b(Then,)39 b Fm(f)47 b Fr(is)38 b(analytic)-5 b(al)5 b(ly)40 b(line)-5 b(arizable)39 b(and)71 3499 y(the)33 b(line)-5 b(arization)35 b Fm(\021)28 b FI(:)e Fq(T)929 3466 y FJ(1)995 3499 y Fo(!)f Fq(T)1170 3466 y FJ(1)1244 3499 y Fr(is)33 b(analytic)g(in)g(the)g(c)-5 b(omplex)34 b(strip)g Fo(A)2578 3514 y Fl(d)2618 3499 y Fr(,)e(with)1512 3755 y Fm(d)25 b Fo(\025)g FI(\001)20 b Fo(\000)1904 3693 y FI(1)p 1877 3734 101 4 v 1877 3817 a(2)p Fm(\031)1987 3755 y FI(\010\()p Fm(\022)s FI(\))g Fo(\000)g Fm(C)2345 3769 y FJ(0)2384 3755 y Fm(;)71 4003 y Fr(and)33 b(veri\014es)g Fm(\021)s FI(\()p Fo(A)716 4018 y Fl(d)757 4003 y FI(\))25 b Fo(\032)g(A)986 4017 y FJ(\001)1049 4003 y Fr(.)71 4215 y Ft(Remark)40 b(2.2.)45 b FI(F)-8 b(or)36 b(the)f(Arnold)f(standard)g(family)g(\(1\))i(a)g (sort)f(of)g(recipro)s(cal)f(is)h(also)g(true.)55 b(Indeed,)36 b(it)71 4328 y(w)m(as)g(sho)m(wn)g(in)f([G])i(that)f(if)f(a)i(mem)m(b)s (er)e(of)i(the)f(Arnold)f(standard)g(family)g(is)g(analytically)f (linearizable,)71 4441 y(then)c(its)g(rotation)g(n)m(um)m(b)s(er)g(m)m (ust)g(b)s(e)f(Brjuno.)212 4652 y(If)f Fm(F)41 b FI(is)28 b(a)h(holomorphic)d(map)i(lea)m(ving)h(the)f(unit)f(circle)h(in)m(v)-5 b(arian)m(t,)28 b(and)g(b)m(y)h(applying)d(Theorem)i(2.1)i(to)71 4765 y(the)g(lift)f Fm(f)40 b FI(of)31 b Fm(F)618 4784 y Fp(j)p Fi(C)697 4793 y Fj(1)735 4765 y FI(,)g(w)m(e)f(can)h(state)h (an)e(analogous)h(result)e(ab)s(out)h(the)h(analytic)f(linearization)e (of)j Fm(F)13 b FI(.)71 4990 y Ft(Theorem)24 b(2.3.)35 b Fr(L)-5 b(et)26 b Fm(\022)h Fr(b)-5 b(e)24 b(a)i(Brjuno)f(numb)-5 b(er)25 b(and)h Fm(R)g(>)f FI(1)h Fr(such)f(that)h Fm(R)g(>)f FI(e)2753 4957 y FJ(\010\()p Fl(\022)r FJ(\)+2)p Fl(\031)r(C)3076 4966 y Fj(0)3116 4990 y Fr(.)39 b(L)-5 b(et)25 b Fm(F)39 b FI(:)25 b Ft(C)3555 5004 y FJ(1)3619 4990 y Fo(!)g Ft(C)3811 5004 y FJ(1)71 5103 y Fr(b)-5 b(e)35 b(an)h(analytic)g (di\013e)-5 b(omorphism)39 b(with)d(r)-5 b(otation)38 b(numb)-5 b(er)36 b Fm(\022)s Fr(.)49 b(We)35 b(assume)h(that)g Fm(F)48 b Fr(is)36 b(holomorphic)i(and)71 5216 y(univalent)f(in)g(the)h (annulus)f Fm(A)p FI(\(1)p Fm(=R)q(;)15 b(R)q FI(\))p Fr(.)57 b(Then,)38 b Fm(F)50 b Fr(is)37 b(analytic)-5 b(al)5 b(ly)39 b(line)-5 b(arizable)39 b(and)f(the)f(line)-5 b(arization)71 5329 y Fm(')26 b FI(:)f Ft(C)282 5343 y FJ(1)346 5329 y Fo(!)g Ft(C)538 5343 y FJ(1)610 5329 y Fr(is)32 b(analytic)i(in)f(the)g(annulus)g Fm(A)p FI(\(1)p Fm(=r)m(;)15 b(r)s FI(\))p Fr(,)35 b(with)1602 5551 y Fm(r)27 b Fo(\025)e Fm(R)q FI(e)1876 5514 y Fp(\000)p FJ(\010\()p Fl(\022)r FJ(\))p Fp(\000)p FJ(2)p Fl(\031)r(C)2254 5523 y Fj(0)2294 5551 y Fm(;)71 5774 y Fr(and)33 b(veri\014es)g Fm(')p FI(\()p Fm(A)p FI(\(1)p Fm(=r)m(;)15 b(r)s FI(\)\))29 b Fo(\032)c Fm(A)p FI(\(1)p Fm(=R)q(;)15 b(R)q FI(\))p Fr(.)p eop %%Page: 8 8 8 7 bop 71 274 a FI(8)1046 b Fs(Asymptotic)30 b(Size)g(of)h(Herman)f (Rings)f(of)i(the)f(Complex)g(Standard)f(F)-8 b(amily)71 573 y Fg(2.2)112 b(Quasiconformal)37 b(mappings)g(and)i(the)e(Beltrami) e(equation)71 747 y FI(In)c(this)f(section)i(w)m(e)h(recall)e(shortly)f (the)i(relev)-5 b(an)m(t)32 b(de\014nitions)e(and)h(results)f(to)j(b)s (e)e(used)g(in)f(the)i Fr(quasic)-5 b(on-)71 860 y(formal)32 b(sur)-5 b(gery)37 b FI(pro)s(cedure,)28 b(whic)m(h)f(is)g(going)i(to)g (b)s(e)e(one)i(of)f(the)h(main)e(to)s(ols)h(to)h(pro)m(v)m(e)g(the)g (results)e(of)h(this)71 973 y(pap)s(er.)49 b(The)33 b(standard)g (reference)h(for)f(quasiconformal)f(mappings)g(is)h([Ah].)50 b(In)33 b(this)g(section,)h Fo(U)9 b Fm(;)15 b Fo(V)38 b(\032)31 b Fq(C)71 1086 y FI(are)g(op)s(en)e(sets.)71 1279 y Ft(De\014nition)37 b(2.4.)43 b FI(Giv)m(en)32 b(a)h(measurable)e(function)g Fm(\026)c FI(:)i Fo(U)37 b(!)28 b Fq(C)18 b FI(,)38 b(w)m(e)33 b(sa)m(y)g(that)f Fm(\026)g FI(is)f(a)i Fm(k)s Fo(\000)p Fr(Beltr)-5 b(ami)35 b(form)71 1391 y FI(of)d Fo(U)40 b FI(if)30 b Fo(j)p Fm(\026)p FI(\()p Fm(z)t FI(\))p Fo(j)f(\024)e Fm(k)j(<)d FI(1)32 b(almost)f(ev)m(erywhere)h(in)e Fo(U)9 b FI(.)44 b(Tw)m(o)32 b(Beltrami)f(forms)g(of)g Fo(U)41 b FI(are)32 b(equiv)-5 b(alen)m(t)31 b(if)f(they)71 1504 y(coincide)f(almost)i(ev)m (erywhere)g(in)e Fo(U)9 b FI(.)212 1697 y(Equiv)-5 b(alen)m(tly)d(,)43 b(a)f(Beltrami)f(form)g(of)h Fo(U)50 b FI(giv)m(es)42 b(an)f Fr(almost)k(c)-5 b(omplex)44 b(structur)-5 b(e)50 b Fm(\033)s FI(,)44 b(whic)m(h)d(means)g(a)71 1810 y(measurable)36 b(\014eld)f(of)i(ellipses)e(in)h(the)h(tangen)m(t)h(space)g(of)f Fo(U)9 b FI(,)38 b(cen)m(tered)g(at)g(0)f(and)f(de\014ned)g(up)g(to)h (m)m(ulti-)71 1923 y(plication)g(b)m(y)h(a)h(non-zero)g(real)f(constan) m(t.)67 b(The)38 b(argumen)m(t)h(of)f(the)h(ma)5 b(jor)38 b(axis)g(of)h(these)g(in\014nitesimal)71 2036 y(ellipses,)28 b(at)i(the)h(p)s(oin)m(t)d Fm(z)i Fo(2)25 b(U)9 b FI(,)30 b(is)f Fm(\031)s(=)p FI(2)20 b(+)f(arg)r(\()p Fm(\026)p FI(\()p Fm(z)t FI(\)\))p Fm(=)p FI(2,)32 b(and)e(the)g(ratio)g(of)g (minor)e(and)h(ma)5 b(jor)30 b(axes)h(equals)71 2149 y(\(1)21 b Fo(\000)f(j)p Fm(\026)p FI(\()p Fm(z)t FI(\))p Fo(j)p FI(\))p Fm(=)p FI(\(1)j(+)d Fo(j)p Fm(\026)p FI(\()p Fm(z)t FI(\))p Fo(j)p FI(\).)71 2342 y Ft(De\014nition)36 b(2.5.)42 b FI(A)32 b(homeomorphism)d Fm(f)36 b FI(:)26 b Fo(U)36 b(!)26 b(V)38 b FI(is)30 b(said)g(to)i(b)s(e)e Fm(k)s Fo(\000)p Fr(quasic)-5 b(onformal)43 b FI(if)30 b(it)h(has)g(lo)s(cally)71 2455 y(in)m(tegrable)f(w)m(eak)h(deriv)-5 b(ativ)m(es)30 b(and)1672 2638 y Fm(\026)1727 2653 y Fl(f)1772 2638 y FI(\()p Fm(z)t FI(\))c(=)2030 2530 y Fl(@)t(f)p 2030 2551 83 4 v 2033 2603 a(@)p 2074 2565 36 3 v 4 w(z)2122 2572 y FI(\()p Fm(z)t FI(\))p 2020 2617 220 4 v 2030 2675 a Fl(@)t(f)p 2030 2695 83 4 v 2033 2747 a(@)t(z)2122 2716 y FI(\()p Fm(z)t FI(\))71 2877 y(is)36 b(a)h Fm(k)s Fo(\000)p FI(Beltrami)f(form.)59 b(In)36 b(this)f(case,)40 b(w)m(e)d(sa)m(y)h(that)f Fm(\026)2116 2892 y Fl(f)2198 2877 y FI(is)e(the)i Fr(c)-5 b(omplex)40 b(dilatation)46 b FI(or)37 b(the)g Fr(Beltr)-5 b(ami)71 2990 y(c)g(o)g(e\016cient)40 b FI(of)30 b Fm(f)10 b FI(.)71 3183 y Ft(Remark)37 b(2.6.)43 b FI(With)32 b(the)g(same)h (de\014nition,)e(but)h(skipping)d(the)k(h)m(yp)s(othesis)e(on)h Fm(f)42 b FI(to)33 b(b)s(e)f(a)h(homeomor-)71 3296 y(phism,)24 b Fm(f)34 b FI(is)24 b(called)h(a)g Fm(k)s Fo(\000)p Fr(quasir)-5 b(e)g(gular)29 b(map)p FI(.)40 b(It)26 b(is)e(easy)h(to)h (c)m(hec)m(k)h(that)e(a)h Fm(k)s Fo(\000)p FI(quasiregular)d(map)i(is)f (lo)s(cally)71 3408 y(the)30 b(comp)s(osition)g Fm(g)23 b Fo(\016)e Fm(h)31 b FI(of)f(a)h(holomorphic)d(map)i Fm(g)k FI(and)c(a)h Fm(k)s Fo(\000)p FI(quasiconformal)e(map)h Fm(h)p FI(.)71 3601 y Ft(De\014nition)40 b(2.7.)k FI(Giv)m(en)34 b(a)h(Beltrami)e(form)h Fm(\026)g FI(of)h Fo(V)41 b FI(and)34 b(a)h(quasiregular)d(map)i Fm(h)e FI(:)g Fo(U)41 b(!)32 b(V)7 b FI(,)35 b(w)m(e)g(de\014ne)71 3714 y(the)30 b Fr(pul)5 b(l-b)-5 b(ack)41 b FI(of)30 b Fm(\026)g FI(b)m(y)h Fm(h)f FI(as)h(the)f(Beltrami)g(form)g(of)h Fo(U)39 b FI(de\014ned)29 b(b)m(y:)1515 4002 y Fm(h)1567 3965 y Fp(\003)1607 4002 y Fm(\026)c FI(=)1803 3900 y Fl(@)t(h)p 1803 3915 82 4 v 1806 3967 a(@)p 1847 3929 36 3 v 4 w(z)1915 3935 y FI(+)20 b(\()p Fm(\026)g Fo(\016)h Fm(h)p FI(\))p 2269 3843 102 4 v 2279 3900 a Fl(@)t(h)p 2279 3915 82 4 v 2282 3967 a(@)t(z)p 1793 3981 579 4 v 1803 4057 a(@)t(h)p 1803 4072 82 4 v 1806 4124 a(@)t(z)1915 4092 y FI(+)f(\()p Fm(\026)g Fo(\016)h Fm(h)p FI(\))p 2269 4000 102 4 v 2279 4057 a Fl(@)t(h)p 2279 4072 82 4 v 2282 4124 a(@)p 2323 4086 36 3 v 4 w(z)2381 4002 y Fm(:)71 4292 y Ft(Remark)33 b(2.8.)42 b FI(Notice)30 b(that)h(if,)d(in)h(the)h(previous)e (de\014nition,)f Fm(\026)e FI(=)g Fm(\026)2541 4307 y Fl(f)2616 4292 y FI(for)k(certain)h(quasiregular)e(map)h Fm(f)10 b FI(,)71 4405 y(then)30 b Fm(h)330 4372 y Fp(\003)370 4405 y Fm(\026)425 4420 y Fl(f)495 4405 y FI(=)25 b Fm(\026)646 4420 y Fl(f)7 b Fp(\016)p Fl(h)767 4405 y FI(.)71 4597 y Ft(Remark)31 b(2.9.)39 b FI(Pulling-bac)m(k)25 b(b)m(y)j(holomorphic) d(functions)h(do)s(es)g(not)i(increase)f(the)g(maximal)f(dilatation,)71 4710 y Fm(k)s FI(,)31 b(of)f(a)h Fm(k)s Fo(\000)p FI(Beltrami)f(form.) 71 4903 y Ft(Remark)48 b(2.10.)g FI(The)42 b(standard)f(complex)h (structure)f(corresp)s(onds)g(to)i Fm(\026)2792 4917 y FJ(0)2875 4903 y Fo(\021)i FI(0,)g(whic)m(h)c(is)g(a)i(\014eld)d(of) 71 5016 y(circles.)g(A)30 b(quasiregular)f(mapping)g Fm(f)39 b FI(is)30 b(holomorphic)e(if)h(and)h(only)g(if)f Fm(f)2660 4983 y Fp(\003)2699 5016 y Fm(\026)2754 5030 y FJ(0)2818 5016 y FI(=)c Fm(\026)2969 5030 y FJ(0)3008 5016 y FI(.)71 5209 y Ft(De\014nition)35 b(2.11.)42 b FI(Giv)m(en)31 b(a)f(Beltrami)g(form)g Fm(\026)p FI(,)g(the)h(partial)e (di\013eren)m(tial)g(equation)1697 5393 y Fm(@)5 b(f)p 1697 5434 108 4 v 1701 5517 a(@)p 1754 5467 47 4 v 5 w(z)1839 5455 y FI(=)25 b Fm(\026)p FI(\()p Fm(z)t FI(\))2116 5393 y Fm(@)5 b(f)p 2116 5434 108 4 v 2120 5517 a(@)g(z)3734 5455 y FI(\(7\))71 5684 y(is)21 b(called)h Fr(the)k(Beltr)-5 b(ami)27 b(e)-5 b(quation)p FI(.)38 b(By)23 b Fr(inte)-5 b(gr)g(ation)31 b FI(of)23 b Fm(\026)f FI(w)m(e)h(mean)g(the)f (construction)g(of)h(a)g(quasiconformal)71 5796 y(map)30 b Fm(f)39 b FI(solving)30 b(this)f(equation)h(almost)g(ev)m(erywhere,)h (or)g(equiv)-5 b(alen)m(tly)d(,)30 b(suc)m(h)g(that)h Fm(\026)3095 5811 y Fl(f)3165 5796 y FI(=)25 b Fm(\026)p FI(.)p eop %%Page: 9 9 9 8 bop 71 274 a Fs(N.)31 b(F)-8 b(agella,)31 b(T.)f(M.)h(Seara)g(and)f (J.)g(Villan)m(uev)-5 b(a)2071 b FI(9)212 573 y(The)30 b(famous)g Fr(Me)-5 b(asur)g(able)34 b(R)n(iemann)f(Mapping)h(The)-5 b(or)g(em)39 b FI(b)m(y)30 b(Morrey)-8 b(,)32 b(Ahlfors,)d(Bers)i(and)f (Bo)5 b(jarski)71 686 y(states)33 b(that)f(ev)m(ery)h(almost)f(complex) f(structure)h(is)e(in)m(tegrable.)45 b(As)32 b(w)m(e)g(are)g(going)g (to)h(use)e(this)g(result)g(for)71 799 y(Beltrami)f(forms)f(with)g Fo(U)35 b FI(=)25 b Fo(V)32 b FI(=)25 b Fq(C)18 b FI(,)36 b(w)m(e)31 b(giv)m(e)g(a)g(statemen)m(t)h(adapted)e(to)h(this)f(con)m (text.)71 985 y Ft(Theorem)38 b(2.12)i(\(Ahlfors-Bers,)h([Ah,)e(BD)q (]\).)k Fr(L)-5 b(et)37 b Fm(\026)e Fr(b)-5 b(e)36 b(a)h Fm(k)s Fo(\000)p Fr(Beltr)-5 b(ami)37 b(form)g(of)g Fq(C)17 b Fr(.)58 b(Then,)37 b(ther)-5 b(e)71 1098 y(exists)42 b(a)g(unique)e(quasic)-5 b(onformal)45 b(map)d Fm(h)g FI(:)g Fq(C)65 b Fo(!)42 b Fq(C)65 b Fr(such)41 b(that)i Fm(h)p FI(\(0\))g(=)f(0)p Fr(,)i Fm(h)p FI(\(1\))f(=)e(1)h Fr(and)g Fm(\026)3569 1113 y Fl(h)3656 1098 y FI(=)f Fm(\026)p Fr(.)71 1211 y(F)-7 b(urthermor)i(e,)39 b(if)c Fm(\026)774 1225 y Fl(t)839 1211 y Fr(is)g(a)h(family)g(of)g(Beltr)-5 b(ami)37 b(forms)g(such)e(that)i Fm(\026)2500 1225 y Fl(t)2530 1211 y FI(\()p Fm(z)t FI(\))f Fr(dep)-5 b(ends)37 b(analytic)-5 b(al)5 b(ly)38 b(on)e Fm(t)p Fr(,)g(for)71 1324 y(any)d Fm(z)d Fo(2)24 b Fq(C)18 b Fr(,)39 b(then)33 b Fm(h)783 1338 y Fl(t)845 1324 y Fr(dep)-5 b(ends)34 b(analytic)-5 b(al)5 b(ly)35 b(on)e Fm(t)p Fr(.)71 1510 y Ft(Remark)e(2.13.)40 b FI(The)27 b(application)f(of)h(Ahlfors-Bers')g (theorem)h(to)g(complex)f(dynamics)f(is)h(the)h(follo)m(wing.)71 1623 y(Let)j Fm(f)40 b FI(and)30 b Fm(\026)g FI(b)s(e,)g(resp)s(ectiv)m (ely)-8 b(,)31 b(a)g(quasiregular)e(mapping)f(of)j Fq(C)54 b FI(and)30 b(a)h(Beltrami)f(form)g(of)h Fq(C)18 b FI(,)36 b(suc)m(h)31 b(that)71 1736 y Fm(f)126 1703 y Fp(\003)165 1736 y Fm(\026)25 b FI(=)g Fm(\026)p FI(.)39 b(If)29 b(w)m(e)g(apply)e(Theorem)i(2.12)h(to)g(in)m(tegrate)g Fm(\026)p FI(,)f(and)f(w)m(e)h(construct)g(a)h(quasiconformal)d (mapping)71 1849 y Fm(h)k FI(suc)m(h)g(that)g Fm(\026)612 1864 y Fl(h)683 1849 y FI(=)25 b Fm(\026)p FI(,)31 b(then)g Fm(g)e FI(=)d Fm(h)21 b Fo(\016)g Fm(f)30 b Fo(\016)21 b Fm(h)1599 1816 y Fp(\000)p FJ(1)1724 1849 y FI(v)m(eri\014es)31 b Fm(g)2075 1816 y Fp(\003)2115 1849 y Fm(\026)2170 1863 y FJ(0)2235 1849 y FI(=)26 b Fm(\026)2387 1863 y FJ(0)2426 1849 y FI(,)31 b(and)f(hence)h Fm(g)k FI(is)30 b(a)h(holomorphic)e(map) 71 1962 y(of)h Fq(C)18 b FI(.)47 b(Moreo)m(v)m(er,)32 b Fm(f)40 b FI(and)30 b Fm(g)k FI(are)d(quasiconformally)d(conjugate,)k (i.)40 b(e.,)31 b(they)f(ha)m(v)m(e)i(the)e(same)h(dynamics.)71 2205 y Fg(2.3)112 b(Hyp)s(erb)s(olic)36 b(geometry)71 2377 y FI(During)19 b(this)h(pap)s(er,)h(b)s(esides)e(quasiconformal)g (surgery)-8 b(,)23 b(w)m(e)e(will)d(use)i(some)h(results)f(of)g Fr(hyp)-5 b(erb)g(olic)26 b(ge)-5 b(ometry)71 2490 y FI(\(see)21 b([Be2)r(])f(for)h(a)f(surv)m(ey\).)38 b(In)20 b(a)g(few)g(w)m(ords,)j(quasiconformal)c(surgery)g(will)f(b)s(e)i(the)h (k)m(ey)g(for)f(the)h(geometrical)71 2603 y(constructions)30 b(w)m(e)h(do)f(and)g(h)m(yp)s(erb)s(olic)d(geometry)32 b(will)c(pro)m(vide)h(some)i(of)f(the)h(quan)m(titativ)m(e)g (estimates.)71 2789 y Ft(De\014nition)45 b(2.14.)i FI(Giv)m(en)40 b Fo(U)48 b(\022)40 b Fq(C)63 b FI(a)40 b(domain)e(\(op)s(en)g(and)h (connected)h(set\))g(and)f(giv)m(en)g(a)h(con)m(tin)m(uous)71 2902 y(function)26 b Fm(\025)f FI(:)h Fo(U)34 b(!)25 b FI([0)p Fm(;)15 b FI(+)p Fo(1)p FI(\),)30 b(with)c(at)i(most)g (isolated)f(zeros,)i(w)m(e)g(de\014ne)d(the)i Fr(R)n(iemannian)j (metric)i Fm(\025)28 b FI(on)f Fo(U)71 3015 y FI(as)k(the)h(metric)f (ha)m(ving)f Fm(\025)p FI(\()p Fm(z)t FI(\))p Fo(j)p Fm(dz)t Fo(j)j FI(as)f(a)f(line)f(elemen)m(t.)44 b(More)31 b(precisely)-8 b(,)31 b(giv)m(en)g(a)h(piecewise)e(di\013eren)m(tiable) 71 3128 y(arc)h Fm(\015)f FI(:)c([)p Fm(a;)15 b(b)p FI(])26 b Fo(!)f(U)9 b FI(,)30 b(the)h Fr(length)38 b FI(of)30 b Fm(\015)36 b FI(with)29 b(resp)s(ect)h(to)h(the)g(metric)f Fm(\025)g FI(is)g(de\014ned)f(b)m(y)1083 3392 y Fm(l)1110 3407 y Fl(\025)1155 3392 y FI(\()p Fm(\015)5 b FI(\))26 b(=)1399 3268 y Fn(Z)1490 3295 y Fl(b)1450 3475 y(a)1539 3392 y Fm(\025)p FI(\()p Fm(\015)5 b FI(\()p Fm(t)p FI(\)\))p Fo(j)p Fm(dz)t Fo(j)28 b FI(=)2084 3268 y Fn(Z)2175 3295 y Fl(b)2135 3475 y(a)2224 3392 y Fm(\025)p FI(\()p Fm(\015)5 b FI(\()p Fm(t)p FI(\)\))p Fo(j)p Fm(\015)2579 3355 y Fp(0)2605 3392 y FI(\()p Fm(t)p FI(\))p Fo(j)p Fm(dt:)71 3636 y Ft(De\014nition)30 b(2.15.)39 b FI(Giv)m(en)26 b(a)h(Riemannian)d(metric)i Fm(\025)g FI(on)h Fo(U)35 b FI(and)25 b(giv)m(en)i(t)m(w)m(o)g(p)s(oin)m(ts)e Fm(z)3097 3650 y FJ(1)3137 3636 y Fm(;)15 b(z)3219 3650 y FJ(2)3284 3636 y Fo(2)25 b(U)9 b FI(,)27 b(w)m(e)g(de\014ne)71 3749 y(the)j Fr(distanc)-5 b(e)39 b FI(d)634 3764 y Fl(\025)679 3749 y FI(\()p Fm(z)756 3763 y FJ(1)796 3749 y Fm(;)15 b(z)878 3763 y FJ(2)918 3749 y FI(\))31 b(b)m(y)977 3952 y(d)1028 3967 y Fl(\025)1073 3952 y FI(\()p Fm(z)1150 3966 y FJ(1)1190 3952 y Fm(;)15 b(z)1272 3966 y FJ(2)1312 3952 y FI(\))26 b(=)f(inf)5 b Fo(f)p Fm(l)1651 3967 y Fl(\025)1697 3952 y FI(\()p Fm(\015)g FI(\))26 b Fo(j)56 b Fm(\015)31 b Fo(\032)25 b Fm(U)40 b FI(arc)31 b(from)e Fm(z)2610 3966 y FJ(1)2680 3952 y FI(to)j Fm(z)2834 3966 y FJ(2)2873 3952 y Fo(g)p Fm(:)71 4154 y FI(In)k(the)h(case)i(when)d (this)g(in\014m)m(um)e(is)i(ac)m(hiev)m(ed)i(b)m(y)f(an)g(arc)g Fm(\015)2291 4121 y Fp(\003)2368 4154 y FI(from)g Fm(z)2632 4168 y FJ(1)2708 4154 y FI(to)h Fm(z)2868 4168 y FJ(2)2908 4154 y FI(,)h(this)d(arc)h Fm(\015)3366 4121 y Fp(\003)3443 4154 y FI(is)f(called)g(a)71 4267 y Fr(ge)-5 b(o)g(desic)36 b FI(b)s(et)m(w)m(een)31 b Fm(z)815 4281 y FJ(1)885 4267 y FI(and)f Fm(z)1104 4281 y FJ(2)1144 4267 y FI(.)212 4454 y(An)m(y)24 b(holomorphic)f(map)h(b)s(et)m(w)m(een)h(t)m(w)m(o)h (domains)d Fo(U)33 b FI(and)24 b Fo(V)31 b FI(can)25 b(b)s(e)e(used)h(to)h(transp)s(ort)f(a)g(Riemannian)71 4566 y(metric)30 b(on)g Fo(V)38 b FI(to)31 b(a)g(Riemannian)d(metric)i (on)g Fo(U)9 b FI(.)71 4753 y Ft(De\014nition)36 b(2.16.)43 b FI(Giv)m(en)31 b(an)g(holomorphic)f(map)h Fm(f)36 b FI(:)27 b Fo(U)35 b(!)27 b(V)38 b FI(and)31 b(a)g(Riemannian)f(metric)h Fm(\025)g FI(on)g Fo(V)7 b FI(,)32 b(w)m(e)71 4866 y(de\014ne)d(the)i Fr(pul)5 b(l-b)-5 b(ack)40 b FI(of)31 b Fm(\025)f FI(b)m(y)g Fm(f)40 b FI(as)31 b(the)f(Riemannian)f(metric)h(on)g Fo(U)39 b FI(giv)m(en)30 b(b)m(y)1618 5069 y Fm(f)1673 5031 y Fp(\003)1712 5069 y Fm(\025)25 b FI(=)g(\()p Fm(\025)c Fo(\016)f Fm(f)10 b FI(\))p Fo(j)p Fm(f)2230 5031 y Fp(0)2253 5069 y Fo(j)p Fm(:)71 5271 y FI(With)40 b(this)f(de\014nition)f Fm(f)50 b FI(is)39 b(a)i(lo)s(cal)f(isometry)g(b)s(et)m(w)m(een)h(\()p Fo(U)9 b Fm(;)15 b(f)2363 5238 y Fp(\003)2402 5271 y Fm(\025)p FI(\))41 b(and)f(\()p Fo(V)7 b Fm(;)15 b(\025)p FI(\),)44 b(i.e.,)f(it)d(preserv)m(es)h(arc-)71 5384 y(lengths.)f(If)30 b Fm(f)39 b FI(is)30 b(biholomorphic,)d(then)j(it)g (is)g(a)g(global)g(isometry)-8 b(.)212 5571 y(The)25 b(example)g(that)h(concerns)g(us)f(is)f(the)i Fr(hyp)-5 b(erb)g(olic)30 b(metric)p FI(,)d(whic)m(h)d(is)h(a)h(Riemannian)d (metric)j(de\014ned)71 5684 y(on)g(domains)f Fo(U)35 b FI(that)27 b(ha)m(v)m(e)h(the)f(unit)d(disk)h Fq(D)48 b FI(:=)25 b Fq(D)1837 5698 y FJ(1)1909 5684 y FI(as)h(a)h(co)m(v)m (ering)h(space,)g(and)d(whic)m(h)g(is)h(preserv)m(ed)g(under)71 5796 y(conformal)k(self-mappings)e(of)j Fo(U)9 b FI(.)40 b(On)30 b Fq(D)53 b FI(the)30 b(h)m(yp)s(erb)s(olic)e(metric)i(tak)m (es)i(the)e(follo)m(wing)f(form.)p eop %%Page: 10 10 10 9 bop 71 274 a FI(10)1001 b Fs(Asymptotic)30 b(Size)g(of)h(Herman)f (Rings)f(of)i(the)f(Complex)g(Standard)f(F)-8 b(amily)71 573 y Ft(De\014nition)35 b(2.17.)42 b FI(The)30 b(h)m(yp)s(erb)s(olic)e (or)i Fr(Poinc)-5 b(ar)n(\023)-44 b(e)38 b FI(metric)31 b(on)f Fq(D)53 b FI(is)29 b(the)i(metric)f(de\014ned)f(b)m(y)1620 828 y Fm(\025)1673 842 y Fh(D)1725 828 y FI(\()p Fm(z)t FI(\))d(=)2097 766 y(2)p 1973 807 293 4 v 1973 890 a(1)21 b Fo(\000)f(j)p Fm(z)t Fo(j)2226 864 y FJ(2)2276 828 y Fm(:)71 1098 y FI(The)31 b(P)m(oincar)m(\023)-43 b(e)33 b(metric)f Fm(\025)971 1112 y Fh(D)1055 1098 y FI(is)f(the)h(unique)d (metric)j(on)g Fq(D)54 b FI(\(up)31 b(to)i(m)m(ultiplication)28 b(b)m(y)k(p)s(ositiv)m(e)f(constan)m(ts\))71 1211 y(in)m(v)-5 b(arian)m(t)29 b(under)g(conformal)h(automorphisms)f(of)h Fq(D)17 b FI(.)212 1422 y(W)-8 b(e)32 b(will)27 b(need)j(an)h(explicit) d(expression)i(for)g(the)g(distance)g(in)f Fq(D)53 b FI(de\014ned)29 b(b)m(y)i(the)f(P)m(oincar)m(\023)-43 b(e)32 b(metric.)71 1633 y Ft(Prop)s(osition)f(2.18.)39 b Fr(Given)28 b Fm(w)1210 1647 y FJ(1)1250 1633 y Fm(;)15 b(w)1355 1647 y FJ(2)1420 1633 y Fo(2)25 b Fq(D)16 b Fr(,)35 b(we)29 b(have)g(the)g(fol)5 b(lowing)30 b(formula)g(for)f(the) g(hyp)-5 b(erb)g(olic)30 b(distanc)-5 b(e)71 1746 y Fm(\016)111 1760 y Fh(D)196 1746 y Fr(in)32 b Fq(D)17 b Fr(:)1083 1924 y FI(sinh)1245 1886 y FJ(2)1300 1796 y Fn(\024)1358 1862 y Fm(\016)1398 1876 y Fh(D)1450 1862 y FI(\()p Fm(w)1550 1876 y FJ(1)1590 1862 y Fm(;)e(w)1695 1876 y FJ(2)1735 1862 y FI(\))p 1358 1903 413 4 v 1541 1986 a(2)1780 1796 y Fn(\025)1854 1924 y FI(=)2153 1862 y(4)p Fo(j)p Fm(w)2288 1876 y FJ(1)2349 1862 y Fo(\000)20 b Fm(w)2505 1876 y FJ(2)2544 1862 y Fo(j)2569 1829 y FJ(2)p 1960 1903 844 4 v 1960 1986 a FI(\(1)h Fo(\000)e(j)p Fm(w)2241 2000 y FJ(1)2281 1986 y Fo(j)2306 1960 y FJ(2)2346 1986 y FI(\)\(1)i Fo(\000)f(j)p Fm(w)2663 2000 y FJ(2)2703 1986 y Fo(j)2728 1960 y FJ(2)2768 1986 y FI(\))2813 1924 y Fm(:)71 2156 y Fr(In)33 b(p)-5 b(articular,)34 b(if)f FI(0)25 b Fo(\024)g Fm(r)j(<)d FI(1)p Fr(,)1490 2332 y Fm(\016)1530 2346 y Fh(D)1583 2332 y FI(\(0)p Fm(;)15 b(r)s FI(\))26 b(=)f(log)2037 2203 y Fn(\022)2114 2270 y FI(1)20 b(+)g Fm(r)p 2114 2311 201 4 v 2114 2394 a FI(1)g Fo(\000)g Fm(r)2324 2203 y Fn(\023)2406 2332 y Fm(:)212 2588 y FI(The)34 b(pull-bac)m(k)g(pro)s(cess)g(allo)m(ws)g(us) g(to)i(transp)s(ort)e(the)h(P)m(oincar)m(\023)-43 b(e)36 b(metric)f(to)g(an)m(y)g(domain)f Fo(U)44 b FI(that)35 b(is)71 2701 y(conformally)23 b(equiv)-5 b(alen)m(t)24 b(to)h Fq(D)17 b FI(.)45 b(Indeed,)25 b(if)e Fm( )29 b FI(:)c Fo(U)35 b(!)25 b Fq(D)47 b FI(is)23 b(a)i(Riemann)e(map,)j (then)e(the)h(h)m(yp)s(erb)s(olic)d(metric)71 2813 y(on)30 b Fo(U)39 b FI(is)30 b(giv)m(en)g(b)m(y)1213 2944 y Fm(\025)1266 2958 y Fp(U)1321 2944 y FI(\()p Fm(z)t FI(\))c(=)f(\()p Fm( )1656 2906 y Fp(\003)1696 2944 y Fm(\025)1749 2958 y Fh(D)1801 2944 y FI(\)\()p Fm(z)t FI(\))i(=)e Fm(\025)2128 2958 y Fh(D)2180 2944 y FI(\()p Fm( )s FI(\()p Fm(z)t FI(\)\))p Fo(j)p Fm( )2515 2906 y Fp(0)2541 2944 y FI(\()p Fm(z)t FI(\))p Fo(j)p Fm(;)71 3131 y FI(or)30 b(equiv)-5 b(alen)m(tly)d(,)30 b(if)f Fm(')d FI(:)g Fq(D)47 b Fo(!)26 b(U)39 b FI(is)29 b(a)i(conformal)f(map,)g(then)1575 3396 y Fm(\025)1628 3410 y Fp(U)1683 3396 y FI(\()p Fm(')p FI(\()p Fm(z)t FI(\)\))d(=)2075 3334 y Fm(\025)2128 3348 y Fh(D)2180 3334 y FI(\()p Fm(z)t FI(\))p 2061 3375 251 4 v 2061 3458 a Fo(j)p Fm(')2145 3432 y Fp(0)2169 3458 y FI(\()p Fm(z)t FI(\))p Fo(j)2321 3396 y Fm(:)1388 b FI(\(8\))71 3678 y(An)36 b(imp)s(ortan)m(t)g(example)g(of)h(this)f(is)g (the)h(upp)s(er)d(half)i(plane,)h Fq(H)18 b FI(,)45 b(for)36 b(whic)m(h)f(w)m(e)j(can)f(tak)m(e)h Fm( )s FI(\()p Fm(z)t FI(\))f(=)3701 3642 y Fl(z)s Fp(\000)p Fl(i)p 3701 3657 115 4 v 3701 3709 a(z)s FJ(+)p Fl(i)3825 3678 y FI(,)71 3791 y(obtaining)29 b(the)h(follo)m(wing)f(result.)71 4002 y Ft(Prop)s(osition)46 b(2.19.)h Fr(The)40 b(hyp)-5 b(erb)g(olic)42 b(metric)f(in)f Fq(H)64 b Fr(is)40 b(given)f(by)i Fm(\025)2583 4016 y Fh(H)2638 4002 y FI(\()p Fm(z)t FI(\))f(=)2984 3966 y FJ(1)p 2914 3981 175 4 v 2914 4034 a(Im)o(\()p Fl(z)s FJ(\))3099 4002 y Fr(.)64 b(In)41 b(this)f(c)-5 b(ase,)43 b(the)71 4115 y Fm(\025)124 4129 y Fh(H)180 4115 y Fr(-ge)-5 b(o)g(desics)33 b(ar)-5 b(e)33 b(vertic)-5 b(al)34 b(se)-5 b(gments)33 b(or)g(ar)-5 b(cs)34 b(of)f(cir)-5 b(cles)33 b(ortho)-5 b(gonal)36 b(to)d(the)g(r)-5 b(e)g(al)34 b(axis.)212 4326 y FI(The)c(h)m(yp)s(erb)s(olic)d(metric)j(can)g(also)g (b)s(e)f(transp)s(orted)g(to)i(non-simply)d(connected)i(domains,)f(b)m (y)h(means)71 4439 y(of)g(an)m(y)h Fr(universal)i(c)-5 b(overing)33 b(map)p FI(.)71 4650 y Ft(De\014nition)39 b(2.20.)44 b FI(A)34 b(domain)e Fo(U)42 b FI(of)34 b(the)g(Riemann)e (sphere)p 2290 4576 66 4 v 33 w Fq(C)57 b FI(is)33 b(called)g Fr(hyp)-5 b(erb)g(olic)76 b FI(if)32 b(it)h(has)h(at)g(least)71 4763 y(three)c(b)s(oundary)f(p)s(oin)m(ts.)71 4974 y Ft(Theorem)24 b(2.21.)36 b Fr(If)25 b Fo(U)34 b Fr(is)25 b(a)h(hyp)-5 b(erb)g(olic)27 b(domain,)h(ther)-5 b(e)26 b(exists)f(a)h(holomorphic)i(c)-5 b(overing)26 b(map)g Fm(')g FI(:)f Fq(D)48 b Fo(!)25 b(U)71 5087 y Fr(\(i.e.,)31 b Fm(')g Fr(is)g(a)g(lo)-5 b(c)g(al)33 b(home)-5 b(omorphism)35 b(at)c(every)g(p)-5 b(oint\).)43 b(Each)31 b(such)g(map)h(is)f(c)-5 b(al)5 b(le)-5 b(d)32 b(a)39 b FI(univ)m(ersal)26 b(co)m(v)m(ering)71 5200 y(map)32 b Fr(and)i(it)e(is)h(uniquely)f(determine)-5 b(d)34 b(if)e(we)h(pr)-5 b(e\014x)34 b Fm(')p FI(\(0\))g Fr(and)g(r)-5 b(e)g(quir)g(e)33 b Fm(')2680 5167 y Fp(0)2704 5200 y FI(\(0\))26 b Fm(>)f FI(0)p Fr(.)212 5411 y FI(Then,)i(if)f Fo(U)36 b FI(is)26 b(a)i(h)m(yp)s(erb)s(olic)c(domain,)j(and)g Fm(')g FI(is)f(a)i(univ)m(ersal)d(co)m(v)m(ering)j(for)f Fo(U)9 b FI(,)28 b(the)f(h)m(yp)s(erb)s(olic)e(metric)71 5524 y Fm(\025)124 5538 y Fp(U)179 5524 y Fo(j)p Fm(dz)t Fo(j)31 b FI(is)f(giv)m(en)g(as)h(ab)s(o)m(v)m(e)g(b)m(y)f(\(8\).)212 5665 y(Using)k(that)h Fm(')p FI(\()p Fm(z)t FI(\))f(=)e(exp)1139 5564 y Fn(\020)1203 5629 y Fl(z)s Fp(\000)p FJ(1)p 1203 5644 126 4 v 1203 5697 a Fl(z)s FJ(+1)1339 5564 y Fn(\021)1428 5665 y FI(is)i(a)h(univ)m(ersal)e(co)m(v)m(ering)i(for)g(the)g (punctured)e(disk)g Fq(D)3351 5632 y Fp(\003)3428 5665 y FI(=)g Fq(D)45 b Fo(n)24 b(f)p FI(0)p Fo(g)p FI(,)71 5796 y(w)m(e)31 b(obtain)f(the)g(follo)m(wing)f(prop)s(erties)g(for)h (the)g(h)m(yp)s(erb)s(olic)e(metric)i(in)f Fq(D)2645 5763 y Fp(\003)2690 5796 y FI(.)p eop %%Page: 11 11 11 10 bop 71 274 a Fs(N.)31 b(F)-8 b(agella,)31 b(T.)f(M.)h(Seara)g (and)f(J.)g(Villan)m(uev)-5 b(a)2025 b FI(11)71 573 y Ft(Prop)s(osition)36 b(2.22.)42 b Fr(The)33 b(hyp)-5 b(erb)g(olic)35 b(metric)e(in)f Fq(D)1958 540 y Fp(\003)2036 573 y Fr(has)i(the)f(form)1505 791 y Fm(\025)1558 805 y Fh(D)1600 787 y Ff(\003)1646 791 y FI(\()p Fm(z)t FI(\))26 b(=)2115 730 y(1)p 1894 770 488 4 v 1894 853 a Fo(j)p Fm(z)t Fo(j)15 b FI(log)r(\(1)p Fm(=)p Fo(j)p Fm(z)t Fo(j)p FI(\))2392 791 y Fm(:)71 1020 y Fr(The)33 b(hyp)-5 b(erb)g(olic)34 b(distanc)-5 b(e)34 b Fm(\016)1069 1034 y Fh(D)1112 1015 y Ff(\003)1190 1020 y Fr(satis\014es)864 1255 y FI(sinh)1026 1218 y FJ(2)1081 1127 y Fn(\024)1139 1194 y Fm(\016)1179 1208 y Fh(D)1221 1189 y Ff(\003)1267 1194 y FI(\()p Fm(z)1344 1208 y FJ(1)1384 1194 y Fm(;)15 b(z)1466 1208 y FJ(2)1506 1194 y FI(\))p 1139 1234 404 4 v 1318 1318 a(2)1552 1127 y Fn(\025)1625 1255 y FI(=)1731 1194 y Fo(j)g FI(log)i Fm(z)1946 1208 y FJ(1)2006 1194 y Fo(\000)j FI(log)c Fm(z)2271 1208 y FJ(2)2311 1194 y Fo(j)2336 1161 y FJ(2)p 1731 1234 645 4 v 1781 1318 a FI(log)g Fo(j)p Fm(z)1980 1332 y FJ(1)2020 1318 y Fo(j)f FI(log)i Fo(j)p Fm(z)2260 1332 y 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b(main)f(reason)h(wh)m(y)g(h)m(yp)s(erb)s(olic)d(geometry)39 b(is)d(v)m(ery)h(useful)f(in)f(complex)i(dynamics)f(is)g(the)h(fact)71 2001 y(that)31 b(all)f(holomorphic)f(maps)h(are)h(con)m(tractiv)m(e,)i (when)d(w)m(e)h(lo)s(ok)f(at)i(them)e(under)f(the)i(h)m(yp)s(erb)s (olic)d(metric.)71 2114 y(This)g(is)i(kno)m(wn)g(as)g(the)h Fr(Big)h(Schwartz-Pick)h(lemma)39 b FI(whic)m(h)29 b(reads)h(as)h (follo)m(ws.)71 2286 y Ft(Theorem)42 b(2.23)j(\(Big)e(Sc)m(h)m(w)m (artz-Pic)m(k)i(lemma\).)e Fr(If)d Fo(U)48 b Fr(and)40 b Fo(V)47 b Fr(ar)-5 b(e)40 b(hyp)-5 b(erb)g(olic)42 b(domains)f(and)f Fm(f)47 b FI(:)71 2398 y Fo(U)34 b(!)25 b(V)40 b Fr(is)32 b(holomorphic,)k(then)1406 2511 y Fm(\016)1446 2525 y Fp(V)1500 2511 y FI(\()p Fm(f)10 b FI(\()p Fm(z)1667 2525 y FJ(1)1706 2511 y FI(\))p Fm(;)15 b(f)10 b FI(\()p Fm(z)1913 2525 y FJ(2)1954 2511 y FI(\)\))25 b Fo(\024)g Fm(\016)2185 2525 y Fp(U)2241 2511 y FI(\()p Fm(z)2318 2525 y FJ(1)2358 2511 y Fm(;)15 b(z)2440 2525 y FJ(2)2480 2511 y FI(\))71 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l 3816 7217 l 3804 7217 l 3791 7229 l 3791 7241 l 3791 7254 l 3779 7254 l 3779 7266 l 3767 7266 l 3767 7280 l 3755 7280 l 3755 7293 l 3742 7293 l 3728 7293 l 3716 7293 l 3703 7305 l 3682 7305 l 3667 7305 l 3641 7305 l 3629 7305 l 3629 7316 l 3615 7316 l 3615 7328 l 3605 7328 l 3595 7328 l 3581 7341 l 3569 7341 l 3555 7341 l 3543 7341 l 3528 7341 l 3517 7341 l 3507 7341 l 3507 7328 l 3507 7316 l 3494 7316 l 3494 7305 l 3483 7305 l 3468 7305 l 3455 7293 l 3442 7293 l 3421 7293 l 3406 7280 l 3381 7280 l 3369 7280 l 3355 7280 l 3342 7266 l 3333 7266 l 3333 7254 l 3321 7254 l 3308 7241 l 3308 7229 l 3295 7229 l 3295 7217 l 3282 7217 l 3270 7217 l 3257 7217 l 3246 7217 l 3235 7217 l 3220 7217 l 3195 7217 l 3169 7217 l 3147 7217 l 3133 7217 l 3120 7217 l 3108 7217 l 3094 7217 l 3083 7217 l 3072 7217 l 3060 7217 l 3046 7217 l 3033 7217 l 3021 7217 l 3021 7206 l 3007 7206 l 2995 7206 l 2984 7194 l 2971 7194 l 2959 7194 l 2959 7181 l 2947 7181 l 2933 7181 l 2933 7169 l 2920 7169 l 2909 7169 l 2899 7155 l 2886 7155 l 2873 7155 l 2873 7169 l 2860 7169 l 2860 7181 l 2848 7181 l 2848 7194 l 2835 7194 l 2835 7206 l 2821 7206 l 2821 7217 l 2811 7217 l 2800 7217 l 2786 7229 l 2773 7229 l 2773 7241 l 2773 7254 l 2773 7266 l 2759 7266 l 2759 7280 l 2759 7293 l 2759 7305 l 2759 7316 l 2759 7328 l 2759 7341 l 2759 7353 l 2747 7353 l 2747 7367 l 2747 7380 l 2734 7380 l 2734 7392 l 2734 7403 l 2724 7403 l 2724 7415 l 2713 7415 l 2713 7427 l 2699 7427 l 2699 7442 l 2687 7442 l 2672 7442 l 2660 7442 l 2646 7454 l 2637 7454 l 2637 7466 l 2627 7466 l 2612 7466 l 2586 7478 l 2574 7478 l 2559 7478 l 2548 7478 l 2548 7489 l 2538 7489 l 2525 7504 l 2513 7504 l 2513 7516 l 2513 7528 l 2513 7540 l 2499 7540 l 2499 7553 l 2487 7553 l 2487 7566 l 2475 7566 l 2475 7577 l 2462 7577 l 2451 7577 l 2438 7577 l 2425 7577 l 2413 7577 l 2400 7577 l 2387 7577 l 2376 7577 l 2364 7577 l 2351 7577 l 2337 7577 l 2337 7591 l 2325 7591 l 2325 7603 l 2312 7617 l 2312 7628 l 2300 7628 l 2300 7641 l 2289 7641 l 2289 7652 l 2276 7652 l 2265 7652 l 2250 7663 l 2238 7663 l 2226 7663 l 2203 7676 l 2190 7676 l 2178 7676 l 2178 7689 l 2178 7703 l 2164 7703 l 2164 7715 l 2164 7729 l 2164 7738 l 2152 7738 l 2137 7738 l 2124 7738 l 2114 7738 l 2102 7738 l 2091 7738 l 2066 7738 l 2052 7738 l 2028 7738 l 2015 7738 l 1992 7738 l 1979 7738 l 1953 7738 l 1940 7738 l 1917 7738 l 1904 7738 l 1891 7738 l 1891 7750 l 1879 7750 l 1879 7763 l 1866 7763 l 1854 7776 l 1842 7776 l 1817 7789 l 1804 7789 l 1804 7802 l 1790 7802 l 1779 7815 l 1766 7827 l 1766 7838 l 1766 7849 l 1766 7862 l 1754 7862 l 1754 7876 l 1754 7889 l 1754 7902 l 1766 7902 l 1766 7915 l 1766 7926 l 1754 7926 l 1754 7938 l 1754 7951 l 1742 7951 l 1742 7962 l 1729 7975 l 1729 7990 l 1729 8001 l 1729 8013 l 1729 8024 l 1729 8037 l 1729 8052 l 1729 8062 l 1729 8075 l 1729 8086 l 1729 8099 l 1729 8111 l 1729 8124 l 1717 8124 l 1717 8137 l 1717 8149 l 1717 8163 l 1717 8174 l 1717 8187 l 1702 8187 l 1702 8198 l 1691 8198 l 1691 8211 l 1691 8224 l 1691 8237 l 1691 8250 l 1691 8263 l 1691 8273 l 1691 8284 l 1691 8298 l 1680 8298 l 1680 8311 l 1668 8311 l 1668 8325 l 1655 8325 l 1655 8337 l 1644 8337 l 1630 8350 l 1618 8350 l 1606 8360 l 1593 8373 l 1568 8385 l 1544 8397 l 1531 8412 l 1518 8412 l 1518 8424 l 1518 8436 l 1518 8447 l 1506 8459 l 1506 8474 l 1506 8484 l 1506 8497 l 1494 8523 l 1494 8534 l 1494 8544 l 1494 8559 l 1482 8559 l 1469 8559 l 1456 8559 l 1444 8571 l 1430 8571 l 1430 8585 l 1420 8585 l 1420 8598 l 1407 8610 l 1420 8610 l 1420 8621 l 1420 8633 l 1430 8633 l 1444 8633 l 1444 8645 l 1456 8645 l 1456 8658 l 1469 8658 l 1482 8658 l 1482 8672 l 1494 8684 l 1494 8696 l 1506 8696 l 1506 8707 l 1506 8719 l 1518 8732 l 1518 8746 l 1518 8758 l 1518 8771 l 1518 8785 l 1518 8796 l 1531 8808 l 1531 8820 l 1531 8832 l 1531 8846 l 1544 8846 l 1544 8859 l 1557 8872 l 1568 8872 l 1580 8883 l 1593 8883 l 1606 8896 l 1618 8896 l 1630 8896 l 1644 8896 l 1655 8906 l 1668 8906 l 1680 8906 l 1691 8906 l 1702 8906 l 1717 8906 l 1729 8906 l 1729 8919 l 1742 8919 l 1754 8919 l 1766 8919 l 1779 8919 l 1779 8931 l 1790 8931 l 1804 8931 l 1817 8931 l 1830 8944 l 1854 8944 l 1866 8958 l 1879 8958 l 1891 8958 l 1904 8968 l 1917 8968 l 1928 8968 l 1940 8968 l 1953 8981 l 1966 8981 l 1966 8993 l 1979 8993 l 1979 9007 l 1992 9020 l 2004 9020 l 2004 9032 l 2015 9032 l 2015 9045 l 2015 9055 l 2028 9068 l 2028 9080 l 2028 9094 l 2040 9094 l 2040 9107 l 2040 9120 l 2040 9131 l 2040 9143 l 2040 9154 l 2052 9154 l 2066 9167 l 2077 9167 l 2091 9180 l 2102 9180 l 2114 9193 l 2124 9193 l 2137 9193 l 2152 9193 l 2164 9193 l 2164 9207 l 2164 9220 l 2178 9220 l 2178 9231 l 2178 9242 l 2190 9254 l 2190 9267 l 2203 9267 l 2203 9280 l 2212 9280 l 2212 9294 l 2226 9306 l 2250 9319 l 2265 9329 l 2276 9341 l 2289 9341 l 2289 9354 l 2300 9354 l 2300 9366 l 2300 9380 l 2312 9380 l 2312 9392 l 2312 9405 l 2312 9417 l 2325 9417 l 2325 9429 l 2337 9442 l 2337 9454 l 2351 9454 l 2364 9454 l 2364 9467 l 2376 9467 l 2387 9479 l 2400 9479 l 2413 9479 l 2413 9491 l 2425 9491 l 2438 9504 l 2451 9516 l 2462 9516 l 2462 9529 l 2475 9529 l 2487 9529 l 2487 9541 l 2499 9541 l 2513 9555 l 2525 9555 l 2538 9555 l 2538 9566 l 2548 9566 l 2559 9566 l 2559 9578 l 2574 9578 l 2586 9591 l 2600 9603 l 2612 9615 l 2627 9628 l 2637 9628 l 2637 9642 l 2646 9642 l 2660 9642 l 2672 9642 l 2687 9642 l 2699 9628 l 2699 9615 l 2713 9615 l 2713 9603 l 2724 9603 l 2734 9603 l 2734 9591 l 2747 9591 l 2759 9591 l 2773 9591 l 2786 9591 l 2786 9578 l 2800 9578 l 2811 9578 l 2821 9578 l 2835 9578 l 2848 9591 l 2860 9603 l 2873 9615 l 2873 9628 l 2886 9628 l 2899 9642 l 2909 9642 l 2920 9642 l 2933 9642 l 2947 9655 l 2971 9655 l 2984 9655 l 2995 9665 l 3007 9665 l 3021 9665 l 3021 9677 l 3033 9677 l 3046 9677 l 3060 9677 l 3060 9689 l 3072 9702 l 3083 9702 l 3094 9717 l 3108 9728 l 3120 9741 l 3133 9751 l 3147 9751 l 3147 9764 l 3159 9764 l 3159 9775 l 3169 9775 l 3169 9787 l 3181 9787 l 3195 9801 l 3208 9801 l 3220 9813 l 3246 9813 l 3257 9813 l 3270 9828 l 3295 9828 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4090 9504 l 4104 9504 l 4117 9504 l 4127 9504 l 4127 9491 l 4140 9491 l 4140 9479 l 4150 9479 l 4150 9467 l 4164 9467 l 4164 9454 l 4177 9442 l 4189 9429 l 4202 9429 l 4214 9417 l 4226 9417 l 4237 9417 l 4249 9417 l 4262 9405 l 4276 9405 l 4288 9405 l 4301 9405 l 4313 9405 l 4324 9405 l 4324 9392 l 4338 9392 l 4350 9392 l 4362 9392 l 4362 9380 l 4375 9380 l 4388 9380 l 4388 9366 l 4399 9366 l 4412 9366 l 4424 9366 l 4437 9366 l 4451 9366 l 4462 9380 l 4475 9380 l 4487 9380 l 4487 9366 l 4500 9366 l 4500 9354 l 4513 9354 l 4524 9354 l 4524 9341 l 4539 9341 l 4551 9341 l 4562 9329 l 4574 9329 l 4586 9329 l 4586 9319 l 4599 9319 l 4599 9306 l 4611 9306 l 4611 9294 l 4611 9280 l 4624 9280 l 4624 9267 l 4636 9254 l 4636 9242 l 4636 9231 l 4636 9220 l 4636 9207 l 4636 9193 l 4636 9180 l 4636 9167 l 4636 9154 l 4636 9143 l 4624 9143 l 4624 9131 l 4624 9120 l 4611 9120 l 4611 9107 l 4599 9094 l 4599 9080 l 4599 9068 l 4599 9055 l 4599 9045 l 4611 9045 l 4611 9032 l 4624 9020 l 4624 9007 l 4636 9007 l 4636 8993 l 4636 8981 l 4636 8968 l 4636 8958 l 4648 8944 l 4648 8931 l 4661 8931 l 4661 8919 l 4661 8906 l 4673 8906 l 4673 8896 l 4684 8896 l 4684 8883 l 4684 8872 l 4684 8859 l 4684 8846 l 4673 8832 l 4673 8820 l 4661 8820 l 4661 8808 l 4661 8796 l 4661 8785 l 4661 8771 l 4661 8758 l 4648 8746 l 4648 8732 l 4648 8719 l 4648 8707 l 4648 8696 l 4648 8684 l 4648 8672 l 4648 8658 l 4661 8645 l 4673 8645 l 4673 8633 l 4684 8633 l 4684 8621 l 4698 8621 l 4710 8621 l 4710 8610 l 4723 8610 l 4723 8598 l 4735 8598 l 4735 8585 l 4735 8571 l 4735 8559 l 4748 8559 l 4748 8544 l 4748 8534 l 4760 8523 l 4760 8509 l 4760 8497 l 4760 8484 l 4772 8484 l 4772 8474 l 4772 8459 l 4785 8436 l 4785 8424 l 4772 8424 l 4772 8436 l 4760 8436 l 4772 8424 l 4772 8412 l 4785 8412 l 4797 8373 l gs col0 s gr % Polyline 7.500 slw n 10562 8692 m 10562 8688 l 10562 8683 l 10567 8683 l 10567 8680 l 10571 8680 l 10575 8680 l 10575 8675 l 10575 8672 l 10578 8672 l 10583 8672 l 10587 8672 l 10591 8672 l 10594 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l 9673 9398 l 9685 9398 l 9685 9386 l 9698 9386 l 9708 9386 l 9717 9386 l 9730 9386 l 9742 9398 l 9753 9409 l 9765 9420 l 9765 9432 l 9777 9432 l 9789 9445 l 9798 9445 l 9808 9445 l 9820 9445 l 9833 9457 l 9855 9457 l 9867 9457 l 9877 9466 l 9888 9466 l 9901 9466 l 9901 9477 l 9912 9477 l 9924 9477 l 9937 9477 l 9937 9488 l 9948 9500 l 9958 9500 l 9969 9514 l 9981 9524 l 9993 9536 l 10005 9545 l 10017 9545 l 10017 9557 l 10028 9557 l 10028 9567 l 10038 9567 l 10038 9578 l 10049 9578 l 10062 9591 l 10074 9591 l 10085 9602 l 10109 9602 l 10119 9602 l 10131 9616 l 10154 9616 l 10166 9616 l 10178 9602 l 10189 9602 l 10197 9602 l 10209 9602 l 10222 9602 l 10233 9602 l 10244 9602 l 10256 9602 l 10270 9602 l 10279 9602 l 10289 9602 l 10301 9602 l 10313 9602 l 10327 9602 l 10337 9602 l 10349 9602 l 10349 9591 l 10358 9591 l 10358 9578 l 10358 9567 l 10368 9567 l 10368 9557 l 10368 9545 l 10382 9545 l 10382 9536 l 10382 9524 l 10393 9524 l 10393 9514 l 10393 9500 l 10406 9500 l 10406 9488 l 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11125 9215 l 11136 9215 l 11136 9203 l 11148 9203 l 11160 9203 l 11160 9191 l 11171 9191 l 11183 9191 l 11194 9191 l 11206 9191 l 11218 9191 l 11229 9203 l 11241 9203 l 11252 9203 l 11252 9191 l 11264 9191 l 11264 9180 l 11276 9180 l 11286 9180 l 11286 9168 l 11299 9168 l 11311 9168 l 11321 9157 l 11332 9157 l 11343 9157 l 11343 9147 l 11355 9147 l 11355 9135 l 11366 9135 l 11366 9124 l 11366 9111 l 11378 9111 l 11378 9099 l 11389 9087 l 11389 9076 l 11389 9066 l 11389 9056 l 11389 9044 l 11389 9031 l 11389 9019 l 11389 9007 l 11389 8995 l 11389 8985 l 11378 8985 l 11378 8974 l 11378 8964 l 11366 8964 l 11366 8952 l 11355 8940 l 11355 8927 l 11355 8916 l 11355 8904 l 11355 8895 l 11366 8895 l 11366 8883 l 11378 8872 l 11378 8860 l 11389 8860 l 11389 8847 l 11389 8836 l 11389 8824 l 11389 8815 l 11400 8802 l 11400 8790 l 11412 8790 l 11412 8779 l 11412 8767 l 11423 8767 l 11423 8758 l 11433 8758 l 11433 8746 l 11433 8736 l 11433 8724 l 11433 8712 l 11423 8699 l 11423 8688 l 11412 8688 l 11412 8677 l 11412 8666 l 11412 8655 l 11412 8642 l 11412 8630 l 11400 8619 l 11400 8607 l 11400 8595 l 11400 8584 l 11400 8573 l 11400 8562 l 11400 8551 l 11400 8538 l 11412 8526 l 11423 8526 l 11423 8515 l 11433 8515 l 11433 8504 l 11446 8504 l 11457 8504 l 11457 8494 l 11469 8494 l 11469 8483 l 11480 8483 l 11480 8471 l 11480 8458 l 11480 8447 l 11492 8447 l 11492 8433 l 11492 8424 l 11503 8414 l 11503 8401 l 11503 8390 l 11503 8378 l 11514 8378 l 11514 8369 l 11514 8355 l 11526 8334 l 11526 8323 l 11514 8323 l 11514 8334 l 11503 8334 l 11514 8323 l 11514 8312 l 11526 8312 l 11537 8276 l gs col0 s gr /Times-Roman ff 255.00 scf sf 4292 3237 m gs 1 -1 sc (1) col0 sh gr 7.500 slw [120] 0 sd % Ellipse n 3448 3179 741 741 0 360 DrawEllipse gs col0 s gr [] 0 sd [120] 0 sd % Ellipse n 3641 8598 741 741 0 360 DrawEllipse gs col0 s gr [] 0 sd % Ellipse n 10242 3162 90 90 0 360 DrawEllipse gs col0 s gr % Polyline [60 30 15 30] 0 sd gs clippath 11326 1661 m 11249 1613 l 11154 1764 l 11273 1662 l 11231 1812 l cp eoclip n 10305 3195 m 11280 1650 l gs col0 s gr gr [] 0 sd % arrowhead n 11231 1812 m 11273 1662 l 11154 1764 l 11231 1812 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 8745 3225 m 8745 3075 l 8508 3075 l 8718 3150 l 8508 3225 l cp eoclip n 5910 3150 m 8730 3150 l gs col0 s gr gr % arrowhead n 8508 3225 m 8718 3150 l 8508 3075 l col0 s % Polyline gs clippath 8250 8505 m 8250 8355 l 8013 8355 l 8223 8430 l 8013 8505 l cp eoclip n 5415 8430 m 8235 8430 l gs col0 s gr gr % arrowhead n 8013 8505 m 8223 8430 l 8013 8355 l col0 s % Polyline gs clippath 3285 7065 m 3405 7065 l 3405 6889 l 3345 7039 l 3285 6889 l cp eoclip n 3345 5325 m 3345 7050 l gs col0 s gr gr % arrowhead n 3285 6889 m 3345 7039 l 3405 6889 l col0 s % Polyline gs clippath 10275 6990 m 10395 6990 l 10395 6814 l 10335 6964 l 10275 6814 l cp eoclip n 10335 5205 m 10335 6975 l gs col0 s gr gr % arrowhead n 10275 6814 m 10335 6964 l 10395 6814 l col0 s % Polyline [60 30 15 30] 0 sd gs clippath 4015 2000 m 3929 1974 l 3878 2146 l 3964 2015 l 3965 2171 l cp eoclip n 3637 3133 m 3968 2002 l gs col0 s gr gr [] 0 sd % arrowhead n 3965 2171 m 3964 2015 l 3878 2146 l 3965 2171 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 9644 1917 m 9764 1922 l 9773 1747 l 9706 1894 l 9653 1741 l cp eoclip n 9330 2100 m 9327 2099 l 9320 2097 l 9307 2093 l 9288 2087 l 9263 2079 l 9232 2070 l 9196 2058 l 9156 2045 l 9114 2031 l 9072 2016 l 9030 2001 l 8990 1986 l 8952 1971 l 8917 1957 l 8885 1943 l 8855 1928 l 8828 1914 l 8804 1900 l 8782 1886 l 8762 1872 l 8744 1857 l 8728 1841 l 8713 1825 l 8699 1809 l 8687 1793 l 8676 1776 l 8666 1758 l 8656 1739 l 8647 1720 l 8639 1699 l 8633 1678 l 8627 1657 l 8622 1634 l 8618 1612 l 8616 1589 l 8614 1565 l 8614 1542 l 8615 1519 l 8617 1496 l 8620 1473 l 8624 1451 l 8630 1429 l 8636 1408 l 8644 1387 l 8652 1368 l 8661 1348 l 8672 1330 l 8683 1312 l 8695 1295 l 8709 1277 l 8725 1259 l 8742 1242 l 8760 1226 l 8780 1209 l 8801 1194 l 8823 1178 l 8847 1164 l 8872 1150 l 8897 1136 l 8924 1124 l 8951 1112 l 8978 1101 l 9006 1092 l 9034 1083 l 9061 1076 l 9089 1070 l 9115 1065 l 9142 1061 l 9167 1058 l 9192 1056 l 9216 1056 l 9240 1056 l 9263 1058 l 9285 1060 l 9307 1064 l 9329 1068 l 9350 1074 l 9372 1081 l 9393 1089 l 9414 1098 l 9435 1108 l 9456 1120 l 9476 1132 l 9495 1146 l 9514 1160 l 9532 1175 l 9550 1190 l 9566 1207 l 9582 1223 l 9596 1240 l 9610 1258 l 9622 1275 l 9634 1293 l 9644 1311 l 9654 1329 l 9662 1347 l 9670 1365 l 9678 1385 l 9684 1406 l 9690 1428 l 9695 1450 l 9700 1474 l 9703 1500 l 9706 1528 l 9708 1558 l 9709 1591 l 9710 1625 l 9711 1662 l 9711 1699 l 9710 1737 l 9710 1774 l 9709 1808 l 9708 1838 l 9707 1863 l 9705 1905 l gs col0 s gr gr % arrowhead 0 slj n 9653 1741 m 9706 1894 l 9773 1747 l 9653 1741 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 4922 7414 m 4947 7531 l 5119 7495 l 4960 7468 l 5094 7378 l cp eoclip n 4635 7260 m 4635 7257 l 4634 7250 l 4633 7237 l 4631 7218 l 4629 7192 l 4627 7162 l 4625 7127 l 4623 7090 l 4621 7052 l 4620 7014 l 4619 6977 l 4619 6942 l 4620 6910 l 4621 6880 l 4624 6853 l 4627 6828 l 4632 6805 l 4637 6783 l 4644 6763 l 4651 6743 l 4660 6725 l 4669 6709 l 4679 6693 l 4690 6677 l 4702 6662 l 4716 6646 l 4730 6632 l 4746 6618 l 4762 6604 l 4780 6591 l 4799 6578 l 4818 6566 l 4839 6555 l 4860 6545 l 4881 6536 l 4903 6527 l 4925 6520 l 4948 6514 l 4970 6509 l 4992 6506 l 5014 6503 l 5037 6501 l 5059 6501 l 5081 6501 l 5103 6503 l 5125 6505 l 5147 6509 l 5170 6513 l 5193 6519 l 5217 6526 l 5240 6534 l 5265 6543 l 5289 6553 l 5313 6565 l 5338 6577 l 5362 6591 l 5385 6605 l 5409 6620 l 5431 6635 l 5452 6652 l 5473 6668 l 5492 6685 l 5510 6703 l 5527 6720 l 5543 6738 l 5557 6756 l 5570 6774 l 5582 6792 l 5593 6810 l 5602 6828 l 5610 6847 l 5618 6867 l 5624 6887 l 5629 6907 l 5633 6928 l 5636 6949 l 5638 6971 l 5638 6993 l 5638 7015 l 5636 7037 l 5633 7059 l 5628 7081 l 5623 7102 l 5617 7123 l 5609 7143 l 5600 7163 l 5591 7181 l 5581 7199 l 5570 7216 l 5558 7232 l 5545 7247 l 5532 7261 l 5518 7275 l 5502 7288 l 5486 7300 l 5469 7312 l 5450 7323 l 5429 7333 l 5407 7344 l 5382 7354 l 5355 7364 l 5325 7374 l 5293 7384 l 5258 7395 l 5222 7405 l 5184 7415 l 5145 7425 l 5108 7434 l 5072 7443 l 5039 7450 l 5010 7457 l 4988 7462 l 4950 7470 l gs col0 s gr gr % arrowhead 0 slj n 5094 7378 m 4960 7468 l 5119 7495 l 5094 7378 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 9014 7392 m 9134 7397 l 9143 7222 l 9076 7369 l 9023 7216 l cp eoclip n 8700 7575 m 8697 7574 l 8690 7572 l 8677 7568 l 8658 7562 l 8633 7554 l 8602 7545 l 8566 7533 l 8526 7520 l 8484 7506 l 8442 7491 l 8400 7476 l 8360 7461 l 8322 7446 l 8287 7432 l 8255 7418 l 8225 7403 l 8198 7389 l 8174 7375 l 8152 7361 l 8132 7347 l 8114 7332 l 8098 7316 l 8083 7300 l 8069 7284 l 8057 7268 l 8046 7251 l 8036 7233 l 8026 7214 l 8017 7195 l 8009 7174 l 8003 7153 l 7997 7132 l 7992 7109 l 7988 7087 l 7986 7064 l 7984 7040 l 7984 7017 l 7985 6994 l 7987 6971 l 7990 6948 l 7994 6926 l 8000 6904 l 8006 6883 l 8014 6862 l 8022 6843 l 8031 6823 l 8042 6805 l 8053 6787 l 8065 6770 l 8079 6752 l 8095 6734 l 8112 6717 l 8130 6701 l 8150 6684 l 8171 6669 l 8193 6653 l 8217 6639 l 8242 6625 l 8267 6611 l 8294 6599 l 8321 6587 l 8348 6576 l 8376 6567 l 8404 6558 l 8431 6551 l 8459 6545 l 8485 6540 l 8512 6536 l 8537 6533 l 8562 6531 l 8586 6531 l 8610 6531 l 8633 6533 l 8655 6535 l 8677 6539 l 8699 6543 l 8720 6549 l 8742 6556 l 8763 6564 l 8784 6573 l 8805 6583 l 8826 6595 l 8846 6607 l 8865 6621 l 8884 6635 l 8902 6650 l 8920 6665 l 8936 6682 l 8952 6698 l 8966 6715 l 8980 6733 l 8992 6750 l 9004 6768 l 9014 6786 l 9024 6804 l 9032 6822 l 9040 6840 l 9048 6860 l 9054 6881 l 9060 6903 l 9065 6925 l 9070 6949 l 9073 6975 l 9076 7003 l 9078 7033 l 9079 7066 l 9080 7100 l 9081 7137 l 9081 7174 l 9080 7212 l 9080 7249 l 9079 7283 l 9078 7313 l 9077 7338 l 9075 7380 l gs col0 s gr gr % arrowhead 0 slj n 9023 7216 m 9076 7369 l 9143 7222 l 9023 7216 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 5102 1819 m 5127 1936 l 5299 1900 l 5140 1873 l 5274 1783 l cp eoclip n 4815 1665 m 4815 1662 l 4814 1655 l 4813 1642 l 4811 1623 l 4809 1597 l 4807 1567 l 4805 1532 l 4803 1495 l 4801 1457 l 4800 1419 l 4799 1382 l 4799 1347 l 4800 1315 l 4801 1285 l 4804 1258 l 4807 1233 l 4812 1210 l 4817 1188 l 4824 1168 l 4831 1148 l 4840 1130 l 4849 1114 l 4859 1098 l 4870 1082 l 4882 1067 l 4896 1051 l 4910 1037 l 4926 1023 l 4942 1009 l 4960 996 l 4979 983 l 4998 971 l 5019 960 l 5040 950 l 5061 941 l 5083 932 l 5105 925 l 5128 919 l 5150 914 l 5172 911 l 5194 908 l 5217 906 l 5239 906 l 5261 906 l 5283 908 l 5305 910 l 5327 914 l 5350 918 l 5373 924 l 5397 931 l 5420 939 l 5445 948 l 5469 958 l 5493 970 l 5518 982 l 5542 996 l 5565 1010 l 5589 1025 l 5611 1040 l 5632 1057 l 5653 1073 l 5672 1090 l 5690 1108 l 5707 1125 l 5723 1143 l 5737 1161 l 5750 1179 l 5762 1197 l 5773 1215 l 5782 1233 l 5790 1252 l 5798 1272 l 5804 1292 l 5809 1312 l 5813 1333 l 5816 1354 l 5818 1376 l 5818 1398 l 5818 1420 l 5816 1442 l 5813 1464 l 5808 1486 l 5803 1507 l 5797 1528 l 5789 1548 l 5780 1568 l 5771 1586 l 5761 1604 l 5750 1621 l 5738 1637 l 5725 1652 l 5712 1666 l 5698 1680 l 5682 1693 l 5666 1705 l 5649 1717 l 5630 1728 l 5609 1738 l 5587 1749 l 5562 1759 l 5535 1769 l 5505 1779 l 5473 1789 l 5438 1800 l 5402 1810 l 5364 1820 l 5325 1830 l 5288 1839 l 5252 1848 l 5219 1855 l 5190 1862 l 5168 1867 l 5130 1875 l gs col0 s gr gr % arrowhead 0 slj n 5274 1783 m 5140 1873 l 5299 1900 l 5274 1783 l cp gs 0.00 setgray ef gr col0 s /Times-Roman ff 180.00 scf sf 5520 3225 m gs 1 -1 sc (Ret) col0 sh gr /Times-Roman ff 180.00 scf sf 11355 1740 m gs 1 -1 sc (1/Re) col0 sh gr /Times-Roman ff 180.00 scf sf 10665 3195 m gs 1 -1 sc (e2) col0 sh gr /Times-Roman ff 180.00 scf sf 10921 8605 m gs 1 -1 sc (e2) col0 sh gr /Times-Roman ff 180.00 scf sf 8250 8850 m gs 1 -1 sc (ce) col0 sh gr /Times-Roman ff 180.00 scf sf 8355 1395 m gs 1 -1 sc (Rt) col0 sh gr /Times-Roman ff 180.00 scf sf 6000 1350 m gs 1 -1 sc (Rt) col0 sh gr /Times-Roman ff 180.00 scf sf 11565 3212 m gs 1 -1 sc (Re) col0 sh gr /Times-Roman ff 180.00 scf sf 7860 6495 m gs 1 -1 sc (f) col0 sh gr /Times-Roman ff 180.00 scf sf 5565 6450 m gs 1 -1 sc (ft) col0 sh gr /Times-Roman ff 180.00 scf sf 6795 2925 m gs 1 -1 sc (z=) col0 sh gr /Times-Roman ff 180.00 scf sf 6780 8220 m gs 1 -1 sc (z=) col0 sh gr /Times-Roman ff 180.00 scf sf 1575 7350 m gs 1 -1 sc (Uet) col0 sh gr /Times-Roman ff 180.00 scf sf 4485 8700 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 180.00 scf sf 2895 5985 m gs 1 -1 sc (phit) col0 sh gr /Times-Roman ff 180.00 scf sf 10515 5970 m gs 1 -1 sc (phi) col0 sh gr /Times-Roman ff 180.00 scf sf 11520 7230 m gs 1 -1 sc (Ue) col0 sh gr /Times-Roman ff 180.00 scf sf 1065 8880 m gs 1 -1 sc (cet) col0 sh gr /Times-Roman ff 180.00 scf sf 3864 1793 m gs 1 -1 sc (1/Ret) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 638 2696 a /End PSfrag 638 2696 a 638 774 a /Hide PSfrag 638 774 a -173 837 a FI(PSfrag)30 b(replacemen)m(ts)p -173 869 811 4 v 638 873 a /Unhide PSfrag 638 873 a 595 949 a { 595 949 a 565 960 a Fd(e)552 977 y Fl(R)605 985 y Fk(")595 949 y } 0/Place PSfrag 595 949 a 589 1046 a { 589 1046 a 574 1027 a Fj(1)p 550 1039 79 3 v 562 1077 a Fc(e)550 1090 y Fk(R)596 1098 y FA(")589 1046 y } 1/Place PSfrag 589 1046 a 620 1190 a { 620 1190 a 602 1211 a FJ(1)620 1190 y } 2/Place PSfrag 620 1190 a 595 1297 a { 595 1297 a 552 1316 a Fl(R)605 1324 y Fk(")595 1297 y } 3/Place PSfrag 595 1297 a 570 1393 a { 570 1393 a 538 1384 a FC(")569 1359 y Fz(2)p 513 1398 116 4 v 513 1446 a FB(4)p FC(R)596 1454 y FA(")570 1393 y } 4/Place PSfrag 570 1393 a 611 1524 a { 611 1524 a 595 1506 a FC(")p 594 1520 34 4 v 594 1567 a FB(2)611 1524 y } 5/Place PSfrag 611 1524 a 511 1637 a { 511 1637 a 384 1652 a FD(z)d FF(=)548 1619 y FC(")p 547 1633 34 4 v 547 1680 a FB(2)590 1652 y FD(u)511 1637 y } 6/Place PSfrag 511 1637 a 579 1758 a { 579 1758 a 520 1780 a Fw(R)590 1792 y FC(!)579 1758 y } 7/Place PSfrag 579 1758 a 520 1850 a { 520 1850 a 419 1855 a FE(e)403 1876 y FD(F)456 1891 y FC(\013)p FB(\()p FC(")p FB(\))p FC(;")520 1850 y } 8/Place PSfrag 520 1850 a 520 1975 a { 520 1975 a 403 1989 a FD(F)456 2004 y FC(\013)p FB(\()p FC(")p FB(\))p FC(;")520 1975 y } 9/Place PSfrag 520 1975 a 573 2094 a { 573 2094 a 539 2118 a Fm(U)601 2132 y Fl(")573 2094 y } 10/Place PSfrag 573 2094 a 573 2193 a { 573 2193 a 552 2208 a Fn(e)539 2231 y Fm(U)601 2245 y Fl(")573 2193 y } 11/Place PSfrag 573 2193 a 593 2319 a { 593 2319 a 559 2341 a FE(e)-57 b FD(')602 2353 y FC(")593 2319 y } 12/Place PSfrag 593 2319 a 593 2444 a { 593 2444 a 548 2454 a FD(')602 2466 y FC(")593 2444 y } 13/Place PSfrag 593 2444 a 638 2583 a { 638 2583 a 638 2583 a } 14/Place PSfrag 638 2583 a 638 2696 a { 638 2696 a 638 2696 a } 15/Place PSfrag 638 2696 a 776 2892 a FI(Figure)30 b(3:)41 b FF(Scaling)27 b(the)h(dynamical)f(plane)g(and)g(the)h (linearizing)f(plane.)71 3182 y Fu(4)135 b(Pro)t(of)45 b(of)g(Theorem)g(A)71 3399 y FI(The)24 b(pro)s(of)f(of)h(Theorem)g(A)h (is)e(based)h(in)f(an)h(explicit)e(\(quan)m(titativ)m(e\))k(v)m(ersion) e(of)g(the)g(\(qualitativ)m(e\))h(surgery)71 3512 y(construction)g([G]) g(that)h(relates)f(a)h(mem)m(b)s(er)e(of)i(the)f(\(non)g(scaled\))g (complex)g(standard)f(family)3393 3489 y Fn(e)3375 3512 y Fm(F)3433 3530 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")3622 3512 y FI(,)j(with)71 3625 y(the)e(semistandard)d(map)j Fm(G)p FI(.)39 b(First,)25 b(in)e(Section)h(4.1)i(w)m(e)f(explain)e (Gey)m(er's)i(construction,)h(sligh)m(tly)c(mo)s(di\014ed)71 3737 y(and)33 b(adapted)h(to)g(the)g(scaled)g(map)f Fm(F)1412 3756 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")1602 3737 y FI(.)50 b(In)33 b(Section)h(4.2)h(w)m(e)f(re-form)m(ulate)g(Theorem)f (A)h(in)f(terms)g(of)71 3850 y(the)d(previous)f(surgery)g (construction.)40 b(Section)30 b(4.3)h(giv)m(es)f(an)g(explicit)f(v)m (ersion)g(of)h(Gey)m(er's)h(construction,)71 3963 y(whic)m(h)h(allo)m 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b(alen)m(t)23 b(\(in)g(the)h (dynamical)e(sense\))71 5571 y(to)35 b(the)f(semistandard)f(map.)52 b(Ho)m(w)m(ev)m(er,)37 b(the)e(map)f Fm(H)1995 5585 y Fl(")2065 5571 y FI(constructed)g(in)f(this)g(w)m(a)m(y)j(will)31 b(b)s(e)j(quasiregular,)71 5684 y(but)39 b(not)h(holomorphic.)66 b(W)-8 b(e)41 b(shall)d(mak)m(e)i(it)g(holomorphic)d(b)m(y)j(means)f (of)h(the)g(Ahlfors-Bers')f(theorem)71 5796 y(\(see)g(Theorem)f (2.12\),)k(as)d(w)m(e)g(explained)d(in)h(Remark)h(2.13.)66 b(So,)41 b(w)m(e)d(will)e(construct)j(a)f(Beltrami)g(form)p eop %%Page: 15 15 15 14 bop 71 274 a Fs(N.)31 b(F)-8 b(agella,)31 b(T.)f(M.)h(Seara)g (and)f(J.)g(Villan)m(uev)-5 b(a)2025 b FI(15)71 573 y Fm(\026)126 587 y Fl(")162 573 y FI(,)30 b(in)m(v)-5 b(arian)m(t)29 b(b)m(y)g Fm(H)799 587 y Fl(")835 573 y FI(,)h(and)e(b)m(y)h(in)m(tegration)h(of)f Fm(\026)1809 587 y Fl(")1875 573 y FI(w)m(e)h(will)c(obtain)j(a)h(quasiconformal)e (map)g Fm(h)3413 587 y Fl(")3480 573 y FI(suc)m(h)h(that)71 686 y Fm(h)123 700 y Fl(")174 686 y Fo(\016)14 b Fm(H)309 700 y Fl(")360 686 y Fo(\016)g Fm(h)471 653 y Fp(\000)p FJ(1)471 708 y Fl(")593 686 y FI(is)27 b(holomorphic,)f(and)h(has)g (the)g(same)h(dynamics)e(than)h Fm(H)2620 700 y Fl(")2683 686 y FI(\()p Fr(holomorphic)34 b(smo)-5 b(othing)8 b FI(\).)42 b(By)71 799 y(c)m(ho)s(osing)30 b Fm(H)516 813 y Fl(")583 799 y FI(appropiatelly)-8 b(,)29 b(w)m(e)h(will)e(pro)m (v)m(e)j(that)g(this)f(new)f(map)i(is)e(the)h(semistandard)f(map)h Fm(G)p FI(.)212 912 y(W)-8 b(e)28 b(no)m(w)e(pro)s(ceed)g(to)h(mak)m(e) g(this)f(construction)g(precise.)38 b(T)-8 b(o)27 b(de\014ne)f(a)h (rotation)f(inside)e(the)j(small)e(disk)71 1025 y Fq(D)142 1015 y Fk(")p 146 1027 31 3 v 146 1068 a Fj(2)191 1025 y FI(,)39 b(w)m(e)e(\014rst)g(c)m(ho)s(ose)h(a)f(\\gluing")g(map)f Fm( )1605 1039 y Fl(")1642 1025 y FI(.)61 b(Let)38 b Fm( )1957 1039 y Fl(")2030 1025 y FI(:)p 2091 951 121 4 v 36 w Fq(D)2162 1015 y Fk(")p 2167 1027 31 3 v 2167 1068 a Fj(2)2248 1025 y Fo(!)p 2375 951 121 4 v 36 w Fq(D)2446 1015 y Fk(")p 2451 1027 31 3 v 2451 1068 a Fj(2)2533 1025 y FI(b)s(e)e(an)m(y)h(quasiconformal)f(map)h(that)71 1138 y(agrees)c(with)d Fm(')613 1152 y Fl(")682 1138 y FI(on)h(the)h(b)s(oundary)e(\(i.e.,)j Fm( )1646 1156 y Fl(")p Fp(j)p Fi(C)1769 1155 y Fk(")p 1767 1163 V 1767 1198 a Fj(2)1843 1138 y FI(=)27 b Fm(')2000 1152 y Fl(")2037 1138 y FI(\))32 b(and)f(sends)g(0)h(to)h(0.)45 b(Since)30 b Fm(')3129 1152 y Fl(")3198 1138 y FI(is)h(the)h(pro)5 b(jection)71 1274 y(of)33 b(a)h(real)f(analytic)g(map,)h(the)f (existence)h(of)g Fm( )1731 1288 y Fl(")1801 1274 y FI(is)e(guaran)m (teed)j(\(see)f([P)m(om)q(]\).)50 b(Then)32 b(w)m(e)i(de\014ne)e(the)i (new)71 1387 y(map)c Fm(H)349 1401 y Fl(")415 1387 y FI(as:)1267 1575 y Fm(H)1343 1589 y Fl(")1404 1575 y FI(=)1500 1420 y Fn(\()1573 1509 y Fm(F)1631 1527 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")2247 1509 y FI(on)g Fq(C)44 b Fo(n)20 b Fq(D)2595 1499 y Fk(")p 2600 1511 V 2600 1552 a Fj(2)1573 1647 y Fm( )1632 1661 y Fl(")1690 1647 y Fo(\016)g(R)1832 1661 y Fl(!)1903 1647 y Fo(\016)h Fm( )2031 1614 y Fp(\000)p FJ(1)2028 1670 y Fl(")2247 1647 y FI(on)p 2373 1574 121 4 v 30 w Fq(D)2444 1637 y Fk(")p 2449 1649 31 3 v 2449 1690 a Fj(2)2493 1647 y Fm(:)71 1816 y FI(See)30 b(Figure)g(4.)212 1929 y(By)k(the)f(c)m (hoice)h(of)f Fm( )955 1943 y Fl(")992 1929 y FI(,)i(the)e(map)g Fm(H)1492 1943 y Fl(")1561 1929 y FI(is)g(con)m(tin)m(uous)g(and)f (quasiregular.)48 b(By)33 b(construction,)h(it)f(has)g(a)71 2042 y(\014xed)h(p)s(oin)m(t)f(at)i Fm(z)i FI(=)31 b(0,)37 b(and)c(it)h(is)g(conjugate)i(to)f(a)g(rotation)g(of)f(angle)h Fm(!)i FI(on)d(the)h(set)g(\(top)s(ological)f(disk\))71 2155 y Fq(D)142 2145 y Fk(")p 146 2157 V 146 2199 a Fj(2)215 2155 y Fo([)23 b Fm(U)361 2169 y Fl(")433 2155 y FI(b)m(y)35 b(means)g(of)g(the)h(conjugacy)g(maps)f Fm(')1848 2169 y Fl(")1920 2155 y FI(and)g Fm( )2161 2169 y Fl(")2198 2155 y FI(,)i(whic)m(h)d(matc)m(h)i(up)e(con)m(tin)m(uously)-8 b(.)55 b(Last,)37 b(w)m(e)71 2268 y(notice)23 b(that)h 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FI(b)m(y)f(means)h(of)f Fm( )877 2687 y Fp(\000)p FJ(1)874 2743 y Fl(")972 2720 y FI(.)39 b(W)-8 b(e)26 b(extend)g(this)e(almost)h(complex)g(structure) g(to)h(ev)m(ery)g(preimage)f(of)g Fq(D)3568 2710 y Fk(")p 3573 2722 V 3573 2763 a Fj(2)3643 2720 y FI(using)71 2833 y Fm(H)147 2847 y Fl(")216 2833 y FI(\(or)33 b(equiv)-5 b(alen)m(tly)31 b Fm(F)930 2852 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")1119 2833 y FI(,)j(as)f(b)s(oth)e(maps)h(coincide)g (outside)g(of)h Fq(D)2591 2823 y Fk(")p 2596 2835 V 2596 2876 a Fj(2)2641 2833 y FI(\);)h(and)e(\014nally)e(w)m(e)j(set)h Fm(\026)3527 2847 y Fl(")3592 2833 y FI(=)29 b(0)k(at)71 2946 y(the)d(remaining)f(p)s(oin)m(ts.)40 b(That)30 b(is,)1086 3294 y Fm(\026)1141 3308 y Fl(")1203 3294 y FI(=)1299 3081 y Fn(8)1299 3163 y(>)1299 3190 y(<)1299 3353 y(>)1299 3381 y(:)1380 3086 y(\000)1421 3159 y Fm( )1483 3126 y Fp(\000)p FJ(1)1480 3182 y Fl(")1578 3086 y Fn(\001)1620 3108 y Fp(\003)1674 3159 y FI(\(0\))129 b(on)31 b Fq(D)2115 3149 y Fk(")p 2120 3161 V 2120 3203 a Fj(2)1380 3297 y FI(\()p Fm(H)1498 3264 y Fl(n)1491 3320 y(")1545 3297 y FI(\))1580 3256 y Fp(\003)1635 3297 y FI(\()p Fm(\026)1725 3311 y Fl(")1762 3297 y FI(\))121 b(on)31 b Fm(H)2128 3264 y Fp(\000)p Fl(n)2121 3320 y(")2229 3297 y FI(\()p Fq(D)2335 3287 y Fk(")p 2340 3299 V 2340 3341 a Fj(2)2385 3297 y FI(\),)g(if)e Fm(n)c Fo(\025)g FI(1)1380 3436 y(0)493 b(on)31 b Fq(C)43 b Fo(n)2196 3367 y Fn(S)2272 3462 y Fl(n)p Fp(\025)p FJ(0)2424 3436 y Fm(H)2507 3403 y Fp(\000)p Fl(n)2500 3458 y(")2609 3436 y FI(\()p Fq(D)2715 3426 y Fk(")p 2720 3438 V 2720 3479 a Fj(2)2765 3436 y FI(\))p Fm(;)71 3659 y FI(where)33 b Fm(H)420 3626 y Fp(\000)p Fl(n)413 3681 y(")522 3659 y FI(\()p Fq(D)628 3649 y Fk(")p 633 3661 V 633 3702 a Fj(2)677 3659 y FI(\))h(should)d(b) s(e)i(understo)s(o)s(d)f(as)i(the)f(set)h(of)g(p)s(oin)m(ts)e(whose)i (n)2768 3626 y FJ(th)2871 3659 y FI(iterate)h(falls)d(\(for)h(the)h (\014rst)71 3772 y(time\))f(in)e Fq(D)495 3762 y Fk(")p 500 3774 V 500 3815 a Fj(2)545 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8828 6833 m gs 1 -1 sc (e2) col0 sh gr % Polyline % % Originally from a GIF File: siegel.gif % Begin Imported PCX File: siegel.gif n gs 795 9195 tr 8.660000 -8.662500 sc 0 -400 tr 0 0 tr sa /showpage {} def % PCX image follows: 400 400 sc %*********************************************************************** %* * %* Object: Image decoding PS-routine Date: 01.02.93 * %* Author: Evgeni CHERNYAEV (chernaev@vxcern.cern.ch) * %* * %* Function: Display a run-length encoded color image. * %* The image is displayed in color on viewers and printers * %* that support color Postscript, otherwise it is displayed * %* as grayscale. * %* * %*********************************************************************** /byte 1 string def /color 3 string def systemdict /colorimage known { /cnt 3 def } { /cnt 1 def } ifelse /String 256 cnt mul string def %*********************************************************************** /DecodePacket % Decode color packet * %*********************************************************************** { currentfile byte readhexstring pop 0 get /Nbyte exch 1 add cnt mul def /color ColorMap currentfile byte readhexstring pop 0 get get def String dup 0 cnt Nbyte 1 sub { color putinterval dup } for pop 0 Nbyte getinterval } bind def %*********************************************************************** /DisplayImage % Display run-length encoded color image * %*********************************************************************** { gsave currentfile String readline pop token { /columns exch def } { } ifelse token { /rows exch def pop } { } ifelse currentfile String readline pop token { /Ncol exch def pop } { } ifelse /ColorMap Ncol array def systemdict /colorimage known { 0 1 Ncol 1 sub { ColorMap exch currentfile 3 string readhexstring pop put } for columns rows 8 [ columns 0 0 rows neg 0 rows ] { DecodePacket } false 3 colorimage }{ 0 1 Ncol 1 sub { ColorMap exch 1 string dup 0 currentfile color readhexstring pop pop color 0 get 0.299 mul color 1 get 0.587 mul add color 2 get 0.114 mul add cvi put put } for columns rows 8 [ columns 0 0 rows neg 0 rows ] { DecodePacket } image } ifelse grestore } bind def %*********************************************************************** %* Image decoding * %*********************************************************************** DisplayImage 400 400 16 000000 ffffff ffffff ffffff ffffff ffffff ffffff ffffff ffffff ffffff ffffff ffffff ffffff ffffff ffffff ffffff ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01 ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01 ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01 ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01ff01 ff01ff01ff01ff01ff0109010000010100005a010300ff012c0100000001060007010800 41010800ff012801000000010000000119003e01030002010400ff012601000000011300 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s % Polyline gs clippath 3090 9105 m 3210 9105 l 3210 8865 l 3150 9075 l 3090 8865 l cp eoclip n 3150 8055 m 3150 9090 l gs col0 s gr gr % arrowhead n 3090 8865 m 3150 9075 l 3210 8865 l col0 s % Polyline gs clippath 8265 9105 m 8385 9105 l 8385 8865 l 8325 9075 l 8265 8865 l cp eoclip n 8325 8055 m 8325 9090 l gs col0 s gr gr % arrowhead n 8265 8865 m 8325 9075 l 8385 8865 l col0 s % Polyline gs clippath 6720 3210 m 6720 3090 l 6480 3090 l 6690 3150 l 6480 3210 l cp eoclip n 4275 3150 m 6705 3150 l gs col0 s gr gr % arrowhead n 6480 3210 m 6690 3150 l 6480 3090 l col0 s % Polyline gs clippath 3045 1980 m 3165 1980 l 3165 1740 l 3105 1950 l 3045 1740 l cp eoclip n 3105 1065 m 3105 1965 l gs col0 s gr gr % arrowhead n 3045 1740 m 3105 1950 l 3165 1740 l col0 s % Polyline gs clippath 8220 1965 m 8340 1965 l 8340 1725 l 8280 1935 l 8220 1725 l cp eoclip n 8280 1050 m 8280 1950 l gs col0 s gr gr % arrowhead n 8220 1725 m 8280 1935 l 8340 1725 l col0 s % Polyline gs clippath 6675 75 m 6675 -45 l 6435 -45 l 6645 15 l 6435 75 l cp eoclip n 4230 15 m 6660 15 l gs col0 s gr gr % arrowhead n 6435 75 m 6645 15 l 6435 -45 l col0 s % Polyline gs clippath 5865 6810 m 5865 6690 l 5625 6690 l 5835 6750 l 5625 6810 l cp eoclip n 4275 6750 m 5850 6750 l gs col0 s gr gr % arrowhead n 5625 6810 m 5835 6750 l 5625 6690 l col0 s % Polyline % % Originally from a GIF File: siegel.gif % Begin Imported PCX File: siegel.gif n gs 6001 9195 tr 8.660000 -8.662500 sc 0 -400 tr 0 0 tr sa /showpage {} def % PCX image follows: 400 400 sc %*********************************************************************** %* * %* Object: Image decoding PS-routine Date: 01.02.93 * %* Author: Evgeni CHERNYAEV (chernaev@vxcern.cern.ch) * %* * %* Function: Display a run-length encoded color image. * %* The image is displayed in color on viewers and printers * %* that support color Postscript, otherwise it is displayed * %* as grayscale. * %* * %*********************************************************************** /byte 1 string def /color 3 string def systemdict /colorimage known { /cnt 3 def } { /cnt 1 def } ifelse /String 256 cnt mul string def %*********************************************************************** /DecodePacket % Decode color packet * %*********************************************************************** { currentfile byte readhexstring pop 0 get /Nbyte exch 1 add cnt mul def /color ColorMap currentfile byte readhexstring pop 0 get get def String dup 0 cnt Nbyte 1 sub { color putinterval dup } for pop 0 Nbyte getinterval } bind def %*********************************************************************** /DisplayImage % Display run-length encoded color image * %*********************************************************************** { gsave currentfile String readline pop token { /columns exch def } { } ifelse token { /rows exch def pop } { } ifelse currentfile String readline pop token { /Ncol exch def pop } { } ifelse /ColorMap Ncol array def systemdict /colorimage known 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5818 l 8790 5855 l 8782 5886 l 8775 5911 l 8770 5930 l 8762 5959 l gs col0 s gr gr % arrowhead n 8755 5757 m 8765 5947 l 8870 5789 l col0 s % Polyline gs clippath 3513 5957 m 3628 5989 l 3683 5789 l 3578 5947 l 3568 5757 l cp eoclip n 3632 3593 m 3633 3595 l 3634 3600 l 3636 3608 l 3640 3622 l 3645 3641 l 3651 3665 l 3659 3696 l 3668 3731 l 3679 3772 l 3690 3816 l 3702 3863 l 3714 3913 l 3726 3964 l 3738 4015 l 3750 4066 l 3761 4116 l 3772 4165 l 3781 4212 l 3790 4257 l 3798 4301 l 3806 4343 l 3812 4383 l 3817 4422 l 3822 4459 l 3826 4495 l 3829 4531 l 3831 4565 l 3833 4599 l 3834 4633 l 3834 4667 l 3833 4701 l 3832 4733 l 3831 4765 l 3828 4798 l 3825 4831 l 3822 4865 l 3818 4900 l 3813 4937 l 3807 4975 l 3800 5014 l 3793 5056 l 3785 5099 l 3775 5144 l 3765 5191 l 3755 5241 l 3743 5292 l 3731 5346 l 3718 5401 l 3704 5457 l 3690 5513 l 3676 5569 l 3662 5625 l 3649 5678 l 3636 5729 l 3624 5775 l 3613 5818 l 3603 5855 l 3595 5886 l 3588 5911 l 3583 5930 l 3575 5959 l gs col0 s gr gr % arrowhead n 3568 5757 m 3578 5947 l 3683 5789 l col0 s % Polyline gs clippath 1376 9683 m 1485 9633 l 1400 9444 l 1420 9633 l 1290 9493 l cp eoclip n 1455 2820 m 1454 2822 l 1452 2825 l 1448 2831 l 1443 2841 l 1434 2856 l 1423 2875 l 1410 2899 l 1393 2928 l 1374 2962 l 1353 3000 l 1329 3042 l 1304 3087 l 1277 3136 l 1249 3186 l 1221 3238 l 1192 3290 l 1163 3343 l 1135 3396 l 1107 3448 l 1080 3499 l 1054 3549 l 1029 3598 l 1006 3645 l 983 3691 l 961 3736 l 941 3779 l 921 3821 l 902 3862 l 885 3903 l 868 3942 l 852 3981 l 837 4020 l 822 4058 l 808 4096 l 794 4135 l 780 4173 l 768 4213 l 756 4248 l 745 4284 l 734 4321 l 723 4358 l 712 4396 l 701 4435 l 691 4474 l 680 4514 l 670 4555 l 660 4596 l 650 4638 l 640 4681 l 630 4725 l 621 4769 l 611 4814 l 602 4859 l 593 4905 l 584 4951 l 576 4998 l 568 5044 l 559 5091 l 552 5138 l 544 5185 l 537 5232 l 530 5279 l 523 5325 l 516 5372 l 510 5417 l 504 5463 l 499 5508 l 493 5552 l 488 5596 l 484 5639 l 479 5682 l 475 5724 l 471 5765 l 467 5806 l 464 5846 l 461 5886 l 458 5925 l 455 5964 l 453 6003 l 450 6043 l 448 6083 l 446 6123 l 444 6163 l 442 6204 l 441 6245 l 440 6285 l 439 6327 l 439 6368 l 439 6410 l 439 6453 l 439 6496 l 440 6539 l 441 6582 l 443 6626 l 445 6670 l 447 6715 l 450 6759 l 453 6804 l 457 6849 l 461 6893 l 465 6938 l 470 6983 l 475 7027 l 481 7071 l 487 7115 l 493 7159 l 500 7202 l 507 7245 l 515 7288 l 523 7330 l 531 7372 l 540 7414 l 549 7455 l 559 7496 l 569 7537 l 579 7578 l 590 7619 l 601 7659 l 613 7700 l 623 7737 l 635 7775 l 647 7813 l 659 7851 l 672 7891 l 686 7931 l 701 7972 l 716 8014 l 732 8057 l 749 8101 l 767 8148 l 786 8195 l 806 8245 l 826 8296 l 848 8350 l 872 8405 l 896 8462 l 921 8521 l 947 8583 l 975 8646 l 1003 8710 l 1032 8776 l 1062 8843 l 1092 8911 l 1122 8979 l 1152 9046 l 1182 9113 l 1211 9178 l 1240 9240 l 1267 9300 l 1292 9356 l 1316 9408 l 1337 9455 l 1357 9496 l 1373 9533 l 1387 9563 l 1399 9589 l 1408 9608 l 1415 9623 l 1425 9645 l gs col0 s gr gr % arrowhead n 1290 9493 m 1420 9633 l 1400 9444 l col0 s % Polyline gs clippath 925 10057 m 1036 10010 l 955 9818 l 970 10008 l 844 9865 l cp eoclip n 1560 -15 m 1559 -14 l 1558 -11 l 1556 -5 l 1552 3 l 1546 16 l 1539 32 l 1529 54 l 1518 80 l 1504 111 l 1488 147 l 1470 188 l 1450 233 l 1428 283 l 1405 336 l 1380 393 l 1355 452 l 1328 514 l 1301 577 l 1273 641 l 1246 706 l 1218 772 l 1191 837 l 1164 902 l 1137 966 l 1112 1029 l 1087 1091 l 1062 1152 l 1038 1212 l 1015 1270 l 993 1328 l 972 1384 l 951 1439 l 931 1494 l 912 1548 l 893 1601 l 875 1653 l 857 1706 l 839 1758 l 822 1810 l 806 1862 l 789 1915 l 773 1967 l 757 2021 l 741 2075 l 725 2130 l 712 2176 l 699 2222 l 686 2269 l 673 2317 l 660 2365 l 647 2414 l 634 2464 l 621 2515 l 608 2567 l 595 2619 l 582 2672 l 568 2727 l 555 2782 l 542 2838 l 529 2894 l 516 2952 l 502 3010 l 489 3070 l 476 3129 l 463 3190 l 450 3251 l 437 3313 l 424 3375 l 411 3437 l 398 3501 l 386 3564 l 373 3627 l 361 3691 l 349 3755 l 337 3819 l 325 3883 l 314 3947 l 303 4011 l 291 4074 l 281 4137 l 270 4200 l 260 4263 l 250 4325 l 240 4386 l 231 4447 l 222 4508 l 213 4567 l 204 4627 l 196 4685 l 188 4743 l 180 4800 l 173 4857 l 166 4913 l 159 4968 l 153 5022 l 147 5076 l 141 5129 l 135 5182 l 130 5234 l 125 5286 l 120 5338 l 115 5395 l 110 5451 l 106 5508 l 102 5565 l 98 5621 l 94 5677 l 91 5734 l 88 5790 l 86 5847 l 83 5904 l 81 5961 l 80 6018 l 78 6075 l 77 6132 l 77 6189 l 76 6247 l 76 6304 l 77 6362 l 77 6419 l 78 6477 l 80 6534 l 82 6592 l 84 6648 l 86 6705 l 89 6762 l 92 6818 l 95 6873 l 99 6928 l 103 6983 l 108 7037 l 112 7090 l 117 7143 l 122 7195 l 128 7247 l 134 7297 l 140 7347 l 146 7396 l 152 7445 l 159 7492 l 166 7539 l 173 7585 l 181 7631 l 188 7675 l 196 7719 l 204 7763 l 212 7805 l 221 7848 l 230 7890 l 238 7931 l 247 7972 l 248 7973 l 258 8017 l 268 8061 l 279 8106 l 290 8150 l 302 8194 l 314 8239 l 327 8284 l 340 8330 l 354 8376 l 368 8423 l 383 8471 l 399 8520 l 416 8571 l 434 8623 l 452 8676 l 471 8730 l 491 8786 l 512 8844 l 534 8903 l 556 8964 l 580 9026 l 604 9089 l 628 9153 l 653 9218 l 679 9284 l 704 9349 l 730 9414 l 755 9478 l 780 9540 l 804 9601 l 828 9659 l 850 9713 l 870 9764 l 889 9811 l 906 9853 l 921 9890 l 935 9922 l 946 9949 l 955 9971 l 962 9988 l 967 10001 l 975 10020 l gs col0 s gr gr % arrowhead n 844 9865 m 970 10008 l 955 9818 l col0 s % Polyline gs clippath 9324 9633 m 9433 9683 l 9519 9493 l 9390 9633 l 9409 9444 l cp eoclip n 9355 2820 m 9356 2822 l 9358 2825 l 9362 2831 l 9367 2841 l 9376 2856 l 9387 2875 l 9400 2899 l 9417 2928 l 9436 2962 l 9457 3000 l 9481 3042 l 9506 3087 l 9533 3136 l 9561 3186 l 9589 3238 l 9618 3290 l 9647 3343 l 9675 3396 l 9703 3448 l 9730 3499 l 9756 3549 l 9781 3598 l 9804 3645 l 9827 3691 l 9849 3736 l 9869 3779 l 9889 3821 l 9908 3862 l 9925 3903 l 9942 3942 l 9958 3981 l 9973 4020 l 9988 4058 l 10002 4096 l 10016 4135 l 10030 4173 l 10043 4213 l 10054 4248 l 10065 4284 l 10076 4321 l 10087 4358 l 10098 4396 l 10109 4435 l 10119 4474 l 10130 4514 l 10140 4555 l 10150 4596 l 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10124 7931 l 10109 7972 l 10094 8014 l 10078 8057 l 10061 8101 l 10043 8148 l 10024 8195 l 10004 8245 l 9984 8296 l 9962 8350 l 9938 8405 l 9914 8462 l 9889 8521 l 9863 8583 l 9835 8646 l 9807 8710 l 9778 8776 l 9748 8843 l 9718 8911 l 9688 8979 l 9658 9046 l 9628 9113 l 9599 9178 l 9570 9240 l 9543 9300 l 9518 9356 l 9494 9408 l 9473 9455 l 9453 9496 l 9437 9533 l 9423 9563 l 9411 9589 l 9402 9608 l 9395 9623 l 9385 9645 l gs col0 s gr gr % arrowhead n 9409 9444 m 9390 9633 l 9519 9493 l col0 s % Polyline gs clippath 9643 10025 m 9754 10072 l 9835 9880 l 9710 10023 l 9724 9833 l cp eoclip n 9405 300 m 9405 301 l 9406 304 l 9408 309 l 9411 317 l 9416 329 l 9421 345 l 9428 366 l 9437 391 l 9448 420 l 9460 454 l 9473 493 l 9488 536 l 9505 582 l 9522 632 l 9540 684 l 9559 739 l 9579 796 l 9598 853 l 9618 912 l 9638 971 l 9658 1030 l 9678 1088 l 9697 1146 l 9716 1204 l 9734 1261 l 9752 1316 l 9770 1371 l 9787 1425 l 9803 1478 l 9819 1530 l 9835 1581 l 9850 1632 l 9865 1682 l 9879 1732 l 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6090 l 10603 6147 l 10603 6204 l 10604 6262 l 10604 6319 l 10603 6377 l 10603 6434 l 10602 6492 l 10600 6549 l 10598 6607 l 10596 6663 l 10594 6720 l 10591 6777 l 10588 6833 l 10585 6888 l 10581 6943 l 10577 6998 l 10572 7052 l 10568 7105 l 10563 7158 l 10558 7210 l 10552 7262 l 10546 7312 l 10540 7362 l 10534 7411 l 10528 7460 l 10521 7507 l 10514 7554 l 10507 7600 l 10499 7646 l 10492 7690 l 10484 7734 l 10476 7778 l 10468 7820 l 10459 7863 l 10450 7905 l 10442 7946 l 10433 7987 l 10433 7988 l 10422 8032 l 10412 8076 l 10401 8121 l 10390 8165 l 10378 8209 l 10366 8254 l 10353 8299 l 10340 8345 l 10326 8391 l 10312 8438 l 10297 8486 l 10281 8535 l 10264 8586 l 10246 8638 l 10228 8691 l 10209 8745 l 10189 8801 l 10168 8859 l 10146 8918 l 10124 8979 l 10100 9041 l 10076 9104 l 10052 9168 l 10027 9233 l 10001 9299 l 9976 9364 l 9950 9429 l 9925 9493 l 9900 9555 l 9876 9616 l 9852 9674 l 9830 9728 l 9810 9779 l 9791 9826 l 9774 9868 l 9759 9905 l 9745 9937 l 9734 9964 l 9725 9986 l 9718 10003 l 9713 10016 l 9705 10035 l gs col0 s gr gr % arrowhead n 9724 9833 m 9710 10023 l 9835 9880 l col0 s /Times-Roman-iso ff 180.00 scf sf 4084 53 m gs 1 -1 sc (R) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 3196 1514 m gs 1 -1 sc (z/be) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 5205 -90 m gs 1 -1 sc (Rt) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 5205 3015 m gs 1 -1 sc (Rt) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 9267 46 m gs 1 -1 sc (R) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 9307 3208 m gs 1 -1 sc (Re) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 2556 4773 m gs 1 -1 sc (psie) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 8651 4766 m gs 1 -1 sc (vphie) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 3921 4766 m gs 1 -1 sc (vphie) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 7287 4773 m gs 1 -1 sc (psie) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 641 6855 m gs 1 -1 sc (ce) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 5851 6855 m gs 1 -1 sc (ce) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 4935 6645 m gs 1 -1 sc (He) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 3255 8580 m gs 1 -1 sc (he) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 8040 8580 m gs 1 -1 sc (he) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 5875 11218 m gs 1 -1 sc (-1) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 639 11199 m gs 1 -1 sc (-1) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 105 3990 m gs 1 -1 sc (phi) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 10515 3990 m gs 1 -1 sc (phi) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 9930 5640 m gs 1 -1 sc (phie) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 630 5640 m gs 1 -1 sc (phie) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 7845 1529 m gs 1 -1 sc (z/be) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 4950 10890 m gs 1 -1 sc (G) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 4245 9930 m gs 1 -1 sc (U) col0 sh gr % Ellipse n 3102 -40 467 467 0 360 DrawEllipse gs col0 s gr % Ellipse n 8280 -40 467 467 0 360 DrawEllipse gs col0 s gr % Ellipse n 4012 3152 20 20 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3554 3152 20 20 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 9230 3152 20 20 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 8772 3152 20 20 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3568 6798 20 20 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 850 6798 20 20 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 8771 6798 20 20 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 6047 6798 20 20 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 4018 1 20 20 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 9195 -6 20 20 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 850 11039 20 20 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 6121 11039 20 20 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3138 11058 20 20 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 8338 11058 20 20 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr $F2psEnd rs %%EndDocument @endspecial 100 5407 a /End PSfrag 100 5407 a 100 3492 a /Hide PSfrag 100 3492 a -710 3555 a FI(PSfrag)30 b(replacemen)m(ts)p -710 3587 811 4 v 100 3591 a /Unhide PSfrag 100 3591 a 73 3681 a { 73 3681 a 46 3704 a Fl(R)73 3681 y } 0/Place PSfrag 73 3681 a 32 3769 a { 32 3769 a 14 3741 a Fl(z)p -27 3756 118 4 v -27 3809 a(b)p FJ(\()p Fl(")p FJ(\))32 3769 y } 1/Place PSfrag 32 3769 a 36 3901 a { 36 3901 a -28 3925 a Fo(R)49 3939 y Fl(!)36 3901 y } 2/Place PSfrag 36 3901 a 57 4025 a { 57 4025 a 14 4043 a Fl(R)67 4051 y Fk(")57 4025 y } 3/Place PSfrag 57 4025 a 76 4135 a { 76 4135 a 62 4124 a FA(")p 62 4132 29 3 v 62 4165 a Fz(2)76 4135 y } 4/Place PSfrag 76 4135 a 52 4237 a { 52 4237 a 4 4260 a Fm( )63 4274 y Fl(")52 4237 y } 5/Place PSfrag 52 4237 a 52 4362 a { 52 4362 a 4 4373 a Fm(')63 4387 y Fl(")52 4362 y } 6/Place PSfrag 52 4362 a 100 4503 a { 100 4503 a 100 4503 a } 7/Place PSfrag 100 4503 a 44 4578 a { 44 4578 a -12 4603 a Fm(H)64 4617 y Fl(")44 4578 y } 8/Place PSfrag 44 4578 a 55 4691 a { 55 4691 a 10 4715 a Fm(h)62 4729 y Fl(")55 4691 y } 9/Place PSfrag 55 4691 a 64 4811 a { 64 4811 a 28 4842 a Fm(G)64 4811 y } 10/Place PSfrag 64 4811 a 100 4955 a { 100 4955 a 100 4955 a } 11/Place PSfrag 100 4955 a 55 5027 a { 55 5027 a 10 5050 a Fm(\036)64 5064 y Fl(")55 5027 y } 12/Place PSfrag 55 5027 a 70 5152 a { 70 5152 a 40 5163 a Fm(')70 5152 y } 13/Place PSfrag 70 5152 a -60 5241 a { -60 5241 a -220 5251 a Fm(U)-158 5265 y Fl(")-101 5251 y Fo([)20 b Fq(D)51 5241 y Fk(")p 56 5253 31 3 v 56 5294 a Fj(2)-60 5241 y } 14/Place PSfrag -60 5241 a 64 5376 a { 64 5376 a 28 5407 a Fm(U)64 5376 y } 15/Place PSfrag 64 5376 a 351 5603 a FI(Figure)30 b(4:)41 b FF(Comm)n(utativ)n(e)27 b(diagram)f(sho)n (wing)h(all)g(maps)g(in)n(v)n(olv)n(ed)f(in)i(the)g(surgery)e (construction.)p eop %%Page: 17 17 17 16 bop 71 274 a Fs(N.)31 b(F)-8 b(agella,)31 b(T.)f(M.)h(Seara)g (and)f(J.)g(Villan)m(uev)-5 b(a)2025 b FI(17)71 573 y(is)22 b(not)i(only)e(quasiconformally)f(conjugate)k(to)f Fm(H)1780 587 y Fl(")1839 573 y FI(but)f(also)g(holomorphic)e(in)h Fq(C)c FI(.)44 b(Moreo)m(v)m(er,)27 b(if)22 b Fo(k)p Fm(\026)3510 587 y Fl(")3547 573 y Fo(k)k Fm(<)f FI(1)p Fm(=)p FI(3)71 686 y(then)40 b(w)m(e)i(ha)m(v)m(e)g(that)f(this)f(map)g (do)s(es)g(not)h(dep)s(end)e(on)i(the)g(parameter)g Fm(")g FI(as)g(sho)m(wn)f(in)g(the)h(follo)m(wing)71 799 y(prop)s(osition.)71 972 y Ft(Prop)s(osition)36 b(4.3.)42 b Fr(With)33 b(the)g(pr)-5 b(evious)34 b(notations,)g(if)e Fo(k)p Fm(\026)2191 986 y Fl(")2228 972 y Fo(k)26 b Fm(<)f FI(1)p Fm(=)p FI(3)p Fr(,)34 b(then)f(for)g(al)5 b(l)33 b Fm(z)d Fo(2)25 b Fq(C)1502 1159 y Fm(G)1573 1173 y Fl(")1610 1159 y FI(\()p Fm(z)t FI(\))h(=)f Fm(G)p FI(\()p Fm(z)t FI(\))i(=)e Fm(z)t FI(e)2244 1121 y FJ(i)p Fl(!)2314 1159 y FI(e)2354 1121 y Fl(z)2394 1159 y Fm(:)212 1345 y FI(The)f(pro)s(of)g(of)h(this)f (prop)s(osition)f(is)g(similar)g(to)i(an)g(analogous)g(result)f(for)g (Gey)m(er's)i(construction)f(in)e([G)q(].)71 1458 y(Ho)m(w)m(ev)m(er,) 30 b(for)e(the)g(sak)m(e)h(of)f(completeness,)g(w)m(e)h(include)c(here) j(a)g(pro)s(of)f(adapted)h(to)g(our)g(con)m(text.)41 b(The)28 b(k)m(ey)71 1571 y(to)s(ol)j(in)f(the)h(pro)s(of)g(is)f(an)h (estimate)h(on)f(the)g(gro)m(wth)h(of)f(a)h(quasiconformal)d(map)i(at)h Fo(1)p FI(.)43 b(This)29 b(estimate)j(is)71 1684 y(a)f(consequence)g (of)f(the)h(follo)m(wing)e(basic)g(prop)s(ert)m(y)h(of)h (quasiconformal)e(maps.)71 1857 y Ft(Theorem)38 b(4.4)h(\([Ah],)g(p.)58 b(71\).)44 b Fr(A)35 b Fm(k)s Fr(-quasic)-5 b(onformal)38 b(mapping)f(in)f(a)g(domain)h Fo(U)j(\032)30 b Fq(C)60 b Fr(is)35 b(uniformly)71 1970 y(H\177)-46 b(older)33 b(c)-5 b(ontinuous)34 b(with)f(exp)-5 b(onent)1411 1934 y FJ(1)p Fp(\000)p Fl(k)p 1411 1949 129 4 v 1411 2001 a FJ(1+)p Fl(k)1583 1970 y Fr(in)32 b(every)h(c)-5 b(omp)g(act)34 b(subset)f(of)g Fo(U)9 b Fr(.)212 2143 y FI(This)29 b(result)g(implies) e(the)k(desired)e(b)s(ound)f(for)i(a)h(quasiconformal)e(map)h(at)h Fo(1)p FI(.)71 2316 y Ft(Lemma)41 b(4.5.)46 b Fr(L)-5 b(et)40 b Fm(\036)f Fr(b)-5 b(e)39 b(a)h Fm(k)s Fr(-quasic)-5 b(onformal)41 b(mapping)g(of)e Fq(C)18 b Fr(,)47 b(\014xing)39 b FI(0)h Fr(and)g Fo(1)p Fr(.)61 b(Then)40 b(ther)-5 b(e)40 b(exists)71 2448 y Fm(C)32 b(>)25 b FI(0)33 b Fr(such)f(that)i(for)f(any)h Fo(j)p Fm(z)t Fo(j)26 b Fm(>>)e FI(1)33 b Fr(we)g(have)g Fo(j)p Fm(\036)p FI(\()p Fm(z)t FI(\))p Fo(j)27 b(\024)e Fm(C)7 b Fo(j)p Fm(z)t Fo(j)2286 2377 y Fj(1+)p Fk(k)p 2285 2390 113 3 v 2285 2431 a Fj(1)p Ff(\000)p Fk(k)2412 2448 y Fr(.)71 2643 y Ft(Pro)s(of)36 b(:)k FI(W)-8 b(e)32 b(set)f Fm(h)25 b FI(=)g Fm(\036)955 2610 y Fp(\000)p FJ(1)1080 2643 y FI(and)30 b Fm(g)s FI(\()p Fm(z)t FI(\))d(=)1635 2607 y FJ(1)p 1552 2622 202 4 v 1552 2675 a Fl(h)p FJ(\(1)p Fl(=z)s FJ(\))1764 2643 y FI(.)40 b(It)31 b(is)e(easy)i(to)g(c)m(hec)m (k)h(that)f Fm(g)j FI(is)c Fm(k)s FI(-quasiconformal,)g(with)71 2768 y Fm(g)s FI(\(0\))35 b(=)e(0)i(and)g Fm(g)s FI(\()p Fo(1)p FI(\))f(=)f Fo(1)p FI(.)55 b(Applying)33 b(Theorem)i(4.4)h(w)m (e)g(ha)m(v)m(e)g(that)g(there)f(exists)g(a)h(constan)m(t)g Fm(m)d(>)g FI(0)71 2881 y(suc)m(h)d(that)948 2994 y Fo(j)p Fm(g)s FI(\()p Fm(z)1096 3008 y FJ(1)1137 2994 y FI(\))21 b Fo(\000)f Fm(g)s FI(\()p Fm(z)1407 3008 y FJ(2)1447 2994 y FI(\))p Fo(j)26 b(\024)f Fm(m)p Fo(j)p Fm(z)1776 3008 y FJ(1)1836 2994 y Fo(\000)20 b Fm(z)1969 3008 y FJ(2)2009 2994 y Fo(j)2044 2923 y Fj(1)p Ff(\000)p Fk(k)p 2044 2936 113 3 v 2045 2977 a Fj(1+)p Fk(k)2171 2994 y Fm(;)197 b FI(if)29 b Fo(j)p Fm(z)2543 3008 y FJ(1)2583 2994 y Fo(j)p Fm(;)15 b Fo(j)p Fm(z)2715 3008 y FJ(2)2756 2994 y Fo(j)25 b(\024)g FI(1)p Fm(:)71 3152 y FI(W)-8 b(e)31 b(tak)m(e)h Fm(z)466 3166 y FJ(1)531 3152 y FI(=)25 b Fm(z)35 b FI(and)30 b Fm(z)923 3166 y FJ(2)988 3152 y FI(=)25 b(0,)31 b(and)e(w)m(e)i(replace)g Fm(z)e FI(=)2038 3116 y FJ(1)p 1981 3131 150 4 v 1981 3185 a Fl(\036)p FJ(\()p Fl(w)r FJ(\))2171 3152 y FI(to)i(obtain)e(the)i(desired)e(b)s (ound:)1603 3389 y Fo(j)p Fm(\036)p FI(\()p Fm(w)r FI(\))p Fo(j)e(\024)e Fm(C)7 b Fo(j)p Fm(w)r Fo(j)2167 3318 y Fj(1+)p Fk(k)p 2167 3331 113 3 v 2167 3372 a Fj(1)p Ff(\000)p Fk(k)2293 3389 y Fm(;)71 3599 y FI(with)29 b Fm(C)k FI(=)25 b Fm(m)563 3528 y Fj(1+)p Fk(k)p 562 3541 V 562 3582 a Fj(1)p Ff(\000)p Fk(k)689 3599 y FI(.)41 b(Ho)m(w)m(ev)m(er,)34 b(this)29 b(estimate)j(holds)d(pro)m(vided)h(that)h Fo(j)p Fm(\036)p FI(\()p Fm(w)r FI(\))p Fo(j)d(\025)d FI(1.)43 b(As)30 b Fo(f)p Fm(w)f Fo(2)d Fq(C)65 b FI(:)41 b Fo(j)p Fm(\036)p FI(\()p Fm(w)r FI(\))p Fo(j)28 b(\024)71 3712 y FI(1)p Fo(g)j FI(is)e(a)i(compact)h(set,)f(w)m(e)g(can)g(assure)f (that)h Fo(j)p Fm(\036)p FI(\()p Fm(w)r FI(\))p Fo(j)c(\025)e FI(1)30 b(if)g Fo(j)p Fm(w)r Fo(j)c Fm(>>)f FI(1.)3780 3885 y Fe(\003)71 4080 y Ft(Pro)s(of)36 b(of)g(Prop)s(osition)h(4.3:)43 b FI(By)31 b(construction)g(w)m(e)g(kno)m(w)g(the)h(follo)m(wing)d (prop)s(erties)h(of)h Fm(G)3434 4094 y Fl(")3471 4080 y FI(:)42 b(a\))32 b Fm(G)3721 4094 y Fl(")3789 4080 y FI(is)71 4193 y(en)m(tire;)d(b\))f Fm(G)535 4207 y Fl(")572 4193 y FI(\()p Fm(z)t FI(\))e(=)f(0)k(if)e(and)h(only)f(if)g Fm(z)i FI(=)c(0;)30 b(c\))f Fm(G)1860 4207 y Fl(")1925 4193 y FI(has)f(a)h(Siegel)e(disk)g Fm(h)2654 4207 y Fl(")2691 4193 y FI(\()p Fq(D)2797 4183 y Fk(")p 2802 4195 31 3 v 2802 4236 a Fj(2)2862 4193 y Fo([)16 b Fm(U)3001 4207 y Fl(")3037 4193 y FI(\))29 b(around)e Fm(z)i FI(=)c(0,)30 b(with)71 4322 y(rotation)j(n)m(um)m(b)s(er)f Fm(\022)s FI(,)h(and)f(hence)h Fm(G)1365 4289 y Fp(0)1365 4345 y Fl(")1402 4322 y FI(\(0\))e(=)e(e)1688 4289 y FJ(i)p Fl(!)1759 4322 y FI(;)34 b(and)f(d\))f Fm(G)2187 4289 y Fp(0)2187 4345 y Fl(")2225 4322 y FI(\()p Fo(\000)p FI(1\))e(=)f(0)34 b(b)s(ecause)f Fm(G)3030 4336 y Fl(")3100 4322 y FI(is)f(not)h(univ)-5 b(alen)m(t)31 b(\(it)71 4435 y(has)f(degree)h(t)m(w)m(o\))h(around)d Fo(\000)p FI(1.)212 4548 y(Com)m(bining)f(the)j(\014rst)e(t)m(w)m(o)j(prop)s (erties)d(of)h Fm(G)1792 4562 y Fl(")1830 4548 y FI(,)g(w)m(e)h(ha)m(v) m(e)h(that)1653 4735 y Fm(G)1724 4749 y Fl(")1761 4735 y FI(\()p Fm(z)t FI(\))27 b(=)e Fm(z)t(g)2089 4749 y Fl(")2126 4735 y FI(\()p Fm(z)t FI(\))p Fm(;)71 4922 y FI(with)i Fm(g)319 4936 y Fl(")385 4922 y FI(en)m(tire)h(and)g (without)g(zeros.)40 b(No)m(w,)30 b(w)m(e)f(can)g(estimate)g(the)g(gro) m(wth)g(order)f(of)h Fm(g)3142 4936 y Fl(")3179 4922 y FI(.)40 b(T)-8 b(o)29 b(this)e(end,)i(w)m(e)71 5035 y(use)h(that)h(if)e Fm(z)h Fo(2)25 b Fq(C)43 b Fo(n)21 b Fq(D)888 5025 y Fk(")p 893 5037 V 893 5078 a Fj(2)968 5035 y FI(then:)1299 5278 y Fm(H)1375 5292 y Fl(")1411 5278 y FI(\()p Fm(z)t FI(\))26 b(=)f Fm(F)1707 5296 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")1897 5278 y FI(\()p Fm(z)t FI(\))h(=)f Fm(z)t FI(e)2221 5240 y FJ(i)p Fl(\013)p FJ(\()p Fl(")p FJ(\))2378 5278 y FI(e)2418 5240 y Fl(z)s Fp(\000)2519 5213 y Fk(")2548 5192 y Fj(2)p 2519 5225 64 3 v 2520 5266 a(4)p Fk(z)2597 5278 y Fm(:)71 5465 y FI(So,)30 b(w)m(e)h(ha)m(v)m(e)h(that)f(if)e Fo(j)p Fm(z)t Fo(j)d Fm(>>)f FI(1)31 b(then:)1132 5729 y Fm(G)1203 5743 y Fl(")1240 5729 y FI(\()p Fm(z)t FI(\))26 b(=)f Fm(h)1530 5743 y Fl(")1583 5573 y Fn( )1655 5729 y Fm(h)1707 5691 y Fp(\000)p FJ(1)1707 5751 y Fl(")1801 5729 y FI(\()p Fm(z)t FI(\)e)1957 5691 y FJ(i)p Fl(\013)p FJ(\()p Fl(")p FJ(\))2115 5729 y FI(e)2155 5661 y Fl(h)2196 5631 y Ff(\000)p Fj(1)2196 5671 y Fk(")2279 5661 y FJ(\()p Fl(z)s FJ(\))p Fp(\000)2517 5634 y Fk(")2546 5613 y Fj(2)p 2434 5646 229 3 v 2434 5705 a(4)p Fk(h)2500 5681 y Ff(\000)p Fj(1)2500 5719 y Fk(")2583 5705 y Fj(\()p Fk(z)r Fj(\))2677 5573 y Fn(!)2764 5729 y Fm(:)p eop %%Page: 18 18 18 17 bop 71 274 a FI(18)1001 b Fs(Asymptotic)30 b(Size)g(of)h(Herman)f (Rings)f(of)i(the)f(Complex)g(Standard)f(F)-8 b(amily)71 573 y FI(From)29 b(Lemma)f(4.5)i(w)m(e)f(ha)m(v)m(e)h(that)f(if)e Fo(j)p Fm(z)t Fo(j)f Fm(>>)f FI(1)k(there)g(exists)f(some)h(constan)m (t)h Fm(C)i(>)24 b FI(0)29 b(suc)m(h)g(that)g Fo(j)p Fm(h)3575 587 y Fl(")3612 573 y FI(\()p Fm(z)t FI(\))p Fo(j)e(\024)71 698 y Fm(C)7 b Fo(j)p Fm(z)t Fo(j)239 665 y Fl(K)299 673 y Fk(")336 698 y FI(,)31 b(where)g Fm(K)733 712 y Fl(")796 698 y FI(=)902 654 y FJ(1+)p Fp(k)p Fl(\026)1069 662 y Fk(")1104 654 y Fp(k)p 902 678 237 4 v 902 731 a FJ(1)p Fp(\000k)p Fl(\026)1069 739 y Fk(")1104 731 y Fp(k)1149 698 y FI(.)42 b(F)-8 b(or)31 b(the)g(other)g(v)-5 b(alues)31 b(of)g Fm(z)k FI(the)c(map)f Fm(h)2638 712 y Fl(")2706 698 y FI(is)g(b)s(ounded.)40 b(This)29 b(is)h(also)h(true)71 832 y(for)i Fm(h)265 799 y Fp(\000)p FJ(1)265 854 y Fl(")359 832 y FI(,)h(and)e(b)s(oth)h (facts)g(can)h(b)s(e)e(summarized)f(b)m(y)i(sa)m(ying)g(that)h(there)f (exists)f Fm(M)40 b(>)29 b FI(0,)34 b(whic)m(h)e(dep)s(ends)71 945 y(only)d(on)i Fm(")p FI(,)g(suc)m(h)f(that)777 1148 y Fo(j)p Fm(h)854 1162 y Fl(")892 1148 y FI(\()p Fm(z)t FI(\))p Fo(j)c(\024)f Fm(M)g FI(max)q Fo(fj)p Fm(z)t Fo(j)1579 1110 y Fl(K)1639 1118 y Fk(")1677 1148 y Fm(;)15 b FI(1)p Fo(g)p Fm(;)198 b Fo(j)p Fm(h)2107 1110 y Fp(\000)p FJ(1)2107 1170 y Fl(")2202 1148 y FI(\()p Fm(z)t FI(\))p Fo(j)27 b(\024)e Fm(M)g FI(max)p Fo(fj)p Fm(z)t Fo(j)2889 1110 y Fl(K)2949 1118 y Fk(")2987 1148 y Fm(;)15 b FI(1)p Fo(g)p Fm(:)71 1351 y FI(Moreo)m(v)m(er,)33 b(w)m(e)d(ma)m(y)h(also)g (ask)f Fo(j)p Fm(h)1245 1318 y Fp(\000)p FJ(1)1245 1373 y Fl(")1340 1351 y FI(\()p Fm(z)t FI(\))p Fo(j)d(\025)e FI(1)30 b(if)g Fo(j)p Fm(z)t Fo(j)c Fm(>>)f FI(1,)31 b(obtaining)1438 1567 y Fo(j)p Fm(G)1534 1581 y Fl(")1571 1567 y FI(\()p Fm(z)t FI(\))p Fo(j)26 b(\024)f Fm(m)1914 1581 y FJ(1)1954 1567 y Fo(j)p Fm(z)t Fo(j)2050 1529 y Fl(K)2115 1506 y Fj(2)2110 1546 y Fk(")2153 1567 y FI(e)2193 1529 y Fl(m)2255 1538 y Fj(2)2291 1529 y Fp(j)p Fl(z)s Fp(j)2367 1506 y Fk(K)2418 1514 y(")2458 1567 y Fm(;)71 1770 y FI(where)i Fm(m)411 1784 y FJ(1)477 1770 y FI(and)f Fm(m)730 1784 y FJ(2)797 1770 y FI(ma)m(y)i(dep)s(end)d (on)i Fm(")h FI(\(of)f(course,)h(the)g(condition)e Fo(j)p Fm(z)t Fo(j)g Fm(>>)e FI(1)k(is)e(not)h(necessarily)g(uniform)71 1883 y(on)h Fm(")p FI(\).)41 b(As)28 b(w)m(e)h(are)f(assuming)f Fo(k)p Fm(\026)1244 1897 y Fl(")1281 1883 y Fo(k)f Fm(<)f FI(1)p Fm(=)p FI(3,)30 b(w)m(e)f(ha)m(v)m(e)g(that)g(1)d Fo(\024)f Fm(K)2416 1897 y Fl(")2478 1883 y Fm(<)g FI(2,)k(and)f(so,)h (w)m(e)g(deduce)e(that)i Fm(g)3653 1897 y Fl(")3719 1883 y FI(has)71 1996 y(gro)m(wth)i(order)f(con)m(trolled)g(b)m(y)1366 2109 y Fo(j)p Fm(g)1434 2123 y Fl(")1471 2109 y FI(\()p Fm(z)t FI(\))p Fo(j)d(\024)e FI(e)1775 2071 y Fp(j)p Fl(z)s Fp(j)1851 2048 y Fk(p)1890 2109 y Fm(;)198 b FI(if)29 b Fo(j)p Fm(z)t Fo(j)d Fm(>>)f FI(1)p Fm(;)71 2275 y FI(with)33 b(1)f Fo(\024)f Fm(p)g(<)g FI(2.)52 b(The)34 b(kno)m(wn)f(prop)s(erties)g(of)h Fm(g)1826 2289 y Fl(")1863 2275 y FI(\()p Fm(z)t FI(\))h(\(en)m(tire)f(function)f(without)g(zeros) i(and)e(with)g(exp)s(o-)71 2388 y(nen)m(tial)c(gro)m(wth)i(of)g(order)f (1)25 b Fo(\024)g Fm(p)g(<)g FI(2\))31 b(imply)d(that)j(it)f(is)g(of)g (the)h(form:)1682 2591 y Fm(g)1725 2605 y Fl(")1762 2591 y FI(\()p Fm(z)t FI(\))c(=)e(e)2041 2554 y Fl(P)2086 2562 y Fk(")2119 2554 y FJ(\()p Fl(z)s FJ(\))2214 2591 y Fm(;)71 2794 y FI(\(see)37 b([D)q(]\))g(with)e Fm(P)722 2808 y Fl(")759 2794 y FI(\()p Fm(z)t FI(\))i(a)f(p)s(olynomial)e(of)i (degree)h(not)g(greater)g(than)f(1.)59 b(No)m(w,)39 b(the)d(prop)s (osition)e(follo)m(ws)71 2907 y(from)c(the)g(remaining)f(prop)s (erties.)3780 3094 y Fe(\003)71 3280 y Ft(Remark)h(4.6.)38 b FI(Assuming)25 b(that)i Fo(k)p Fm(\026)1371 3294 y Fl(")1408 3280 y Fo(k)e Fm(<)g FI(1)p Fm(=)p FI(3,)k(w)m(e)e(just)f (pro)m(v)m(ed)g(that)h Fm(G)f FI(=)f Fm(h)2798 3294 y Fl(")2847 3280 y Fo(\016)12 b Fm(H)2980 3294 y Fl(")3029 3280 y Fo(\016)g Fm(h)3138 3247 y Fp(\000)p FJ(1)3138 3303 y Fl(")3233 3280 y FI(,)27 b(and)f(so)g Fm(H)3641 3294 y Fl(")3704 3280 y FI(and)71 3393 y Fm(G)35 b FI(are)g(conjugated) h(b)m(y)f Fm(h)983 3407 y Fl(")1020 3393 y FI(.)54 b(Then,)35 b(as)g(the)g(in)m(v)-5 b(arian)m(t)34 b(curv)m(es)h(are)g(preserv)m(ed) g(b)m(y)g(conjugation,)h(w)m(e)f(ha)m(v)m(e)71 3506 y(that)f(the)h (rotation)f(domain)f Fq(D)1182 3496 y Fk(")p 1186 3508 31 3 v 1186 3550 a Fj(2)1254 3506 y Fo([)22 b Fm(U)1399 3520 y Fl(")1470 3506 y FI(of)34 b Fm(H)1653 3520 y Fl(")1723 3506 y FI(is)f(mapp)s(ed)g(b)m(y)h Fm(h)2350 3520 y Fl(")2421 3506 y FI(to)g(the)h(Siegel)e(Disk)g Fm(U)44 b FI(of)34 b Fm(G)h FI(\(see)g(\(5\)\).)71 3619 y(That)30 b(is:)1611 3732 y Fm(U)36 b FI(=)25 b Fm(h)1857 3746 y Fl(")1894 3732 y FI(\()p Fq(D)2000 3722 y Fk(")p 2005 3734 V 2005 3775 a Fj(2)2070 3732 y Fo([)19 b Fm(U)2212 3746 y Fl(")2249 3732 y FI(\))p Fm(:)212 3919 y FI(This)31 b(concludes)h(the)h(surgery)g (construction)f(relating)g(the)h(\(scaled\))h(standard)e(map)g Fm(F)3322 3937 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")3545 3919 y FI(and)g(the)71 4032 y(semistandard)26 b(map)h Fm(G)p FI(.)41 b(In)27 b(the)h(follo)m(wing)e(section)i(w)m(e)g(shall)e (see)i(whic)m(h)e(quan)m(tities)h(w)m(e)i(need)e(to)i(estimate)71 4145 y(in)g(order)h(to)h(obtain)f(quan)m(titativ)m(e)h(information)e (from)g(the)i(surgery)f(w)m(e)h(just)e(p)s(erformed.)71 4388 y Fg(4.2)112 b(Restatemen)m(t)37 b(of)g(the)h(problem)71 4559 y FI(Using)25 b(the)h(same)h(notation)f(of)g(Section)g(4.1)h(and)f (assuming)e Fo(k)p Fm(\026)2275 4573 y Fl(")2312 4559 y Fo(k)i Fm(<)f FI(1)p Fm(=)p FI(3,)j(w)m(e)f(observ)m(e)f(\(see)h (Figure)f(4\))h(that)71 4672 y(the)j(map)g(de\014ned)g(as)1163 4859 y Fm(\036)1217 4873 y Fl(")1254 4859 y FI(\()p Fm(z)t FI(\))c(=)1492 4704 y Fn(\()1565 4793 y FI(\()p Fm(h)1652 4807 y Fl(")1710 4793 y Fo(\016)21 b Fm(')1835 4807 y Fl(")1872 4793 y FI(\)\()p Fm(z)t FI(\))122 b(if)30 b Fm(z)f Fo(2)c Fq(D)2446 4807 y Fl(R)2499 4815 y Fk(")2563 4793 y Fo(n)20 b Fq(D)2699 4783 y Fk(")p 2704 4795 V 2704 4836 a Fj(2)1565 4931 y FI(\()p Fm(h)1652 4945 y Fl(")1710 4931 y Fo(\016)h Fm( )1835 4945 y Fl(")1872 4931 y FI(\)\()p Fm(z)t FI(\))122 b(if)30 b Fm(z)f Fo(2)c Fq(D)2457 4921 y Fk(")p 2462 4933 V 2462 4975 a Fj(2)71 5105 y FI(is)i(holomorphic)f(in)h Fq(D)842 5119 y Fl(R)895 5127 y Fk(")938 5105 y FI(.)40 b(F)-8 b(rom)29 b(Remark)f(4.2,)i(this)d (assertion)h(is)f(ob)m(viously)g(true)h(in)f Fq(D)3145 5119 y Fl(R)3198 5127 y Fk(")3258 5105 y Fo(n)16 b Fq(D)3389 5095 y Fk(")p 3394 5107 V 3394 5148 a Fj(2)3439 5105 y FI(.)40 b(T)-8 b(o)28 b(pro)m(v)m(e)71 5217 y(the)g(analyticit)m(y)g (of)h Fm(\036)831 5231 y Fl(")896 5217 y FI(in)e Fq(D)1071 5207 y Fk(")p 1076 5219 V 1076 5261 a Fj(2)1120 5217 y FI(,)i(w)m(e)g(can)f(c)m(hec)m(k)i(that)f Fm(\036)1962 5184 y Fp(\003)1962 5240 y Fl(")2002 5217 y Fm(\026)2057 5231 y FJ(0)2121 5217 y FI(=)c Fm(\026)2272 5231 y FJ(0)2340 5217 y FI(\(see)k(Remark)f(2.10\),)j(whic)m(h)c(follo)m(ws)h(from)71 5345 y(the)i(fact)i(that)f Fm(\026)659 5359 y Fl(")720 5345 y FI(=)25 b(\()p Fm( )913 5312 y Fp(\000)p FJ(1)1009 5345 y FI(\))1044 5312 y Fp(\003)1083 5345 y Fm(\026)1138 5359 y FJ(0)1208 5345 y FI(in)k Fq(D)1385 5335 y Fk(")p 1390 5347 V 1390 5388 a Fj(2)1434 5345 y FI(.)212 5458 y(By)f(construction,)g Fm(\036)950 5472 y Fl(")1015 5458 y FI(conjugates)h(the)f(semistandard)e(map)i Fm(G)g FI(on)g(the)g (Siegel)f(disk)f Fm(U)36 b FI(=)25 b Fm(h)3433 5472 y Fl(")3470 5458 y FI(\()p Fq(D)3576 5448 y Fk(")p 3581 5460 V 3581 5501 a Fj(2)3640 5458 y Fo([)15 b Fm(U)3778 5472 y Fl(")3815 5458 y FI(\))71 5571 y(to)31 b(the)g(rotation)f Fo(R)764 5585 y Fl(!)845 5571 y FI(on)g Fq(D)1031 5585 y Fl(R)1084 5593 y Fk(")1128 5571 y FI(.)212 5684 y(But)g(w)m(e)h(can)f (not)g(b)s(e)g(sure)f(that)i(the)f(map)g Fm(\036)1778 5698 y Fl(")1845 5684 y FI(is)f(the)h(normalized)f(linearizing)e(map)j Fm(')g FI(of)g Fm(G)h FI(\(see)g(\(5\)\),)71 5796 y(since,)e(among)h (other)f(reasons,)h(w)m(e)f(exp)s(ect)h(the)f(radius)f Fm(R)2106 5810 y Fl(")2172 5796 y FI(to)h(mo)m(v)m(e)i(with)d Fm(")h FI(while)f Fm(G)p FI(,)i Fm(U)39 b FI(and)28 b Fm(R)3500 5810 y FJ(0)3569 5796 y FI(do)h(not.)p eop %%Page: 19 19 19 18 bop 71 274 a Fs(N.)31 b(F)-8 b(agella,)31 b(T.)f(M.)h(Seara)g (and)f(J.)g(Villan)m(uev)-5 b(a)2025 b FI(19)71 573 y(This)29 b(is)i(equiv)-5 b(alen)m(t)31 b(to)h(sa)m(y)g(that)g Fm(\036)1327 540 y Fp(0)1327 595 y Fl(")1364 573 y FI(\(0\))c Fo(6)p FI(=)f(1.)44 b(Then,)31 b(to)h(recup)s(erate)g(the)f (\(normalized\))g(linearizing)e(map)71 686 y Fm(')i FI(let)f(us)g (de\014ne)1708 799 y Fm(b)p FI(\()p Fm(")p FI(\))c(=)f Fm(\036)2035 761 y Fp(0)2035 821 y Fl(")2072 799 y FI(\(0\))p Fm(;)71 948 y FI(and)30 b(so)g(b)m(y)h(the)f(preceding)f(argumen)m(t,) 1586 1161 y Fm(')p FI(\()p Fm(z)t FI(\))d(=)f Fm(\036)1937 1175 y Fl(")1989 1033 y Fn(\022)2119 1100 y Fm(z)p 2066 1141 153 4 v 2066 1224 a(b)p FI(\()p Fm(")p FI(\))2228 1033 y Fn(\023)2310 1161 y Fm(;)71 1416 y FI(giv)m(en)k(that)g Fm(\036)556 1430 y Fl(")608 1316 y Fn(\020)713 1381 y Fl(z)p 672 1396 118 4 v 672 1449 a(b)p FJ(\()p Fl(")p FJ(\))800 1316 y Fn(\021)883 1416 y FI(satis\014es)f(b)s(oth)g (normalization)f(conditions)h(\()p Fm(')p FI(\(0\))f(=)e(0,)k Fm(')2927 1383 y Fp(0)2951 1416 y FI(\(0\))e(=)e(1\).)40 b(See)29 b(Figure)g(4.)212 1567 y(From)k(here,)g(it)f(is)f(clear)i (that)g Fm(R)1360 1581 y Fl(")1426 1567 y FI(=)1570 1529 y Fl(R)1623 1538 y Fj(0)p 1535 1546 157 4 v 1535 1599 a Fp(j)p Fl(b)p FJ(\()p Fl(")p FJ(\))p Fp(j)1735 1567 y FI(so)f(that,)i(to)g(relate)f Fm(R)2511 1581 y FJ(0)2582 1567 y FI(and)f Fm(R)2830 1581 y Fl(")2900 1567 y FI(w)m(e)h(need)f(to) h(ha)m(v)m(e)h(con)m(trol)71 1694 y(o)m(v)m(er)e Fm(b)p FI(\()p Fm(")p FI(\),)f(i.e.)41 b(o)m(v)m(er)31 b(\()p Fm(h)911 1708 y Fl(")969 1694 y Fo(\016)21 b Fm( )1094 1708 y Fl(")1131 1694 y FI(\))1166 1661 y Fp(0)1189 1694 y FI(\(0\).)71 1852 y Ft(Remark)26 b(4.7.)37 b FI(Let)24 b(us)f(observ)m(e)h(that)g(\()p Fm(h)1519 1866 y Fl(")1563 1852 y Fo(\016)7 b Fm( )1674 1866 y Fl(")1711 1852 y FI(\))1746 1819 y Fp(0)1770 1852 y FI(\(0\))25 b(do)s(es)e(not)h(dep)s (end)e(on)h(the)h(particular)e(quasiconformal)71 1965 y(map)39 b Fm( )341 1979 y Fl(")418 1965 y FI(used)f(in)h(the)g (surgery)g(construction)h(\(whic)m(h)e(is)h(of)h(course)f(not)h (unique\).)67 b(This)38 b(allo)m(ws)h(us)g(to)71 2078 y(compute)31 b(this)e(deriv)-5 b(ativ)m(e)30 b(b)m(y)g(constructing)g (explicitly)e(a)j(con)m(v)m(enien)m(t)g Fm( )2659 2092 y Fl(")2696 2078 y FI(.)212 2236 y(From)e(the)g(previous)e(observ)-5 b(ations,)29 b(Theorem)f(A)h(follo)m(ws)f(immediately)f(from)h(the)h (next)f(prop)s(osition.)71 2394 y Ft(Prop)s(osition)36 b(4.8.)42 b Fr(With)33 b(the)g(pr)-5 b(evious)34 b(notation,)g(we)f (have)1243 2563 y Fm(b)p FI(\()p Fm(")p FI(\))26 b(=)f(\()p Fm(h)1603 2577 y Fl(")1661 2563 y Fo(\016)c Fm( )1786 2577 y Fl(")1822 2563 y FI(\))1857 2525 y Fp(0)1881 2563 y FI(\(0\))27 b(=)d(1)d(+)f Fo(O)s FI(\()p Fm(")15 b FI(log)i Fm(")p FI(\))p Fm(:)212 2731 y FI(T)-8 b(o)31 b(pro)m(v)m(e)g(Prop)s(osition)d(4.8,)k(w)m(e)f(shall)e(study)g(the)i (quan)m(tit)m(y)1547 2876 y Fm(d)p 1523 2917 94 4 v 1523 3000 a(dz)1642 2938 y FI(\()q Fm(h)1730 2952 y Fl(")1787 2938 y Fo(\016)21 b Fm( )1912 2952 y Fl(")1948 2938 y FI(\()p Fm(z)t FI(\))h Fo(\000)e Fm(z)t FI(\))2258 2965 y Fp(j)p Fl(z)s FJ(=0)71 3128 y FI(b)m(y)34 b(means)g(of)g(the)g(Cauc)m (h)m(y)g(in)m(tegral)g(form)m(ula.)51 b(This)32 b(will)f(b)s(e)j(done)g (in)e(Section)i(4.4.)53 b(The)33 b(estimates)i(w)m(e)71 3240 y(use)d(come)i(from)f(studying)e(the)i(quan)m(tities)f Fo(j)p Fm( )1722 3254 y Fl(")1760 3240 y FI(\()p Fm(z)t FI(\))22 b Fo(\000)g Fm(z)t Fo(j)33 b FI(and)g Fo(j)p Fm(h)2352 3254 y Fl(")2389 3240 y FI(\()p Fm(z)t FI(\))23 b Fo(\000)e Fm(z)t Fo(j)34 b FI(or,)g(equiv)-5 b(alen)m(tly)d(,)33 b(ho)m(w)f(far)h(the)71 3353 y(maps)d Fm( )368 3367 y Fl(")435 3353 y FI(and)g Fm(h)664 3367 y Fl(")731 3353 y FI(are)h(from)f(the)g(iden)m(tit)m(y)g(map)g(in)f(a)i(neigh)m(b)s (orho)s(o)s(d)d(of)j(zero.)212 3466 y(T)-8 b(o)36 b(obtain)e(suc)m(h)h (an)g(estimate)h(for)f Fm( )1551 3480 y Fl(")1623 3466 y FI(w)m(e)g(construct)h Fm( )2225 3480 y Fl(")2297 3466 y FI(explicitly)c(in)i(Section)h(4.3.)56 b(The)35 b(estimate)71 3579 y(for)30 b Fm(h)262 3593 y Fl(")329 3579 y FI(is)g(a)h(direct)e (application)g(of)i(Prop)s(osition)d(B.)71 3817 y Fg(4.3)112 b(Explicit)34 b(surgery)k(construction)71 3989 y FI(The)26 b(main)g(purp)s(ose)g(of)h(this)f(section)h(is)f(to)i(construct)g (explicitly)c(the)k(quasiconformal)d(extension)i Fm( )3609 4003 y Fl(")3673 3989 y FI(used)71 4102 y(in)j(the)i(surgery)f (construction)g(of)h(Section)f(4.1,)i(and)e(to)i(giv)m(e)f(the)g (explicit)d(estimates)k(that)f(measure)f(ho)m(w)71 4214 y(far)f Fm( )269 4228 y Fl(")336 4214 y FI(is)g(from)g(the)g(iden)m (tit)m(y)g(map.)212 4327 y(Let)c(us)f(recall)h(that)g(w)m(e)g(ha)m(v)m (e)h(a)g(circle)e(of)h(radius)e Fm("=)p FI(2)j(on)f(whic)m(h)e(the)i (real)g(analytic)f(\(scaled\))h(conjugacy)71 4440 y Fm(')130 4454 y Fl(")197 4440 y FI(is)k(de\014ned.)39 b(Our)29 b(goal)i(is)f(to)h(\014nd)e(a)h(quasi-conformal)g(map)g Fm( )2395 4454 y Fl(")2457 4440 y FI(:)25 b Fq(D)2578 4430 y Fk(")p 2583 4442 31 3 v 2583 4484 a Fj(2)2653 4440 y Fo(!)g Fq(D)2840 4430 y Fk(")p 2845 4442 V 2845 4484 a Fj(2)2920 4440 y FI(that)31 b(extends)f Fm(')3507 4454 y Fl(")3544 4440 y FI(.)212 4553 y(W)-8 b(e)42 b(de\014ne)d(the)i (\\gluing)e(map")h Fm( )1462 4567 y Fl(")1540 4553 y FI(to)h(b)s(e)e(the)i(most)g(natural)e(extension:)60 b(the)41 b(radial)e(one.)71 b(More)71 4666 y(explicitly)-8 b(,)28 b(giv)m(en)j Fm(z)e Fo(2)c Fq(D)948 4656 y Fk(")p 953 4668 V 953 4709 a Fj(2)1028 4666 y FI(w)m(e)31 b(de\014ne)1456 4905 y Fm( )1515 4919 y Fl(")1552 4905 y FI(\()p Fm(z)t FI(\))26 b(=)1800 4844 y(2)p 1800 4884 46 4 v 1802 4968 a Fm(")1855 4905 y Fo(j)p Fm(z)t Fo(j)47 b Fm(')2057 4919 y Fl(")2109 4777 y Fn(\022)2187 4844 y Fm(")p 2186 4884 V 2186 4968 a FI(2)2277 4844 y Fm(z)p 2251 4884 97 4 v 2251 4968 a Fo(j)p Fm(z)t Fo(j)2358 4777 y Fn(\023)2440 4905 y Fm(:)1224 b FI(\(13\))71 5119 y(This)29 b(map)i(is)g(clearly)f (con)m(tin)m(uous,)i(it)e(agrees)j(with)d Fm(')1981 5133 y Fl(")2049 5119 y FI(on)h(the)h(b)s(oundary)d(of)i Fq(D)2919 5109 y Fk(")p 2924 5121 31 3 v 2924 5162 a Fj(2)3000 5119 y FI(and)f(sends)h(0)g(to)h(0.)44 b(W)-8 b(e)71 5232 y(observ)m(e)34 b(also)f(that)h(it)f(lea)m(v)m(es)i(all)d(circles) h(in)f Fq(D)1727 5222 y Fk(")p 1732 5234 V 1732 5275 a Fj(2)1810 5232 y FI(in)m(v)-5 b(arian)m(t.)49 b(In)32 b(Prop)s(osition)g(4.9)i(w)m(e)g(will)d(pro)m(v)m(e)j(that)g Fm( )3813 5246 y Fl(")71 5345 y FI(is)h(a)h(quasiconformal)f(mapping)f (if)h Fm(")h FI(is)f(small)f(enough)i(\(ev)m(en)h(more,)g(it)f(is)f Fo(C)2841 5312 y Fp(1)2952 5345 y FI(at)h(all)f(p)s(oin)m(ts)g(except)i (at)71 5458 y Fm(z)43 b FI(=)c(0\).)66 b(Moreo)m(v)m(er,)43 b(this)37 b(result)h(will)e(sho)m(w)i(that)h(for)g(small)e(v)-5 b(alues)38 b(of)g Fm(")h FI(w)m(e)g(ha)m(v)m(e)h Fo(k)p Fm(\026)3262 5472 y Fl(")3299 5458 y Fo(k)g Fm(<)e FI(1)p Fm(=)p FI(3)i(\(this)71 5571 y(condition)33 b(is)g(needed)h(to)h(sho)m (w)g(that)f Fm(G)1492 5585 y Fl(")1562 5571 y FI(=)d Fm(G)p FI(,)36 b(as)f(stated)g(in)e(Prop)s(osition)f(4.3\),)37 b(and)d(so)h(all)e(the)i(results)71 5684 y(deriv)m(ed)29 b(from)h(the)h(quasiconformal)e(construction)h(\(see)h(Sections)f(4.1)i (and)d(4.2\))j(hold.)212 5796 y(Our)d(goal)i(in)e(the)i(remainder)d(of) j(this)e(section)i(is)e(to)i(pro)m(v)m(e)h(the)e(follo)m(wing)f (result.)p eop %%Page: 20 20 20 19 bop 71 274 a FI(20)1001 b Fs(Asymptotic)30 b(Size)g(of)h(Herman)f (Rings)f(of)i(the)f(Complex)g(Standard)f(F)-8 b(amily)71 573 y Ft(Prop)s(osition)37 b(4.9.)43 b Fr(Ther)-5 b(e)34 b(exists)g(a)g(c)-5 b(onstant)35 b Fm(C)1862 587 y FJ(1)1929 573 y Fm(>)26 b FI(0)p Fr(,)34 b(indep)-5 b(endent)35 b(of)f Fm(")p Fr(,)g(such)f(that)i(for)f(any)g Fm(")g Fr(smal)5 b(l)71 686 y(enough)33 b Fm( )439 700 y Fl(")508 686 y Fr(is)g(a)g FI(\()p Fm(C)785 700 y FJ(1)825 686 y Fm(")p FI(\))p Fr(-quasic)-5 b(onformal)35 b(mapping)f(in)f Fq(D)2110 676 y Fk(")p 2115 688 31 3 v 2115 729 a Fj(2)2160 686 y Fr(,)f(and)i(it)e(veri\014es)1555 881 y Fo(j)p Fm( )1639 895 y Fl(")1676 881 y FI(\()p Fm(z)t FI(\))22 b Fo(\000)d Fm(z)t Fo(j)26 b(\024)f Fm(C)2162 895 y FJ(1)2202 881 y Fm(")p Fo(j)p Fm(z)t Fo(j)p Fm(:)1324 b FI(\(14\))71 1076 y Ft(Remark)23 b(4.10.)34 b FI(One)20 b(could)f(also)h(use)g(an)g (alternativ)m(e,)j(more)e(dynamically)d(meaningful,)i(quasi-conformal) 71 1189 y(extension)28 b(of)g Fm(')631 1203 y Fl(")696 1189 y FI(giv)m(en)h(b)m(y)f Fm( )1115 1203 y Fl(")1152 1189 y FI(\()p Fm(z)t FI(\))e(=)f Fm(')1449 1208 y FJ(2)p Fp(j)p Fl(z)s Fp(j)1564 1189 y FI(\()p Fm(z)t FI(\).)41 b(In)27 b(this)g(case,)j(the)f(map)e Fm(H)2678 1203 y Fl(")2743 1189 y FI(w)m(ould)g(b)s(e)g(explicitly)f(giv)m(en)i(b)m(y)71 1315 y Fm(H)147 1329 y Fl(")183 1315 y FI(\()p Fm(z)t FI(\))33 b(=)e Fm(F)492 1334 y Fl(\013)p FJ(\(2)p Fp(j)p Fl(z)s Fp(j)p FJ(\))p Fl(;)p FJ(2)p Fp(j)p Fl(z)s Fp(j)837 1315 y FI(\()p Fm(z)t FI(\))h(=)f Fm(z)t FI(e)1173 1282 y FJ(i)p Fl(\013)p FJ(\(2)p Fp(j)p Fl(z)s Fp(j)p FJ(\))1408 1315 y FI(e)1448 1282 y Fl(z)s Fp(\000)p 1539 1245 36 3 v Fl(z)1613 1315 y FI(on)j Fq(D)1814 1305 y Fk(")p 1818 1317 31 3 v 1818 1359 a Fj(2)1863 1315 y FI(.)52 b(This)32 b(extension)i(also)g(preserv)m(es)g(circles)g(in)e Fq(D)3595 1305 y Fk(")p 3600 1317 V 3600 1359 a Fj(2)3679 1315 y FI(and,)71 1428 y(on)c(eac)m(h)h(of)f(these)g(circles,)g(it)g (is)f(the)h(linearizing)d(map)j(of)g(a)g(scaled)g(standard)f(map.)40 b(In)27 b(this)g(case,)j(one)e(can)71 1541 y(sho)m(w)i(that)h Fo(j)p Fm( )577 1555 y Fl(")614 1541 y FI(\()p Fm(z)t FI(\))21 b Fo(\000)f Fm(z)t Fo(j)26 b(\024)f Fm(C)1100 1555 y FJ(1)1140 1541 y Fo(j)p Fm(z)t Fo(j)1236 1508 y FJ(2)1276 1541 y FI(.)212 1721 y(The)30 b(pro)s(of)f(of)i(this)e (prop)s(osition)f(will)g(b)s(e)i(an)g(easy)h(consequence)g(of)g(the)f (follo)m(wing)f(lemma.)71 1931 y Ft(Lemma)21 b(4.11.)34 b Fr(The)24 b(line)-5 b(arization)25 b Fm(')1428 1945 y Fl(")1489 1931 y Fr(\(se)-5 b(e)23 b(\(11\)\))i(of)e(the)h(sc)-5 b(ale)g(d)25 b(standar)-5 b(d)26 b(map)f Fm(F)2993 1949 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")3182 1931 y FI(\()p Fm(z)t FI(\))i(=)e Fm(z)t FI(e)3507 1898 y FJ(i)p Fl(\013)p FJ(\()p Fl(")p FJ(\))3664 1931 y FI(e)3704 1895 y Fl(z)s Fp(\000)3805 1868 y Fk(")3834 1847 y Fj(2)p 3805 1880 64 3 v 3806 1921 a(4)p Fk(z)3883 1931 y Fr(,)71 2044 y(veri\014es)32 b(the)h(fol)5 b(lowing)34 b(b)-5 b(ounds)34 b(if)e Fm(")h Fr(is)f(smal)5 b(l)34 b(enough)f(and)h Fo(j)p Fm(z)t Fo(j)26 b FI(=)2437 2008 y Fl(")p 2436 2023 36 4 v 2436 2075 a FJ(2)2481 2044 y Fr(:)1121 2239 y Fo(j)q Fm(')1206 2253 y Fl(")1243 2239 y FI(\()p Fm(z)t FI(\))21 b Fo(\000)f Fm(z)t Fo(j)26 b(\024)f Fm(C)1729 2253 y FJ(2)1768 2239 y Fm(")1810 2201 y FJ(2)1850 2239 y Fm(;)2076 2162 y Fn(\014)2076 2216 y(\014)2107 2239 y Fm(')2166 2201 y Fp(0)2166 2261 y Fl(")2203 2239 y FI(\()p Fm(z)t FI(\))c Fo(\000)f FI(1)2476 2162 y Fn(\014)2476 2216 y(\014)2532 2239 y Fo(\024)25 b Fm(C)2693 2253 y FJ(2)2732 2239 y Fm(";)890 b FI(\(15\))71 2434 y Fr(wher)-5 b(e)34 b Fm(C)393 2448 y FJ(2)465 2434 y Fr(is)e(a)h(c)-5 b(onstant)35 b(indep)-5 b(endent)34 b(of)f Fm(")p Fr(.)212 2614 y FI(The)d(pro)s(of)f(of)i(Lemma)g(4.11)g(is)f(deferred)f(to)i (the)g(end)f(of)g(the)h(section.)71 2818 y Ft(Pro)s(of)45 b(of)g(Prop)s(osition)h(4.9:)58 b FI(First)38 b(of)h(all,)h(w)m(e)f (stress)g(that)g(the)g(estimates)g(giv)m(en)g(b)m(y)g(Lemma)g(4.11)71 2931 y(are)32 b(only)g(v)-5 b(alid)30 b(if)i(w)m(e)g(ev)-5 b(aluate)33 b Fm(')1285 2945 y Fl(")1322 2931 y FI(\()p Fm(z)t FI(\))h(for)e Fo(j)p Fm(z)t Fo(j)d FI(=)f Fm("=)p FI(2.)47 b(F)-8 b(rom)33 b(the)f(de\014nition)e(of)j Fm( )3011 2945 y Fl(")3048 2931 y FI(\()p Fm(z)t FI(\))g(\(see)g (\(13\))r(\))f(this)f(is)71 3044 y(precisely)e(the)h(case)i(whic)m(h)d (w)m(e)i(are)g(in)m(terested)f(in.)212 3157 y(Let)35 b(us)f(see)i(that)f Fm( )912 3171 y Fl(")984 3157 y FI(is)f(a)h (quasiconformal)e(mapping,)i(and)f(obtain)g(a)h(b)s(ound)e(for)h(its)h (distortion.)52 b(In)71 3270 y Fq(D)142 3260 y Fk(")p 146 3272 31 3 v 146 3313 a Fj(2)221 3270 y FI(w)m(e)31 b(ha)m(v)m(e:)1140 3391 y Fm(@)5 b( )1252 3405 y Fl(")p 1140 3432 150 4 v 1165 3515 a Fm(@)g(z)1299 3453 y FI(\()p Fm(z)t FI(\))27 b(=)1548 3391 y(1)p 1548 3432 46 4 v 1548 3515 a(2)1603 3453 y Fm(')1662 3415 y Fp(0)1662 3475 y Fl(")1714 3325 y Fn(\022)1793 3391 y Fm(")p 1791 3432 V 1791 3515 a FI(2)1882 3391 y Fm(z)p 1857 3432 97 4 v 1857 3515 a Fo(j)p Fm(z)t Fo(j)1963 3325 y Fn(\023)2050 3453 y FI(+)2151 3391 y(1)p 2151 3432 46 4 v 2153 3515 a Fm(")2247 3391 y FI(\026)-50 b Fm(z)p 2217 3432 97 4 v 2217 3515 a Fo(j)p Fm(z)t Fo(j)2323 3453 y Fm(')2382 3467 y Fl(")2435 3325 y Fn(\022)2513 3391 y Fm(")p 2512 3432 46 4 v 2512 3515 a FI(2)2602 3391 y Fm(z)p 2577 3432 97 4 v 2577 3515 a Fo(j)p Fm(z)t Fo(j)2684 3325 y Fn(\023)2766 3453 y Fm(;)71 3660 y FI(and)1027 3744 y Fm(@)5 b( )1139 3758 y Fl(")p 1027 3785 150 4 v 1052 3868 a Fm(@)11 b FI(\026)-51 b Fm(z)1186 3806 y FI(\()p Fm(z)t FI(\))26 b(=)f Fo(\000)1505 3744 y FI(1)p 1505 3785 46 4 v 1505 3868 a(2)1595 3744 y Fm(z)1641 3711 y FJ(2)p 1570 3785 137 4 v 1570 3881 a Fo(j)p Fm(z)t Fo(j)1667 3839 y FJ(2)1716 3806 y Fm(')1775 3768 y Fp(0)1775 3828 y Fl(")1828 3678 y Fn(\022)1906 3744 y Fm(")p 1905 3785 46 4 v 1905 3868 a FI(2)1995 3744 y Fm(z)p 1970 3785 97 4 v 1970 3868 a Fo(j)p Fm(z)t Fo(j)2077 3678 y Fn(\023)2164 3806 y FI(+)2265 3744 y(1)p 2265 3785 46 4 v 2267 3868 a Fm(")2355 3744 y(z)p 2330 3785 97 4 v 2330 3868 a Fo(j)p Fm(z)t Fo(j)2437 3806 y Fm(')2496 3820 y Fl(")2548 3678 y Fn(\022)2627 3744 y Fm(")p 2625 3785 46 4 v 2625 3868 a FI(2)2716 3744 y Fm(z)p 2690 3785 97 4 v 2690 3868 a Fo(j)q Fm(z)t Fo(j)2797 3678 y Fn(\023)2879 3806 y Fm(:)71 4024 y FI(Then,)k(applying)g(Lemma)h (4.11,)i(w)m(e)f(can)g(b)s(ound:)572 4133 y Fn(\014)572 4187 y(\014)572 4242 y(\014)572 4296 y(\014)613 4203 y Fm(@)5 b( )725 4217 y Fl(")p 613 4243 150 4 v 637 4327 a Fm(@)g(z)772 4264 y FI(\()p Fm(z)t FI(\))21 b Fo(\000)f FI(1)1045 4133 y Fn(\014)1045 4187 y(\014)1045 4242 y(\014)1045 4296 y(\014)1159 4264 y FI(=)1312 4133 y Fn(\014)1312 4187 y(\014)1312 4242 y(\014)1312 4296 y(\014)1353 4203 y FI(1)p 1353 4243 46 4 v 1353 4327 a(2)1423 4136 y Fn(\022)1490 4264 y Fm(')1549 4227 y Fp(0)1549 4287 y Fl(")1601 4136 y Fn(\022)1680 4203 y Fm(")p 1678 4243 V 1678 4327 a FI(2)1769 4203 y Fm(z)p 1744 4243 97 4 v 1744 4327 a Fo(j)p Fm(z)t Fo(j)1850 4136 y Fn(\023)1938 4264 y Fo(\000)g FI(1)2074 4136 y Fn(\023)2161 4264 y FI(+)2262 4203 y(1)p 2262 4243 46 4 v 2264 4327 a Fm(")2358 4203 y FI(\026)-50 b Fm(z)p 2327 4243 97 4 v 2327 4327 a Fo(j)q Fm(z)t Fo(j)2449 4136 y Fn(\022)2516 4264 y Fm(')2575 4278 y Fl(")2628 4136 y Fn(\022)2706 4203 y Fm(")p 2704 4243 46 4 v 2704 4327 a FI(2)2795 4203 y Fm(z)p 2770 4243 97 4 v 2770 4327 a Fo(j)p Fm(z)t Fo(j)2877 4136 y Fn(\023)2964 4264 y Fo(\000)3066 4203 y Fm(")p 3065 4243 46 4 v 3065 4327 a FI(2)3155 4203 y Fm(z)p 3130 4243 97 4 v 3130 4327 a Fo(j)p Fm(z)t Fo(j)3237 4136 y Fn(\023)3304 4133 y(\014)3304 4187 y(\014)3304 4242 y(\014)3304 4296 y(\014)1159 4504 y Fo(\024)1322 4443 y FI(1)p 1322 4483 46 4 v 1322 4566 a(2)1378 4504 y Fm(C)1443 4518 y FJ(2)1482 4504 y Fm(")21 b FI(+)1646 4443 y(1)p 1646 4483 V 1648 4566 a Fm(")1701 4504 y(C)1766 4518 y FJ(2)1805 4504 y Fm(")1847 4467 y FJ(2)1913 4504 y Fo(\024)2018 4443 y FI(3)p 2018 4483 V 2018 4566 a(2)2074 4504 y Fm(C)2139 4518 y FJ(2)2178 4504 y Fm(";)71 4720 y FI(and)537 4816 y Fn(\014)537 4871 y(\014)537 4925 y(\014)537 4980 y(\014)577 4887 y Fm(@)5 b( )689 4901 y Fl(")p 577 4927 150 4 v 602 5011 a Fm(@)11 b FI(\026)-51 b Fm(z)737 4948 y FI(\()p Fm(z)t FI(\))853 4816 y Fn(\014)853 4871 y(\014)853 4925 y(\014)853 4980 y(\014)967 4948 y FI(=)1121 4816 y Fn(\014)1121 4871 y(\014)1121 4925 y(\014)1121 4980 y(\014)1151 4948 y Fo(\000)1232 4887 y FI(1)p 1232 4927 46 4 v 1232 5011 a(2)1322 4887 y Fm(z)1368 4854 y FJ(2)p 1297 4927 137 4 v 1297 5023 a Fo(j)p Fm(z)t Fo(j)1394 4981 y FJ(2)1458 4820 y Fn(\022)1525 4948 y Fm(')1584 4911 y Fp(0)1584 4971 y Fl(")1637 4820 y Fn(\022)1715 4887 y Fm(")p 1713 4927 46 4 v 1713 5011 a FI(2)1804 4887 y Fm(z)p 1779 4927 97 4 v 1779 5011 a Fo(j)p Fm(z)t Fo(j)1886 4820 y Fn(\023)1973 4948 y Fo(\000)20 b FI(1)2109 4820 y Fn(\023)2196 4948 y FI(+)2297 4887 y(1)p 2297 4927 46 4 v 2299 5011 a Fm(")2388 4887 y(z)p 2362 4927 97 4 v 2362 5011 a Fo(j)q Fm(z)t Fo(j)2484 4820 y Fn(\022)2551 4948 y Fm(')2610 4962 y Fl(")2663 4820 y Fn(\022)2741 4887 y Fm(")p 2740 4927 46 4 v 2740 5011 a FI(2)2830 4887 y Fm(z)p 2805 4927 97 4 v 2805 5011 a Fo(j)p Fm(z)t Fo(j)2912 4820 y Fn(\023)2999 4948 y Fo(\000)3101 4887 y Fm(")p 3100 4927 46 4 v 3100 5011 a FI(2)3190 4887 y Fm(z)p 3165 4927 97 4 v 3165 5011 a Fo(j)p Fm(z)t Fo(j)3272 4820 y Fn(\023)3339 4816 y(\014)3339 4871 y(\014)3339 4925 y(\014)3339 4980 y(\014)967 5199 y Fo(\024)1131 5137 y FI(1)p 1131 5178 46 4 v 1131 5261 a(2)1186 5199 y Fm(C)1251 5213 y FJ(2)1290 5199 y Fm(")h FI(+)1454 5137 y(1)p 1454 5178 V 1456 5261 a Fm(")1509 5199 y(C)1574 5213 y FJ(2)1614 5199 y Fm(")1656 5161 y FJ(2)1721 5199 y Fo(\024)1827 5137 y FI(3)p 1827 5178 V 1827 5261 a(2)1882 5199 y Fm(C)1947 5213 y FJ(2)1987 5199 y Fm(":)71 5431 y FI(So,)44 b(if)c(w)m(e)i (assume)f Fm(")g FI(to)h(b)s(e)e(small)g(enough)h(in)f(order)h(to)g(ha) m(v)m(e)i(that)2638 5395 y FJ(3)p 2638 5410 36 4 v 2638 5462 a(2)2683 5431 y Fm(C)2748 5445 y FJ(2)2787 5431 y Fm(")h Fo(\024)2997 5395 y FJ(1)p 2997 5410 V 2997 5462 a(2)3042 5431 y FI(,)g(w)m(e)e(can)f(b)s(ound)e(the)71 5543 y(distortion)29 b(of)h Fm( )648 5557 y Fl(")716 5543 y FI(b)m(y:)1372 5569 y Fn(\014)1372 5624 y(\014)1372 5679 y(\014)1372 5733 y(\014)1372 5788 y(\014)1423 5621 y Fl(@)t( )1510 5629 y Fk(")p 1423 5641 121 4 v 1445 5693 a Fl(@)8 b FJ(\026)-39 b Fl(z)1553 5662 y FI(\()p Fm(z)t FI(\))p 1413 5708 258 4 v 1423 5765 a Fl(@)t( )1510 5773 y Fk(")p 1423 5785 121 4 v 1445 5837 a Fl(@)t(z)1553 5806 y FI(\()p Fm(z)t FI(\))1680 5569 y Fn(\014)1680 5624 y(\014)1680 5679 y(\014)1680 5733 y(\014)1680 5788 y(\014)1736 5729 y Fo(\024)1930 5626 y FJ(3)p 1930 5641 36 4 v 1930 5693 a(2)1975 5662 y Fm(C)2040 5676 y FJ(2)2080 5662 y Fm(")p 1842 5708 359 4 v 1842 5797 a FI(1)20 b Fo(\000)2008 5761 y FJ(3)p 2008 5776 36 4 v 2008 5828 a(2)2053 5797 y Fm(C)2118 5811 y FJ(2)2158 5797 y Fm(")2235 5729 y Fo(\024)25 b FI(3)p Fm(C)2441 5743 y FJ(2)2481 5729 y Fm(":)p eop %%Page: 21 21 21 20 bop 71 274 a Fs(N.)31 b(F)-8 b(agella,)31 b(T.)f(M.)h(Seara)g (and)f(J.)g(Villan)m(uev)-5 b(a)2025 b FI(21)71 573 y(Hence,)31 b Fm( )425 587 y Fl(")493 573 y FI(is)e(a)i(\()p Fm(C)760 587 y FJ(1)800 573 y Fm(")p FI(\)-quasiconformal)f(mapping)e(with)h Fm(C)2180 587 y FJ(1)2245 573 y FI(=)c(3)p Fm(C)2451 587 y FJ(2)2491 573 y FI(.)212 686 y(In)g(order)g(to)h(estimate)h(ho)m (w)f(far)f Fm( )1398 700 y Fl(")1460 686 y FI(is)g(from)g(the)h(iden)m (tit)m(y)f(map,)i(w)m(e)f(apply)e(the)i(\014rst)f(inequalit)m(y)f(of)i (\(15\))71 799 y(\(see)31 b(Lemma)g(4.11\),)h(obtaining)696 1039 y Fo(j)p Fm( )780 1053 y Fl(")817 1039 y FI(\()p Fm(z)t FI(\))21 b Fo(\000)f Fm(z)t Fo(j)26 b FI(=)1238 907 y Fn(\014)1238 962 y(\014)1238 1016 y(\014)1238 1071 y(\014)1278 977 y FI(2)p 1278 1018 46 4 v 1280 1101 a Fm(")1349 1039 y Fo(j)p Fm(z)t Fo(j)1461 911 y Fn(\022)1528 1039 y Fm(')1587 1053 y Fl(")1639 911 y Fn(\022)1717 977 y Fm(")p 1716 1018 V 1716 1101 a FI(2)1807 977 y Fm(z)p 1781 1018 97 4 v 1781 1101 a Fo(j)q Fm(z)t Fo(j)1888 911 y Fn(\023)1975 1039 y Fo(\000)2078 977 y Fm(")p 2076 1018 46 4 v 2076 1101 a FI(2)2167 977 y Fm(z)p 2141 1018 97 4 v 2141 1101 a Fo(j)q Fm(z)t Fo(j)2248 911 y Fn(\023)2315 907 y(\014)2315 962 y(\014)2315 1016 y(\014)2315 1071 y(\014)2371 1039 y Fo(\024)2477 977 y FI(2)p 2477 1018 46 4 v 2479 1101 a Fm(")2547 1039 y Fo(j)p Fm(z)t Fo(j)16 b Fm(C)2724 1053 y FJ(2)2763 1039 y Fm(")2805 1001 y FJ(2)2871 1039 y Fo(\024)25 b Fm(C)3032 1053 y FJ(1)3071 1039 y Fm(")15 b Fo(j)q Fm(z)t Fo(j)71 1274 y FI(whic)m(h)29 b(concludes)h(the)g(pro)s(of.)3780 1450 y Fe(\003)71 1626 y FI(The)39 b(rest)g(of)h(the)f(section)h(is)e(dedicated)h(to)h (pro)m(v)m(e)g(Lemma)g(4.11.)69 b(F)-8 b(or)40 b(this)e(purp)s(ose,)i (w)m(e)g(will)d(need)i(a)71 1739 y(Diophan)m(tine-lik)m(e)34 b(b)s(ound)g(for)i(Brjuno)f(n)m(um)m(b)s(ers,)h(that)h(is)e(w)m(eak)m (er)i(than)f(the)g(Brjuno)f(condition.)56 b(More)71 1852 y(precisely)-8 b(,)33 b(if)f Fm(\022)j FI(is)d(a)i(Brjuno)e(n)m(um)m(b) s(er,)h(there)g(exist)g(constan)m(ts)h Fm(c)2365 1866 y FJ(1)2405 1852 y FI(,)g Fm(c)2503 1866 y FJ(2)2542 1852 y FI(,)g(dep)s(ending)d(only)h(on)h Fm(\022)s FI(,)g(suc)m(h)g (that)71 1965 y(for)d(an)m(y)h Fm(k)d Fo(2)d Fq(Z)16 b Fo(n)k(f)p FI(0)p Fo(g)p FI(,)32 b(the)f(follo)m(wing)e(inequalit)m (y)g(is)g(satis\014ed)g(\(see)j([Br1,)f(p.)40 b(140]\))1457 2155 y Fo(j)p FI(e)1522 2117 y FJ(2)p Fl(\031)r FJ(i)p Fl(k)r(\022)1718 2155 y Fo(\000)20 b FI(1)p Fo(j)1879 2117 y Fp(\000)p FJ(1)1999 2155 y Fo(\024)25 b Fm(c)2134 2169 y FJ(1)2174 2155 y FI(e)2214 2117 y FJ(2)p Fl(\031)r(c)2323 2126 y Fj(2)2358 2117 y Fp(j)p Fl(k)r Fp(j)2439 2155 y Fm(:)1225 b FI(\(16\))71 2345 y(This)28 b(inequalit)m(y)h(leads)h(to) h(the)g(follo)m(wing)d(Lemma:)71 2521 y Ft(Lemma)43 b(4.12.)k Fr(L)-5 b(et)41 b Fm(m)p FI(\()p Fm(x)p FI(\))g Fr(b)-5 b(e)41 b(a)g FI(1)p Fr(-p)-5 b(erio)g(dic)42 b(function)f(with)h(zer)-5 b(o)42 b(aver)-5 b(age,)43 b(and)f Fm(\022)g Fo(2)e Fq(R)49 b Fr(b)-5 b(e)41 b(a)g(Brjuno)71 2634 y(numb)-5 b(er,)37 b(henc)-5 b(e)37 b(verifying)e(\(16\).)54 b(We)35 b(assume)i(that)h Fm(m)d Fr(is)h(analytic)i(in)e(the)g(c)-5 b(omplex)38 b(strip)f Fo(A)3458 2648 y Fl(c)3489 2657 y Fj(3)3559 2634 y FI(=)31 b Fo(f)p Fm(x)h Fo(2)71 2747 y Fq(C)18 b Fm(=)p Fq(Z)27 b FI(:)58 b Fo(j)p FI(Im\()p Fm(x)p FI(\))p Fo(j)26 b Fm(<)f(c)793 2761 y FJ(3)833 2747 y Fo(g)p Fr(,)32 b(b)-5 b(eing)33 b Fm(c)1210 2761 y FJ(3)1275 2747 y Fm(>)25 b(c)1410 2761 y FJ(2)1450 2747 y Fr(,)32 b(and)i(that)f Fm(B)d FI(=)25 b(sup)2204 2769 y Fl(x)p Fp(2A)2347 2777 y Fk(c)2375 2792 y Fj(3)2433 2747 y Fo(j)p Fm(m)p FI(\()p Fm(x)p FI(\))p Fo(j)h Fm(<)f FI(+)p Fo(1)p Fr(.)212 2869 y(Consider)34 b(the)f(1-p)-5 b(erio)g(dic)34 b(solution)g Fm(\030)t FI(\()p Fm(x)p FI(\))f Fr(of)g(the)g(di\013er)-5 b(enc)g(e)33 b(e)-5 b(quation:)1209 3059 y Fm(\030)t FI(\()p Fm(x)21 b FI(+)f Fm(\022)s FI(\))f Fo(\000)h Fm(\030)t FI(\()p Fm(x)p FI(\))26 b(=)f Fm(m)p FI(\()p Fm(x)p FI(\))p Fm(;)202 b(\030)t FI(\(0\))26 b(=)f(0)p Fm(:)978 b FI(\(17\))71 3249 y Fr(Then,)33 b Fm(\030)j Fr(is)d(analytic)h(in)e Fo(A)1038 3263 y Fl(c)1069 3272 y Fj(3)1103 3263 y Fp(\000)p Fl(c)1189 3272 y Fj(2)1227 3249 y Fr(,)g(and)i(veri\014es:)921 3502 y Fo(j)p Fm(\030)t FI(\()p Fm(x)p FI(\))p Fo(j)26 b(\024)f FI(4)p Fm(B)5 b(c)1417 3516 y FJ(1)1545 3440 y FI(e)1585 3407 y Fp(\000)p FJ(2)p Fl(\031)r FJ(\()p Fl(c)1776 3416 y Fj(3)1811 3407 y Fp(\000)p Fl(c)1897 3416 y Fj(2)1931 3407 y Fp(\000j)p FJ(Im)o(\()p Fl(x)p FJ(\))p Fp(j)p FJ(\))p 1467 3481 848 4 v 1467 3568 a FI(1)20 b Fo(\000)g FI(e)1663 3542 y Fp(\000)p FJ(2)p Fl(\031)r FJ(\()p Fl(c)1854 3551 y Fj(3)1889 3542 y Fp(\000)p Fl(c)1975 3551 y Fj(2)2009 3542 y Fp(\000j)p FJ(Im)o(\()p Fl(x)p FJ(\))p Fp(j)p FJ(\))2324 3502 y Fm(;)201 b(x)25 b Fo(2)g(A)2786 3516 y Fl(c)2817 3525 y Fj(3)2851 3516 y Fp(\000)p Fl(c)2937 3525 y Fj(2)2975 3502 y Fm(:)71 3733 y Ft(Pro)s(of)36 b(:)k FI(First,)30 b(w)m(e)h(expand)f Fm(m)g FI(in)f(F)-8 b(ourier)30 b(series:)1541 3932 y Fm(m)p FI(\()p Fm(x)p FI(\))c(=)1865 3845 y Fn(X)1866 4043 y Fl(k)r Fp(2)p Fh(Z)2012 3932 y Fm(m)2092 3947 y Fl(k)2134 3932 y FI(e)2174 3894 y FJ(2)p Fl(\031)r FJ(i)p Fl(k)r(x)2355 3932 y Fm(:)71 4210 y FI(Then,)37 b(using)d(the)i(b)s(ound)e(of)j Fo(j)p Fm(m)p Fo(j)f FI(in)f(the)h(strip)f Fo(A)1874 4224 y Fl(c)1905 4233 y Fj(3)1943 4210 y FI(,)i(and)f(that)g(it)g(has)g(zero)h (a)m(v)m(erage,)j(one)c(has)g(that)h(its)71 4323 y(F)-8 b(ourier)31 b(co)s(e\016cien)m(ts)g(v)m(erify)g Fm(m)1181 4337 y FJ(0)1247 4323 y FI(=)26 b(0)31 b(and)g Fo(j)p Fm(m)1703 4338 y Fl(k)1746 4323 y Fo(j)26 b(\024)h Fm(B)5 b FI(e)2009 4290 y Fp(\000)p FJ(2)p Fl(\031)r(c)2173 4299 y Fj(3)2207 4290 y Fp(j)o Fl(k)r Fp(j)2320 4323 y FI(if)30 b Fm(k)f Fo(6)p FI(=)e(0.)43 b(On)30 b(the)h(other)h(hand,)e (writing)f Fm(\030)71 4436 y FI(also)h(in)f(F)-8 b(ourier)30 b(series,)g(w)m(e)h(can)g(solv)m(e)f(equation)h(\(17\))g(for)g(the)f (co)s(e\016cien)m(ts)h(of)g Fm(\030)t FI(,)f(obtaining)1434 4645 y Fm(\030)1474 4660 y Fl(k)1542 4645 y FI(=)1773 4583 y Fm(m)1853 4598 y Fl(k)p 1648 4624 373 4 v 1648 4707 a FI(e)1688 4681 y FJ(2)p Fl(\031)r FJ(i)p Fl(k)r(\022)1884 4707 y Fo(\000)20 b FI(1)2030 4645 y Fm(;)107 b FI(if)29 b Fm(k)f Fo(6)p FI(=)d(0)p Fm(:)71 4869 y FI(Moreo)m(v)m(er,)33 b(condition)c Fm(\030)t FI(\(0\))d(=)f(0)31 b(giv)m(es)f Fm(\030)1514 4883 y FJ(0)1579 4869 y FI(=)25 b Fo(\000)1761 4800 y Fn(P)1857 4896 y Fl(k)r Fp(2)p Fh(Z)-7 b Fp(nf)p FJ(0)p Fp(g)2147 4869 y Fm(\030)2187 4884 y Fl(k)2229 4869 y FI(.)212 4982 y(Using)30 b(\(16\))h(w)m(e)g(obtain)f(that)947 5172 y Fo(j)p Fm(\030)1012 5187 y Fl(k)1055 5172 y Fo(j)25 b(\024)g Fm(c)1240 5186 y FJ(1)1295 5172 y Fo(j)p Fm(m)1400 5187 y Fl(k)1443 5172 y Fo(j)15 b FI(e)1523 5134 y FJ(2)p Fl(\031)r(c)1632 5143 y Fj(2)1667 5134 y Fp(j)o Fl(k)r Fp(j)1774 5172 y Fo(\024)25 b Fm(B)5 b(c)1983 5186 y FJ(1)2022 5172 y FI(e)2062 5134 y Fp(\000)p FJ(2)p Fl(\031)r FJ(\()p Fl(c)2253 5143 y Fj(3)2288 5134 y Fp(\000)p Fl(c)2374 5143 y Fj(2)2408 5134 y FJ(\))p Fp(j)p Fl(k)r Fp(j)2518 5172 y Fm(;)106 b FI(if)29 b Fm(k)g Fo(6)p FI(=)c(0)p Fm(;)71 5362 y FI(and,)30 b(for)g Fm(\030)452 5376 y FJ(0)491 5362 y FI(,)1156 5532 y Fo(j)q Fm(\030)1222 5546 y FJ(0)1261 5532 y Fo(j)25 b(\024)1477 5446 y Fn(X)1407 5647 y Fl(k)r Fp(2)p Fh(Z)-6 b Fp(nf)p FJ(0)p Fp(g)1693 5532 y Fo(j)q Fm(\030)1759 5547 y Fl(k)1801 5532 y Fo(j)26 b(\024)f FI(2)p Fm(B)5 b(c)2106 5546 y FJ(1)2234 5471 y FI(e)2274 5438 y Fp(\000)p FJ(2)p Fl(\031)r FJ(\()p Fl(c)2465 5447 y Fj(3)2500 5438 y Fp(\000)p Fl(c)2586 5447 y Fj(2)2620 5438 y FJ(\))p 2155 5511 575 4 v 2155 5599 a FI(1)21 b Fo(\000)f FI(e)2352 5572 y Fp(\000)p FJ(2)p Fl(\031)r FJ(\()p Fl(c)2543 5581 y Fj(3)2578 5572 y Fp(\000)p Fl(c)2664 5581 y Fj(2)2698 5572 y FJ(\))2740 5532 y Fm(:)71 5796 y FI(Joining)28 b(these)j(b)s(ounds,)e(the)h (estimate)h(for)f Fo(j)p Fm(\030)t FI(\()p Fm(x)p FI(\))p Fo(j)i FI(in)d Fo(A)2051 5810 y Fl(c)2082 5819 y Fj(3)2116 5810 y Fp(\000)p Fl(c)2202 5819 y Fj(2)2270 5796 y FI(follo)m(ws)h (immediately)-8 b(.)p eop %%Page: 22 22 22 21 bop 71 274 a FI(22)1001 b Fs(Asymptotic)30 b(Size)g(of)h(Herman)f (Rings)f(of)i(the)f(Complex)g(Standard)f(F)-8 b(amily)3780 573 y Fe(\003)71 742 y FI(No)m(w,)31 b(w)m(e)g(ha)m(v)m(e)h(all)d(the)h (ingredien)m(ts)f(to)i(pro)m(v)m(e)h(Lemma)e(4.11.)71 912 y Ft(Pro)s(of)j(of)g(Lemma)e(4.11:)40 b FI(T)-8 b(o)29 b(obtain)f(the)h(estimates)g(of)f(this)g(Lemma,)h(it)f(will)e(b)s(e)h (more)i(con)m(v)m(enien)m(t)h(for)71 1025 y(us)i(to)h(w)m(ork)g(with)e (the)i(lift)e(on)h(the)h(circle)f(of)h(the)g(scaled)f(standard)g(map,)h (that)g(is,)f(the)h(Arnold)e(standard)71 1137 y(family)j(\(1\))q(.)54 b(T)-8 b(o)35 b(this)e(end,)j(w)m(e)f(mak)m(e)h(the)e(transformations)g (w)m(e)h(ha)m(v)m(e)h(done)f(bac)m(kw)m(ards)g(in)e(order)h(to)h(go)71 1263 y(from)304 1239 y Fn(e)287 1263 y Fm(f)332 1282 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")521 1263 y FI(\()p Fm(x)p FI(\))28 b(=)f Fm(x)22 b FI(+)944 1218 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p 944 1242 133 4 v 972 1294 a(2)p Fl(\031)1108 1263 y FI(+)1233 1227 y Fl(")p 1210 1242 79 4 v 1210 1294 a FJ(2)p Fl(\031)1313 1263 y FI(sin)o(\(2)p Fm(\031)s(x)p FI(\))33 b(to)f Fm(F)1850 1282 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")2040 1263 y FI(\()p Fm(z)t FI(\).)46 b(Then,)31 b(if)g(w)m(e)h(mak)m(e)h(the)f(c)m(hange)h Fm(z)f FI(=)3594 1227 y Fl(")p 3592 1242 36 4 v 3592 1294 a FJ(2)3637 1263 y Fm(u)p FI(,)h(w)m(e)71 1395 y(ha)m(v)m(e)38 b Fn(e)-63 b Fm(')334 1409 y Fl(")371 1395 y FI(\()p Fm(u)p FI(\))26 b(=)625 1359 y FJ(2)p 625 1374 V 626 1426 a Fl(")670 1395 y Fm(')729 1409 y Fl(")766 1395 y FI(\()812 1359 y Fl(")p 811 1374 V 811 1426 a FJ(2)857 1395 y Fm(u)p FI(\),)g(where)37 b Fn(e)-63 b Fm(')1312 1409 y Fl(")1374 1395 y FI(is)24 b(the)g(linearization)f(of)i(the)g (standard)f(map)2963 1372 y Fn(e)2945 1395 y Fm(F)3003 1414 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")3217 1395 y FI(giv)m(en)h(in)e(\(3\).)40 b(On)71 1528 y(the)34 b(other)g(hand,)h(writing)d Fm(u)f FI(=)g(e)1272 1495 y FJ(i)p Fl(x)1370 1528 y FI(w)m(e)k(ha)m(v)m(e)47 b Fn(e)-63 b Fm(')1780 1542 y Fl(")1817 1528 y FI(\(e)1892 1495 y FJ(2)p Fl(\031)r FJ(i)p Fl(x)2035 1528 y FI(\))31 b(=)g(e)2243 1495 y FJ(2)p Fl(\031)r FJ(i)s Fd(e)-42 b Fl(\021)2376 1503 y Fk(")2410 1495 y FJ(\()p Fl(x)p FJ(\))2509 1528 y FI(,)35 b(b)s(eing)i Fn(e)-55 b Fm(\021)2863 1542 y Fl(")2934 1528 y FI(a)34 b(conjugation)h(of)3634 1504 y Fn(e)3617 1528 y Fm(f)3662 1547 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")71 1641 y FI(to)c(the)g(rotation)f Fo(T)737 1656 y Fl(\022)806 1641 y FI(in)f Fq(T)971 1608 y FJ(1)1043 1641 y FI(\(see)i(\(2\)\).)212 1754 y(Then,)f(w)m(e)g (obtain)g(that:)590 1950 y Fm(')649 1964 y Fl(")686 1950 y FI(\()p Fm(z)t FI(\))21 b Fo(\000)f Fm(z)30 b FI(=)1093 1889 y Fm(")p 1092 1929 46 4 v 1092 2012 a FI(2)1147 1950 y(\()12 b Fn(e)-63 b Fm(')1241 1964 y Fl(")1279 1950 y FI(\()p Fm(u)p FI(\))21 b Fo(\000)e Fm(u)p FI(\))26 b(=)1733 1889 y Fm(")p 1731 1929 V 1731 2012 a FI(2)1786 1950 y(\(e)1861 1912 y FJ(2)p Fl(\031)r FJ(i)s Fd(e)-42 b Fl(\021)1994 1920 y Fk(")2029 1912 y FJ(\()p Fl(x)p FJ(\))2148 1950 y Fo(\000)20 b FI(e)2279 1912 y FJ(2)p Fl(\031)r FJ(i)p Fl(x)2421 1950 y FI(\))25 b(=)g Fm(z)t FI(\(e)2698 1912 y FJ(2)p Fl(\031)r FJ(i\()s Fd(e)-42 b Fl(\021)2858 1920 y Fk(")2893 1912 y FJ(\()p Fl(x)p FJ(\))p Fp(\000)p Fl(x)p FJ(\))3134 1950 y Fo(\000)20 b FI(1\))p Fm(:)359 b FI(\(18\))71 2158 y(Our)28 b(purp)s(ose)g(is)g (to)j(b)s(ound)g Fn(e)-55 b Fm(\021)1123 2172 y Fl(")1160 2158 y FI(\()p Fm(x)p FI(\))19 b Fo(\000)f Fm(x)30 b FI(using)e(the)h(Lo)s(cal)h(Conjugacy)f(Theorem)h(2.1,)h(and)d(to)j (deriv)m(e)e(from)71 2271 y(these)i(b)s(ounds)d(the)i(ones)h(for)f Fm(')1177 2285 y Fl(")1214 2271 y FI(\()p Fm(z)t FI(\))21 b Fo(\000)f Fm(z)35 b FI(and)30 b(its)g(deriv)-5 b(ativ)m(e.)212 2384 y(First)33 b(of)h(all,)g(w)m(e)g(observ)m(e)h(that)1382 2360 y Fn(e)1365 2384 y Fm(f)1410 2403 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")1599 2384 y FI(\()p Fm(x)p FI(\))f(is)f(univ)-5 b(alen)m(t)32 b(in)h(the)h(strip)e Fo(A)2835 2385 y Fj(1)p 2816 2397 69 3 v 2816 2438 a(2)p Fk(\031)2906 2412 y FJ(log)q(\()p Fl(\033)r FJ(\()p Fl(")p FJ(\))p Fl(=")p FJ(\))3254 2384 y FI(,)j(where)e Fm(\033)s FI(\()p Fm(")p FI(\))g(=)71 2530 y(1)21 b(+)227 2454 y Fo(p)p 303 2454 239 4 v 76 x FI(1)g Fo(\000)f Fm(")502 2504 y FJ(2)572 2530 y FI(is)29 b(de\014ned)g(in)g(\(9\).)212 2643 y(In)43 b(order)h(to)h(apply)d(Theorem)i(2.1,)49 b(let)44 b(us)f(observ)m(e)h (that,)49 b(if)43 b Fm(")h FI(is)f(small)g(enough,)k(w)m(e)d(ha)m(v)m (e)i(that)102 2720 y FJ(1)p 81 2735 79 4 v 81 2788 a(2)p Fl(\031)184 2756 y FI(log)r(\()p Fm(\033)s FI(\()p Fm(")p FI(\))p Fm(=")p FI(\))28 b Fm(>)781 2720 y FJ(1)p 760 2735 V 760 2788 a(2)p Fl(\031)848 2756 y FI(\010\()p Fm(\022)s FI(\))20 b(+)f Fm(C)1205 2770 y FJ(0)1245 2756 y FI(.)40 b(Th)m(us,)30 b(Theorem)g(2.1)i(ensures)d(that)35 b Fn(e)-55 b Fm(\021)2658 2770 y Fl(")2695 2756 y FI(\()p Fm(x)p FI(\))31 b(is)e(analytic)h(if)1142 2985 y Fo(j)p FI(Im\()p Fm(x)p FI(\))p Fo(j)c Fm(<)1583 2924 y FI(1)p 1555 2964 101 4 v 1555 3048 a(2)p Fm(\031)1681 2985 y FI(log)q(\()p Fm(\033)s FI(\()p Fm(")p FI(\))p Fm(=")p FI(\))d Fo(\000)2274 2924 y FI(1)p 2246 2964 V 2246 3048 a(2)p Fm(\031)2357 2985 y FI(\010\()p Fm(\022)s FI(\))c Fo(\000)h Fm(C)2714 2999 y FJ(0)2754 2985 y Fm(:)71 3220 y FI(In)41 b(particular,)j(if)d(w)m(e)i(tak)m(e)h(an)m(y)e(constan)m(t) i(0)h Fm(<)g(c)1913 3234 y FJ(4)1998 3220 y Fo(\024)2234 3175 y Fl(\033)r FJ(\()p Fl(")p FJ(\))p 2123 3199 352 4 v 2123 3262 a(e)2154 3240 y Fj(\010\()p Fk(\022)r Fj(\)+2)p Fk(\031)r(C)2435 3255 y Fj(0)2485 3220 y FI(,)g(the)e(linearization)h Fn(e)-55 b Fm(\021)45 b FI(is)d(de\014ned)f(in)71 3333 y Fo(A)172 3334 y Fj(1)p 153 3346 69 3 v 153 3387 a(2)p Fk(\031)244 3361 y FJ(log)q(\()p Fl(c)393 3370 y Fj(4)427 3361 y Fl(=")p FJ(\))567 3333 y FI(and)f(its)f(range)i(is)e(con)m (tained)i(in)e Fo(A)1888 3334 y Fj(1)p 1869 3346 V 1869 3387 a(2)p Fk(\031)1959 3361 y FJ(log)q(\()p Fl(\033)r FJ(\()p Fl(")p FJ(\))p Fl(=")p FJ(\))2308 3333 y FI(.)70 b(Moreo)m(v)m(er,)45 b(in)39 b(this)g(domain)g(it)h(v)m(eri\014es)88 3461 y Fn(e)71 3485 y Fm(f)116 3503 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")325 3485 y Fo(\016)24 b Fn(e)-55 b Fm(\021)435 3499 y Fl(")497 3485 y FI(=)29 b Fn(e)-55 b Fm(\021)638 3499 y Fl(")696 3485 y Fo(\016)20 b(T)811 3500 y Fl(\022)850 3485 y FI(.)212 3598 y(No)m(w,)31 b(calling)e Fm(\030)t FI(\()p Fm(x)p FI(\))d(=)j Fn(e)-55 b Fm(\021)1065 3612 y Fl(")1102 3598 y FI(\()p Fm(x)p FI(\))21 b Fo(\000)f Fm(x)p FI(,)30 b(it)g(is)g(straigh)m(tforw)m(ard)g (to)h(c)m(hec)m(k)h(that)1486 3780 y Fm(\030)t FI(\()p Fm(x)20 b FI(+)g Fm(\022)s FI(\))g Fo(\000)g Fm(\030)t FI(\()p Fm(x)p FI(\))26 b(=)f Fm(m)p FI(\()p Fm(x)p FI(\))p Fm(;)71 3962 y FI(where)35 b Fm(m)p FI(\()p Fm(x)p FI(\))f(=)696 3938 y Fn(e)679 3962 y Fm(f)724 3981 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")913 3962 y FI(\()t Fn(e)-55 b Fm(\021)993 3976 y Fl(")1030 3962 y FI(\()p Fm(x)p FI(\)\))24 b Fo(\000)k Fn(e)-55 b Fm(\021)1351 3976 y Fl(")1387 3962 y FI(\()p Fm(x)p FI(\))25 b Fo(\000)e Fm(\022)s FI(.)55 b(Then,)35 b Fm(\030)t FI(\()p Fm(x)p FI(\))h(v)m(eri\014es)f(the)g(h)m(yp)s (otheses)h(of)f(Lemma)g(4.12)i(if)71 4096 y(w)m(e)h(de\014ne)e Fm(c)520 4110 y FJ(3)597 4096 y Fo(\021)g Fm(c)743 4110 y FJ(3)783 4096 y FI(\()p Fm(")p FI(\))i(=)1072 4060 y FJ(1)p 1050 4075 79 4 v 1050 4127 a(2)p Fl(\031)1153 4096 y FI(log)r(\()p Fm(c)1345 4110 y FJ(4)1385 4096 y Fm(=")p FI(\))g(and)f(w)m(e)h(tak)m(e)h Fm(B)i FI(a)d(b)s(ound)d(of)j Fo(j)2707 4072 y Fn(e)2690 4096 y Fm(f)2735 4114 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")2924 4096 y FI(\()p Fm(x)p FI(\))25 b Fo(\000)g Fm(x)p Fo(j)37 b FI(in)f(the)h(domain)71 4218 y Fo(A)172 4219 y Fj(1)p 153 4231 69 3 v 153 4271 a(2)p Fk(\031)244 4245 y FJ(log)q(\()p Fl(\033)r FJ(\()p Fl(")p FJ(\))p Fl(=")p FJ(\))592 4218 y FI(,)f(whic)m(h)d(can)i(b)s(e)f (tak)m(en)i(indep)s(enden)m(t)c(of)j Fm(")p FI(.)54 b(W)-8 b(e)36 b(remark)e(that)i(b)s(eing)d Fm(c)3182 4232 y FJ(3)3254 4218 y FI(=)3389 4182 y FJ(1)p 3367 4197 79 4 v 3367 4249 a(2)p Fl(\031)3470 4218 y FI(log)r(\()p Fm(c)3662 4232 y FJ(4)3702 4218 y Fm(=")p FI(\),)71 4356 y(the)d(h)m(yp)s(othesis)f Fm(c)714 4370 y FJ(3)779 4356 y Fm(>)c(c)914 4370 y FJ(2)984 4356 y FI(of)31 b(Lemma)g(4.12)g(alw)m (a)m(ys)g(hold)e(if)h(w)m(e)h(assume)f Fm(")25 b(<)g(c)2831 4370 y FJ(4)2871 4356 y FI(e)2911 4323 y Fp(\000)p FJ(2)p Fl(\031)r(c)3075 4332 y Fj(2)3114 4356 y FI(.)212 4469 y(Applying)36 b(the)j(conclusions)e(of)h(Lemma)h(4.12,)j(w)m(e)d (obtain)f(the)h(follo)m(wing)e(b)s(ound)g(for)h(the)h(function)71 4581 y Fm(\030)t FI(\()p Fm(x)p FI(\))31 b(when)e Fo(j)p FI(Im\()p Fm(x)p FI(\))p Fo(j)d(\024)f Fm(c)947 4595 y FJ(3)1007 4581 y Fo(\000)20 b Fm(c)1137 4595 y FJ(2)1202 4581 y FI(=)1329 4546 y FJ(1)p 1308 4561 V 1308 4613 a(2)p Fl(\031)1411 4581 y FI(log)r(\()p Fm(c)1603 4595 y FJ(4)1643 4581 y Fm(=")p FI(\))h Fo(\000)f Fm(c)1916 4595 y FJ(2)1956 4581 y FI(,)640 4835 y Fo(j)p Fm(\030)t FI(\()p Fm(x)p FI(\))p Fo(j)26 b(\024)f FI(4)p Fm(B)5 b(c)1136 4849 y FJ(1)1264 4773 y FI(e)1304 4740 y Fp(\000)p FJ(\(log)r(\()p Fl(c)1536 4749 y Fj(4)1571 4740 y Fl(=")p FJ(\))p Fp(\000)p FJ(2)p Fl(\031)r FJ(\()p Fl(c)1857 4749 y Fj(2)1892 4740 y FJ(+)p Fp(j)o FJ(Im\()p Fl(x)p FJ(\))p Fp(j)p FJ(\)\))p 1186 4814 1117 4 v 1186 4901 a FI(1)21 b Fo(\000)f FI(e)1383 4875 y Fp(\000)p FJ(\(log)q(\()p Fl(c)1614 4884 y Fj(4)1649 4875 y Fl(=")p FJ(\))p Fp(\000)p FJ(2)p Fl(\031)r FJ(\()p Fl(c)1935 4884 y Fj(2)1970 4875 y FJ(+)p Fp(j)p FJ(Im)o(\()p Fl(x)p FJ(\))p Fp(j)p FJ(\)\))2337 4835 y FI(=)2725 4773 y(4)p Fm(B)5 b(c)2883 4787 y FJ(1)2922 4773 y Fm(")p 2443 4814 803 4 v 2443 4901 a(c)2482 4915 y FJ(4)2522 4901 y FI(e)2562 4875 y Fp(\000)p FJ(2)p Fl(\031)r FJ(\()p Fl(c)2753 4884 y Fj(2)2788 4875 y FJ(+)p Fp(j)p FJ(Im)o(\()p Fl(x)p FJ(\))p Fp(j)p FJ(\))3113 4901 y Fo(\000)20 b Fm(")3256 4835 y(:)212 5071 y FI(Let)33 b(us)e(no)m(w)i(tak)m(e)h Fm(c)925 5085 y FJ(5)997 5071 y FI(an)m(y)e(constan)m(t,)j(indep)s(enden)m(t)30 b(of)i Fm(")p FI(,)i(v)m(erifying)d(0)e Fm(<)f(c)2876 5085 y FJ(5)2944 5071 y Fm(<)3075 5035 y FJ(1)p 3053 5050 79 4 v 3053 5102 a(2)p Fl(\031)3156 5071 y FI(log)3289 4997 y Fn(\000)3342 5034 y Fl(c)3373 5043 y Fj(4)p 3341 5050 68 4 v 3341 5102 a FJ(2)p Fl(")3419 4997 y Fn(\001)3482 5071 y Fo(\000)21 b Fm(c)3613 5085 y FJ(2)3653 5071 y FI(.)46 b(W)-8 b(e)71 5184 y(observ)m(e)38 b(that)f Fm(c)644 5198 y FJ(5)721 5184 y FI(can)g(b)s(e)g(tak)m(en)h(arbitrarily)d (large,)k(pro)m(vided)c(that)j Fm(")g FI(is)e(su\016cien)m(tly)f (small,)j(and)e(that,)71 5297 y(with)29 b(this)g(de\014nition)f(of)j Fm(c)1002 5311 y FJ(5)1042 5297 y FI(,)f(w)m(e)h(alw)m(a)m(ys)g(ha)m(v) m(e)g(the)g(inequalit)m(y)e Fm(")c Fo(\024)2483 5259 y Fl(c)2514 5268 y Fj(4)p 2483 5276 66 4 v 2498 5328 a FJ(2)2558 5297 y FI(e)2598 5264 y Fp(\000)p FJ(2)p Fl(\031)r FJ(\()p Fl(c)2789 5273 y Fj(2)2825 5264 y FJ(+)p Fl(c)2911 5273 y Fj(5)2945 5264 y FJ(\))2976 5297 y FI(.)212 5410 y(Then,)j(the)g(previous)e(considerations)h(imply)f(that)i(the)h (function)h Fn(e)-55 b Fm(\021)2576 5424 y Fl(")2613 5410 y FI(\()p Fm(x)p FI(\))16 b Fo(\000)f Fm(x)26 b FI(=)f Fm(\030)t FI(\()p Fm(x)p FI(\))j(is)f(de\014ned)g(in)f Fo(A)3781 5424 y Fl(c)3812 5433 y Fj(5)71 5523 y FI(and)k(v)m (eri\014es,)g(in)f(this)g(domain,)g(the)i(b)s(ound)971 5762 y Fo(j)5 b Fn(e)-55 b Fm(\021)1042 5776 y Fl(")1078 5762 y FI(\()p Fm(x)p FI(\))21 b Fo(\000)f Fm(x)p Fo(j)26 b(\024)1726 5701 y FI(4)p Fm(B)5 b(c)1884 5715 y FJ(1)1923 5701 y Fm(")p 1521 5741 651 4 v 1521 5829 a(c)1560 5843 y FJ(4)1599 5829 y FI(e)1639 5802 y Fp(\000)p FJ(2)p Fl(\031)r FJ(\()p Fl(c)1830 5811 y Fj(2)1865 5802 y FJ(+)p Fl(c)1951 5811 y Fj(5)1986 5802 y FJ(\))2037 5829 y Fo(\000)20 b Fm(")2206 5762 y Fo(\024)2312 5701 y FI(8)p Fm(B)5 b(c)2470 5715 y FJ(1)2509 5701 y FI(e)2549 5668 y FJ(2)p Fl(\031)r FJ(\()p Fl(c)2685 5677 y Fj(2)2721 5668 y FJ(+)p Fl(c)2807 5677 y Fj(5)2841 5668 y FJ(\))p 2312 5741 561 4 v 2553 5825 a Fm(c)2592 5839 y FJ(4)2882 5762 y Fm(":)740 b FI(\(19\))p eop %%Page: 23 23 23 22 bop 71 274 a Fs(N.)31 b(F)-8 b(agella,)31 b(T.)f(M.)h(Seara)g (and)f(J.)g(Villan)m(uev)-5 b(a)2025 b FI(23)212 573 y(Using)20 b(the)i(c)m(hanges)g Fm(u)j FI(=)g(e)1146 540 y FJ(2)p Fl(\031)r FJ(i)p Fl(x)1309 573 y FI(and)20 b Fm(z)30 b FI(=)1655 537 y Fl(")p 1654 552 36 4 v 1654 604 a FJ(2)1699 573 y Fm(u)p FI(,)23 b(w)m(e)f(ha)m(v)m(e)g(that)g(if)e Fm(x)25 b Fo(2)g(A)2622 587 y Fl(c)2653 596 y Fj(5)2712 573 y FI(then)c Fm(u)k Fo(2)g Fm(A)p FI(\(e)3216 540 y Fp(\000)p FJ(2)p Fl(\031)r(c)3380 549 y Fj(5)3419 573 y Fm(;)15 b FI(e)3499 540 y FJ(2)p Fl(\031)r(c)3608 549 y Fj(5)3648 573 y FI(\))21 b(and)71 711 y Fm(z)30 b Fo(2)25 b Fm(A)312 637 y Fn(\000)365 675 y Fl(")p 364 690 V 364 742 a FJ(2)409 711 y FI(e)449 678 y Fp(\000)p FJ(2)p Fl(\031)r(c)613 687 y Fj(5)652 711 y Fm(;)704 675 y Fl(")p 702 690 V 702 742 a FJ(2)747 711 y FI(e)787 678 y FJ(2)p Fl(\031)r(c)896 687 y Fj(5)935 637 y Fn(\001)1008 711 y FI(\(see)31 b(\(4\))q(\).)42 b(No)m(w,)31 b(w)m(e)g(assume)g Fm(")f FI(small)g(enough)g(suc)m(h)g(that)3125 673 y FJ(8)p Fl(B)s(c)3247 682 y Fj(1)3282 673 y FJ(e)3313 650 y Fj(2)p Fk(\031)r Fj(\()p Fk(c)3433 665 y Fj(2)3468 650 y(+)p Fk(c)3543 665 y Fj(5)3577 650 y(\))p 3125 690 481 4 v 3333 742 a Fl(c)3364 751 y Fj(4)3615 711 y Fm(")c Fo(\024)g FI(1.)71 824 y(So,)k(equation)h(\(19\))h(implies)27 b Fo(j)t Fn(e)-55 b Fm(\021)1164 838 y Fl(")1201 824 y FI(\()p Fm(x)p FI(\))21 b Fo(\000)f Fm(x)p Fo(j)25 b(\024)g FI(1.)42 b(Then,)29 b(for)h(these)h(v)-5 b(alues)30 b(of)g Fm(z)t FI(,)h(inequalit)m(y)f(\(18\))i(leads)e(to)694 1070 y Fo(j)p Fm(')778 1084 y Fl(")815 1070 y FI(\()p Fm(z)t FI(\))21 b Fo(\000)f Fm(z)t Fo(j)26 b(\024)f(j)p Fm(z)t Fo(j)p FI(e)1372 1033 y FJ(2)p Fl(\031)r Fp(j)s Fd(e)-42 b Fl(\021)1505 1041 y Fk(")1540 1033 y FJ(\()p Fl(x)p FJ(\))p Fp(\000)p Fl(x)p Fp(j)1753 1070 y Fo(j)t Fn(e)-55 b Fm(\021)1823 1084 y Fl(")1860 1070 y FI(\()p Fm(x)p FI(\))21 b Fo(\000)f Fm(x)p Fo(j)25 b(\024)2302 1009 y FI(16)p Fm(\031)s(B)5 b(c)2560 1023 y FJ(1)2600 1009 y FI(e)2640 976 y FJ(2)p Fl(\031)r FJ(\()p Fl(c)2776 985 y Fj(2)2811 976 y FJ(+)p Fl(c)2897 985 y Fj(5)2931 976 y FJ(+1\))p 2302 1049 752 4 v 2638 1133 a Fm(c)2677 1147 y FJ(4)3063 1070 y Fm(")p Fo(j)p Fm(z)t Fo(j)p Fm(;)463 b FI(\(20\))212 1294 y(So,)47 b(the)c(\014rst)g(b)s(ound)e(in)h(\(15\)) j(follo)m(ws)d(from)h(\(20\))i(as)e(a)h(particular)e(case)i(when)e Fo(j)p Fm(z)t Fo(j)48 b FI(=)3485 1258 y Fl(")p 3483 1273 36 4 v 3483 1325 a FJ(2)3529 1294 y FI(,)e(taking)71 1432 y Fm(C)136 1446 y FJ(2)201 1432 y Fo(\025)306 1394 y FJ(8)p Fl(\031)r(B)s(c)471 1403 y Fj(1)506 1394 y FJ(e)537 1371 y Fj(2)p Fk(\031)r Fj(\()p Fk(c)657 1386 y Fj(2)693 1371 y(+)p Fk(c)768 1386 y Fj(5)802 1371 y(+1\))p 306 1411 601 4 v 574 1463 a Fl(c)605 1472 y Fj(4)917 1432 y FI(.)212 1558 y(Using)28 b(the)h(b)s(ound)e(of)i(the)g(function)f Fm(')1571 1572 y Fl(")1608 1558 y FI(\()p Fm(z)t FI(\))18 b Fo(\000)f Fm(z)33 b FI(in)28 b(the)h(ann)m(ulus)e Fm(A)2583 1484 y Fn(\000)2636 1522 y Fl(")p 2635 1537 36 4 v 2635 1589 a FJ(2)2680 1558 y FI(e)2720 1525 y Fp(\000)p FJ(2)p Fl(\031)r(c)2884 1534 y Fj(5)2923 1558 y Fm(;)2975 1522 y Fl(")p 2973 1537 V 2973 1589 a FJ(2)3019 1558 y FI(e)3059 1525 y FJ(2)p Fl(\031)r(c)3168 1534 y Fj(5)3207 1484 y Fn(\001)3248 1558 y FI(,)j(w)m(e)f(pro)s(ceed)g(to)71 1671 y(b)s(ound)23 b(its)i(deriv)-5 b(ativ)m(e)25 b(for)g Fo(j)p Fm(z)t Fo(j)h FI(=)1244 1635 y Fl(")p 1242 1650 V 1242 1702 a FJ(2)1288 1671 y FI(.)39 b(T)-8 b(o)26 b(this)e(end,)i(w)m(e)g(pic)m(k)f(a)h(particular)d(v)-5 b(alue)25 b(of)h Fm(z)t FI(,)h(and)e(w)m(e)h(consider)e(the)71 1784 y(disk)29 b(of)i(cen)m(ter)h Fm(z)j FI(and)30 b(radius)f Fm(r)g FI(=)1342 1748 y Fl(")p 1341 1763 V 1341 1815 a FJ(2)1386 1784 y FI(\(1)21 b Fo(\000)f FI(e)1618 1751 y Fp(\000)p FJ(2)p Fl(\031)r(c)1782 1760 y Fj(5)1821 1784 y FI(\),)32 b(whic)m(h)d(w)m(e)j(denote)f(b)m(y)g Ft(C)2804 1798 y Fl(r)2841 1784 y FI(\()p Fm(z)t FI(\).)43 b(One)31 b(can)g(easily)f(c)m(hec)m(k)71 1897 y(that)38 b(if)e(w)m(e)i(tak)m(e)h(a)f(p)s(oin)m(t)f Fm(t)g Fo(2)f Ft(C)1281 1911 y Fl(r)1319 1897 y FI(\()p Fm(z)t FI(\),)k(then)d Fm(t)h FI(b)s(elongs)e(to)i Fm(A)2319 1823 y Fn(\000)2372 1861 y Fl(")p 2371 1876 V 2371 1928 a FJ(2)2416 1897 y FI(e)2456 1864 y Fp(\000)p FJ(2)p Fl(\031)r(c)2620 1873 y Fj(5)2659 1897 y Fm(;)2711 1861 y Fl(")p 2709 1876 V 2709 1928 a FJ(2)2755 1897 y FI(e)2795 1864 y FJ(2)p Fl(\031)r(c)2904 1873 y Fj(5)2943 1823 y Fn(\001)2984 1897 y FI(,)i(and)d(then)g Fo(j)p Fm(')3531 1911 y Fl(")3568 1897 y FI(\()p Fm(t)p FI(\))26 b Fo(\000)e Fm(t)p Fo(j)71 2010 y FI(v)m(eri\014es)30 b(\(20\))i(with)d Fm(z)g Fo(\021)c Fm(t)p FI(.)212 2123 y(Then,)i(w)m(e)g(can)g(b)s(ound)d(the)j(deriv)-5 b(ativ)m(e)26 b(of)h Fm(')1766 2137 y Fl(")1803 2123 y FI(\()p Fm(z)t FI(\))13 b Fo(\000)g Fm(z)31 b FI(using)25 b(the)i(Cauc)m(h)m(y)g(in)m(tegral)f(form)m(ula.)39 b(W)-8 b(e)28 b(ha)m(v)m(e:)886 2354 y Fm(')945 2317 y Fp(0)945 2377 y Fl(")982 2354 y FI(\()p Fm(z)t FI(\))21 b Fo(\000)f FI(1)26 b(=)1403 2293 y Fm(d)p 1387 2334 81 4 v 1387 2417 a(dt)1477 2354 y FI(\()p Fm(')1571 2368 y Fl(")1608 2354 y FI(\()p Fm(t)p FI(\))21 b Fo(\000)f Fm(t)p FI(\))1891 2373 y Fp(j)p Fl(t)p FJ(=)p Fl(z)2056 2354 y FI(=)2202 2293 y(1)p 2162 2334 126 4 v 2162 2417 a(2)p Fm(\031)s FI(i)2313 2231 y Fn(Z)2363 2437 y Fi(C)2422 2445 y Fk(r)2457 2437 y FJ(\()p Fl(z)s FJ(\))2577 2293 y Fm(')2636 2307 y Fl(")2673 2293 y FI(\()p Fm(t)p FI(\))h Fo(\000)e Fm(t)p 2577 2334 344 4 v 2598 2417 a FI(\()p Fm(t)i Fo(\000)f Fm(z)t FI(\))2859 2391 y FJ(2)2930 2354 y Fm(dt:)71 2601 y FI(In)27 b(order)g(to)h(b)s(ound)e(the)i(in)m(tegral)f(ab)s(o)m(v)m (e,)j(w)m(e)e(ma)m(y)g(apply)e(inequalit)m(y)g(\(20\))j(to)g(an)m(y)f Fm(t)d Fo(2)g Ft(C)3244 2615 y Fl(r)3281 2601 y FI(\()p Fm(z)t FI(\),)k(obtaining:)480 2785 y Fn(\014)480 2840 y(\014)511 2863 y Fm(')570 2825 y Fp(0)570 2885 y Fl(")607 2863 y FI(\()p Fm(z)t FI(\))21 b Fo(\000)f FI(1)880 2785 y Fn(\014)880 2840 y(\014)994 2863 y Fo(\024)1157 2801 y FI(1)p 1157 2842 46 4 v 1158 2925 a Fm(r)1272 2863 y FI(max)1228 2928 y Fl(t)p Fp(2)p Fi(C)1359 2936 y Fk(r)1394 2928 y FJ(\()p Fl(z)s FJ(\))1500 2863 y Fo(j)p Fm(')1584 2877 y Fl(")1621 2863 y FI(\()p Fm(t)p FI(\))h Fo(\000)f Fm(t)p Fo(j)25 b(\024)2025 2801 y FI(1)p 2025 2842 V 2026 2925 a Fm(r)2139 2863 y FI(max)2096 2928 y Fl(t)p Fp(2)p Fi(C)2227 2936 y Fk(r)2262 2928 y FJ(\()p Fl(z)s FJ(\))2367 2707 y Fn(\()2450 2801 y FI(16)p Fm(\031)s(B)5 b(c)2708 2815 y FJ(1)2749 2801 y FI(e)2789 2768 y FJ(2)p Fl(\031)r FJ(\()p Fl(c)2925 2777 y Fj(2)2960 2768 y FJ(+)p Fl(c)3046 2777 y Fj(5)3080 2768 y FJ(+1\))p 2450 2842 752 4 v 2787 2925 a Fm(c)2826 2939 y FJ(4)3212 2863 y Fm(")15 b Fo(j)p Fm(t)p Fo(j)3352 2707 y Fn(\))994 3154 y Fo(\024)1157 3092 y FI(2)21 b Fo(\000)f FI(e)1354 3059 y Fp(\000)p FJ(2)p Fl(\031)r(c)1518 3068 y Fj(5)p 1157 3133 400 4 v 1157 3216 a FI(1)h Fo(\000)f FI(e)1354 3190 y Fp(\000)p FJ(2)p Fl(\031)r(c)1518 3199 y Fj(5)1577 3092 y FI(16)p Fm(\031)s(B)5 b(c)1835 3106 y FJ(1)1875 3092 y FI(e)1915 3059 y FJ(2)p Fl(\031)r FJ(\()p Fl(c)2051 3068 y Fj(2)2086 3059 y FJ(+)p Fl(c)2172 3068 y Fj(5)2206 3059 y FJ(+1\))p 1577 3133 752 4 v 1913 3216 a Fm(c)1952 3230 y FJ(4)2338 3154 y Fm(";)71 3408 y FI(whic)m(h)29 b(giv)m(es)i(the)f(second)h(part)f(of)g(\(15\))i(taking)f Fm(C)1842 3422 y FJ(2)1906 3408 y Fo(\025)2012 3372 y FJ(2)p Fp(\000)p FJ(e)2133 3348 y Ff(\000)p Fj(2)p Fk(\031)r(c)2277 3363 y Fj(5)p 2012 3387 305 4 v 2012 3444 a FJ(1)p Fp(\000)p FJ(e)2133 3422 y Ff(\000)p Fj(2)p Fk(\031)r(c)2277 3437 y Fj(5)2337 3370 y FJ(16)p Fl(\031)r(B)s(c)2537 3379 y Fj(1)2572 3370 y FJ(e)2603 3347 y Fj(2)p Fk(\031)r Fj(\()p Fk(c)2723 3362 y Fj(2)2758 3347 y(+)p Fk(c)2833 3362 y Fj(5)2867 3347 y(+1\))p 2337 3387 636 4 v 2622 3439 a Fl(c)2653 3448 y Fj(4)2982 3408 y FI(.)3780 3578 y Fe(\003)71 3819 y Fg(4.4)112 b(Pro)s(of)37 b(of)h(Theorem)f(A)71 3990 y FI(From)d(the)g(restatemen)m(t)h(of)f(the)g(problem)e(in)h (Section)g(4.1,)j(w)m(e)f(only)d(need)i(to)g(pro)m(v)m(e)h(Prop)s (osition)d(4.8)j(in)71 4103 y(order)30 b(to)h(pro)s(of)e(Theorem)i(A.) 71 4274 y Ft(Pro)s(of)38 b(of)f(Prop)s(osition)i(4.8:)45 b FI(As)32 b(w)m(e)h(kno)m(w)f(that)h Fm(h)2020 4288 y Fl(")2079 4274 y Fo(\016)22 b Fm( )2205 4288 y Fl(")2274 4274 y FI(is)31 b(holomorphic,)h(w)m(e)h(can)f(express)g Fm(b)p FI(\()p Fm(")p FI(\))23 b Fo(\000)e FI(1)71 4387 y(in)29 b(terms)h(of)h(the)f(Cauc)m(h)m(y)h(in)m(tegral)f(form)m(ula:) 641 4609 y Fm(b)p FI(\()p Fm(")p FI(\))21 b Fo(\000)f FI(1)84 b(=)1220 4547 y Fm(d)p 1196 4588 94 4 v 1196 4671 a(dz)1315 4609 y FI(\()p Fm(h)1402 4623 y Fl(")1460 4609 y Fo(\016)20 b Fm( )1584 4623 y Fl(")1621 4609 y FI(\()p Fm(z)t FI(\))i Fo(\000)d Fm(z)t FI(\))1931 4636 y Fp(j)p Fl(z)s FJ(=0)1033 4834 y FI(=)1237 4773 y(1)p 1196 4813 126 4 v 1196 4897 a(2)p Fm(\031)s FI(i)1347 4711 y Fn(Z)1398 4917 y Fi(C)1457 4925 y Fk(r)1520 4773 y Fm(h)1572 4787 y Fl(")1630 4773 y Fo(\016)h Fm( )1754 4787 y Fl(")1791 4773 y FI(\()p Fm(z)t FI(\))i Fo(\000)d Fm(z)p 1520 4813 546 4 v 1750 4897 a(z)1796 4870 y FJ(2)2076 4834 y Fm(dz)1033 5088 y FI(=)1237 5026 y(1)p 1196 5067 126 4 v 1196 5150 a(2)p Fm(\031)s FI(i)1347 4964 y Fn(Z)1398 5170 y Fi(C)1457 5178 y Fk(r)1520 5026 y Fm(h)1572 5040 y Fl(")1630 5026 y Fo(\016)h Fm( )1754 5040 y Fl(")1791 5026 y FI(\()p Fm(z)t FI(\))i Fo(\000)d Fm( )2078 5040 y Fl(")2115 5026 y FI(\()p Fm(z)t FI(\))p 1520 5067 712 4 v 1834 5150 a Fm(z)1880 5124 y FJ(2)2242 5088 y Fm(dz)25 b FI(+)2497 5026 y(1)p 2457 5067 126 4 v 2457 5150 a(2)p Fm(\031)s FI(i)2608 4964 y Fn(Z)2658 5170 y Fi(C)2717 5178 y Fk(r)2781 5026 y Fm( )2840 5040 y Fl(")2877 5026 y FI(\()p Fm(z)t FI(\))c Fo(\000)f Fm(z)p 2781 5067 371 4 v 2923 5150 a(z)2969 5124 y FJ(2)3161 5088 y Fm(dz)t(;)71 5339 y FI(where)30 b Ft(C)410 5353 y Fl(r)478 5339 y FI(is)f(an)m(y)i(circle)e(con)m(tained)i(in)p 1496 5265 121 4 v 29 w Fq(D)1567 5329 y Fk(")p 1572 5341 31 3 v 1572 5382 a Fj(2)1616 5339 y FI(.)41 b(F)-8 b(rom)31 b(no)m(w)f(on)g(w)m(e)h(tak)m(e)h Fm(r)c FI(=)2743 5303 y Fl(")p 2742 5318 36 4 v 2742 5370 a FJ(2)2787 5339 y FI(.)212 5452 y(W)-8 b(e)32 b(can)e(b)s(ound)f(the)h(second)h(in)m (tegral)f(b)m(y)g(using)f(inequalit)m(y)f(\(14\))k(of)f(Prop)s(osition) d(4.9,)k(obtaining:)915 5554 y Fn(\014)915 5609 y(\014)915 5663 y(\014)915 5718 y(\014)915 5772 y(\014)996 5652 y FI(1)p 955 5692 126 4 v 955 5776 a(2)p Fm(\031)s FI(i)1106 5590 y Fn(Z)1157 5796 y Fi(C)1227 5795 y Fk(")p 1225 5803 31 3 v 1225 5837 a Fj(2)1299 5652 y Fm( )1358 5666 y Fl(")1395 5652 y FI(\()p Fm(z)t FI(\))21 b Fo(\000)f Fm(z)p 1299 5692 371 4 v 1442 5776 a(z)1488 5749 y FJ(2)1680 5713 y Fm(dz)1773 5554 y Fn(\014)1773 5609 y(\014)1773 5663 y(\014)1773 5718 y(\014)1773 5772 y(\014)1829 5713 y Fo(\024)1935 5652 y FI(2)p 1935 5692 46 4 v 1937 5776 a Fm(")2035 5713 y FI(sup)2005 5792 y Fl(z)s Fp(2)p Fi(C)2158 5791 y Fk(")p 2157 5799 31 3 v 2157 5833 a Fj(2)2201 5713 y Fo(fj)p Fm( )2330 5727 y Fl(")2368 5713 y FI(\()p Fm(z)t FI(\))h Fo(\000)f Fm(z)t Fo(jg)26 b(\024)f Fm(C)2899 5727 y FJ(1)2939 5713 y Fm(":)p eop %%Page: 24 24 24 23 bop 71 274 a FI(24)1001 b Fs(Asymptotic)30 b(Size)g(of)h(Herman)f (Rings)f(of)i(the)f(Complex)g(Standard)f(F)-8 b(amily)71 573 y FI(F)g(or)31 b(the)g(\014rst)e(in)m(tegral,)i(w)m(e)f(use)g(that) h Fo(j)p Fm( )1506 587 y Fl(")1544 573 y FI(\()p Fm(z)t FI(\))p Fo(j)26 b FI(=)f Fo(j)p Fm(z)t Fo(j)31 b FI(if)e Fo(j)p Fm(z)t Fo(j)d(\024)2247 537 y Fl(")p 2245 552 36 4 v 2245 604 a FJ(2)2321 573 y FI(\(see)31 b(\(13\)\),)i(and)d(so,) 882 736 y Fn(\014)882 791 y(\014)882 845 y(\014)882 900 y(\014)882 954 y(\014)962 834 y FI(1)p 922 874 126 4 v 922 957 a(2)p Fm(\031)s FI(i)1073 771 y Fn(Z)1123 978 y Fi(C)1193 977 y Fk(")p 1192 985 31 3 v 1192 1019 a Fj(2)1266 834 y Fm(h)1318 848 y Fl(")1375 834 y Fo(\016)21 b Fm( )1500 848 y Fl(")1537 834 y FI(\()p Fm(z)t FI(\))g Fo(\000)f Fm( )1824 848 y Fl(")1861 834 y FI(\()p Fm(z)t FI(\))p 1266 874 712 4 v 1579 957 a Fm(z)1625 931 y FJ(2)1988 895 y Fm(dz)2081 736 y Fn(\014)2081 791 y(\014)2081 845 y(\014)2081 900 y(\014)2081 954 y(\014)2137 895 y Fo(\024)2243 834 y FI(2)p 2243 874 46 4 v 2245 957 a Fm(")2343 895 y FI(sup)2313 973 y Fl(z)s Fp(2)p Fi(C)2466 972 y Fk(")p 2465 980 31 3 v 2465 1015 a Fj(2)2509 895 y Fo(fj)p Fm(h)2631 909 y Fl(")2669 895 y FI(\()p Fm(z)t FI(\))h Fo(\000)f Fm(z)t Fo(jg)p Fm(:)71 1224 y FI(In)g(order)g(to)h(b)s(ound)e Fm(h)827 1238 y Fl(")864 1224 y FI(\()p Fm(z)t FI(\))q Fo(\000)q Fm(z)24 b FI(w)m(e)d(\014rst)f(recall)g(that)h Fm( )1900 1238 y Fl(")1937 1224 y FI(\()p Fm(z)t FI(\))h(is)d(\()p Fm(C)2256 1238 y FJ(1)2296 1224 y Fm(")p FI(\)-quasiconformal)h(\(see)i (Prop)s(osition)d(4.9\).)71 1337 y(Hence,)43 b(the)c(maximal)f (dilatation)g(of)i Fm(\026)1514 1351 y Fl(")1590 1337 y FI(is)e(b)s(ounded)f(b)m(y)j(the)f(same)h(constan)m(t,)j(that)d(is,)h Fo(k)p Fm(\026)3445 1351 y Fl(")3482 1337 y Fo(k)f(\024)g Fm(C)3743 1351 y FJ(1)3783 1337 y Fm(")p FI(.)71 1450 y(No)m(w,)32 b(w)m(e)g(w)m(an)m(t)g(to)g(apply)e(Prop)s(osition)g(B.)i (Ho)m(w)m(ev)m(er,)h(w)m(e)f(can)g(not)f(do)g(it)g(directly)f(b)s (ecause)i Fm(h)3435 1464 y Fl(")3503 1450 y FI(has)f(b)s(een)71 1563 y(constructed)e(in)f(suc)m(h)g(a)h(w)m(a)m(y)h(that)g Fm(h)1373 1577 y Fl(")1410 1563 y FI(\(0\))c(=)f(0)k(and)f Fm(h)1948 1577 y Fl(")1986 1563 y FI(\()p Fm(c)2060 1577 y Fp(\000)2119 1563 y FI(\()p Fm(")p FI(\)\))f(=)e Fo(\000)p FI(1,)k(where)g Fm(c)2860 1577 y Fp(\000)2919 1563 y FI(\()p Fm(")p FI(\))e(=)d(\()p Fo(\000)p FI(1)18 b Fo(\000)3410 1486 y(p)p 3486 1486 239 4 v 77 x FI(1)i Fo(\000)g Fm(")3684 1536 y FJ(2)3724 1563 y FI(\))p Fm(=)p FI(2)71 1676 y(is)33 b(the)g(critical)g(p)s(oin)m(t)f(of)i Fm(F)1037 1694 y Fl(\013)p FJ(\()p Fl(")p FJ(\))p Fl(;")1260 1676 y FI(giv)m(en)g(in)e(\(10\).)52 b(W)-8 b(e)35 b(can)f(arrange)g Fm(h)2561 1690 y Fl(")2632 1676 y FI(to)g(apply)e(Prop)s(osition)g(B)i (simply)71 1809 y(b)m(y)28 b(de\014ning)536 1785 y Fn(e)537 1809 y Fm(h)589 1823 y Fl(")626 1809 y FI(\()p Fm(z)t FI(\))f(=)e Fo(\000)p Fm(h)988 1823 y Fl(")1024 1809 y FI(\()p Fm(c)1098 1823 y Fp(\000)1158 1809 y FI(\()p Fm(")p FI(\))p Fm(z)t FI(\).)42 b(W)-8 b(e)30 b(observ)m(e)f(that)2089 1785 y Fn(e)2091 1809 y Fm(h)2143 1823 y Fl(")2208 1809 y FI(is)f(a)h(quasiconformal)f(map)g(of)h Fq(C)17 b FI(,)36 b(solving)27 b(the)71 1922 y(Beltrami)22 b(equation)h(\(7\))h(with)d (the)i(Beltrami)f(form)27 b Fn(e)-55 b Fm(\026)1925 1936 y Fl(")1961 1922 y FI(\()p Fm(z)t FI(\))26 b(=)f Fm(\026)2253 1938 y Fd(e)2254 1956 y Fl(h)2295 1964 y Fk(")2332 1922 y FI(\()p Fm(z)t FI(\))h(=)f Fm(\026)2625 1936 y Fl(")2662 1922 y FI(\()p Fm(c)2736 1936 y Fp(\000)2796 1922 y FI(\()p Fm(")p FI(\))p Fm(z)t FI(\))f(and)e(v)m(eri\014es)3477 1898 y Fn(e)3478 1922 y Fm(h)3530 1936 y Fl(")3568 1922 y FI(\(0\))k(=)f(0)71 2062 y(and)250 2038 y Fn(e)251 2062 y Fm(h)303 2076 y Fl(")341 2062 y FI(\(1\))32 b(=)g(1.)52 b(Then,)35 b(if)e(w)m(e)i(assume)f(that)g Fm(")h FI(is)e(small)g (enough)h(suc)m(h)g(that)h Fo(k)5 b Fn(e)-56 b Fm(\026)2965 2076 y Fl(")3001 2062 y Fo(k)33 b FI(=)e Fo(k)p Fm(\026)3281 2076 y Fl(")3318 2062 y Fo(k)h(\024)f Fm(C)3562 2076 y FJ(1)3602 2062 y Fm(")h Fo(\024)f Fm(\032)p FI(,)71 2175 y(w)m(e)f(can)h(apply)d(Prop)s(osition)h(B)h(to)35 b Fn(e)-55 b Fm(\026)1370 2189 y Fl(")1406 2175 y FI(,)31 b(obtaining)d(that)j(if)e Fo(j)p Fm(z)t Fo(j)d(\024)f Fm(\032)30 b FI(and)f Fo(k)5 b Fn(e)-56 b Fm(\026)2716 2189 y Fl(")2753 2175 y Fo(kj)15 b FI(log)j Fo(j)p Fm(z)t Fo(jj)26 b(\024)f Fm(C)3280 2189 y FJ(1)3320 2175 y Fm(")p Fo(j)15 b FI(log)i Fo(j)p Fm(z)t Fo(jj)26 b(\024)f Fm(\032)p FI(,)71 2288 y(then)992 2433 y Fo(j)1015 2409 y Fn(e)1017 2433 y Fm(h)1069 2447 y Fl(")1106 2433 y FI(\()p Fm(z)t FI(\))c Fo(\000)f Fm(z)t Fo(j)26 b(\024)f Fm(C)7 b Fo(k)e Fn(e)-56 b Fm(\026)1699 2447 y Fl(")1736 2433 y Fo(kj)p Fm(z)t Fo(jj)15 b FI(log)j Fo(j)p Fm(z)t Fo(jj)26 b(\024)f Fm(C)7 b(C)2431 2447 y FJ(1)2470 2433 y Fm(")p Fo(j)p Fm(z)t Fo(jj)15 b FI(log)j Fo(j)p Fm(z)t Fo(jj)p Fm(:)761 b FI(\(21\))71 2660 y(T)-8 b(aking)30 b(in)m(to)g(accoun)m(t)i(that) 1091 2555 y Fn(\014)1091 2610 y(\014)1091 2664 y(\014)1199 2624 y FJ(1)p 1131 2639 171 4 v 1131 2692 a Fl(c)1162 2701 y Ff(\000)1214 2692 y FJ(\()p Fl(")p FJ(\))1332 2660 y FI(+)20 b(1)1468 2555 y Fn(\014)1468 2610 y(\014)1468 2664 y(\014)1524 2660 y Fo(\024)25 b Fm(")1662 2627 y FJ(2)1701 2660 y FI(,)31 b(w)m(e)g(ha)m(v)m(e)g(that)g(if)f Fo(j)p Fm(z)t Fo(j)c FI(=)2610 2624 y Fl(")p 2609 2639 36 4 v 2609 2691 a FJ(2)2684 2660 y FI(with)j Fm(")i FI(small,)e(then)454 2851 y Fn(\014)454 2905 y(\014)454 2960 y(\014)454 3014 y(\014)577 2921 y Fm(z)p 494 2962 212 4 v 494 3045 a(c)533 3059 y Fp(\000)592 3045 y FI(\()p Fm(")p FI(\))715 2851 y Fn(\014)715 2905 y(\014)715 2960 y(\014)715 3014 y(\014)771 2983 y Fo(\024)878 2921 y Fm(")p 877 2962 46 4 v 877 3045 a FI(2)932 2983 y(\(1)21 b(+)f Fm(")1166 2945 y FJ(2)1206 2983 y FI(\))26 b Fo(\024)f Fm(\032;)197 b Fo(k)p Fm(\026)1732 2997 y Fl(")1769 2983 y Fo(k)1829 2851 y Fn(\014)1829 2905 y(\014)1829 2960 y(\014)1829 3014 y(\014)1860 2983 y FI(log)1992 2851 y Fn(\014)1992 2905 y(\014)1992 2960 y(\014)1992 3014 y(\014)2115 2921 y Fm(z)p 2033 2962 212 4 v 2033 3045 a(c)2072 3059 y Fp(\000)2131 3045 y FI(\()p Fm(")p FI(\))2254 2851 y Fn(\014)2254 2905 y(\014)2254 2960 y(\014)2254 3014 y(\014)2284 2851 y(\014)2284 2905 y(\014)2284 2960 y(\014)2284 3014 y(\014)2340 2983 y Fo(\024)25 b Fm(C)2501 2997 y FJ(1)2540 2983 y Fm(")2597 2878 y Fn(\014)2597 2933 y(\014)2597 2987 y(\014)2628 2983 y FI(log)2761 2882 y Fn(\020)2826 2921 y Fm(")p 2825 2962 46 4 v 2825 3045 a FI(2)2880 2983 y(\(1)c(+)f Fm(")3114 2945 y FJ(2)3154 2983 y FI(\))3189 2882 y Fn(\021)3244 2878 y(\014)3244 2933 y(\014)3244 2987 y(\014)3299 2983 y Fo(\024)25 b Fm(\032:)71 3269 y FI(So,)30 b(w)m(e)h(can)g(apply)e(form)m(ula)g (\(21\))j(to)1487 3233 y Fl(z)p 1419 3248 171 4 v 1419 3301 a(c)1450 3310 y Ff(\000)1502 3301 y FJ(\()p Fl(")p FJ(\))1600 3269 y FI(,)e(obtaining:)451 3582 y Fo(j)p Fm(h)528 3596 y Fl(")566 3582 y FI(\()p Fm(z)t FI(\))21 b Fo(\000)f Fm(z)t Fo(j)83 b FI(=)1102 3450 y Fn(\014)1102 3504 y(\014)1102 3559 y(\014)1102 3613 y(\014)1131 3558 y(e)1132 3582 y Fm(h)1184 3596 y Fl(")1237 3453 y Fn(\022)1396 3520 y Fm(z)p 1313 3561 212 4 v 1313 3644 a(c)1352 3658 y Fp(\000)1412 3644 y FI(\()p Fm(")p FI(\))1535 3453 y Fn(\023)1622 3582 y FI(+)20 b Fm(z)1759 3450 y Fn(\014)1759 3504 y(\014)1759 3559 y(\014)1759 3613 y(\014)1815 3582 y Fo(\024)1911 3450 y Fn(\014)1911 3504 y(\014)1911 3559 y(\014)1911 3613 y(\014)1939 3558 y(e)1941 3582 y Fm(h)1993 3596 y Fl(")2045 3453 y Fn(\022)2205 3520 y Fm(z)p 2122 3561 V 2122 3644 a(c)2161 3658 y Fp(\000)2220 3644 y FI(\()p Fm(")p FI(\))2344 3453 y Fn(\023)2431 3582 y Fo(\000)2614 3520 y Fm(z)p 2532 3561 V 2532 3644 a(c)2571 3658 y Fp(\000)2630 3644 y FI(\()p Fm(")p FI(\))2753 3450 y Fn(\014)2753 3504 y(\014)2753 3559 y(\014)2753 3613 y(\014)2803 3582 y FI(+)g Fo(j)p Fm(z)t Fo(j)3005 3450 y Fn(\014)3005 3504 y(\014)3005 3559 y(\014)3005 3613 y(\014)3037 3582 y FI(1)g(+)3286 3520 y(1)p 3203 3561 V 3203 3644 a Fm(c)3242 3658 y Fp(\000)3302 3644 y FI(\()p Fm(")p FI(\))3425 3450 y Fn(\014)3425 3504 y(\014)3425 3559 y(\014)3425 3613 y(\014)948 3833 y Fo(\024)83 b Fm(C)7 b(C)1239 3847 y FJ(1)1278 3833 y Fm(")1335 3701 y Fn(\014)1335 3756 y(\014)1335 3810 y(\014)1335 3865 y(\014)1458 3772 y Fm(z)p 1376 3812 V 1376 3895 a(c)1415 3909 y Fp(\000)1474 3895 y FI(\()p Fm(")p FI(\))1597 3701 y Fn(\014)1597 3756 y(\014)1597 3810 y(\014)1597 3865 y(\014)1643 3833 y FI(log)1775 3705 y Fn(\022)1842 3701 y(\014)1842 3756 y(\014)1842 3810 y(\014)1842 3865 y(\014)1965 3772 y Fm(z)p 1882 3812 V 1882 3895 a(c)1921 3909 y Fp(\000)1981 3895 y FI(\()p Fm(")p FI(\))2104 3701 y Fn(\014)2104 3756 y(\014)2104 3810 y(\014)2104 3865 y(\014)2134 3705 y(\023)2221 3833 y FI(+)20 b Fo(j)p Fm(z)t Fo(j)p Fm(")2450 3795 y FJ(2)2516 3833 y Fo(\024)25 b Fm(C)2684 3795 y Fp(0)2707 3833 y Fm(")2749 3795 y FJ(2)2788 3833 y Fo(j)15 b FI(log)i Fm(")p Fo(j)p Fm(;)71 4121 y FI(with)29 b Fm(C)350 4088 y Fp(0)403 4121 y FI(dep)s(ending)f (on)i Fm(C)37 b FI(and)30 b Fm(C)1310 4135 y FJ(1)1349 4121 y FI(.)41 b(As)30 b(a)h(consequence)g(of)g(this,)e(w)m(e)i(can)g (b)s(ound:)1061 4270 y Fn(\014)1061 4325 y(\014)1061 4380 y(\014)1061 4434 y(\014)1061 4489 y(\014)1141 4368 y FI(1)p 1101 4409 126 4 v 1101 4492 a(2)p Fm(\031)s FI(i)1252 4306 y Fn(Z)1302 4512 y Fi(C)1372 4511 y Fk(")p 1371 4519 31 3 v 1371 4553 a Fj(2)1445 4368 y Fm(h)1497 4382 y Fl(")1554 4368 y Fo(\016)21 b Fm( )1679 4382 y Fl(")1716 4368 y FI(\()p Fm(z)t FI(\))g Fo(\000)f Fm( )2003 4382 y Fl(")2040 4368 y FI(\()p Fm(z)t FI(\))p 1445 4409 712 4 v 1758 4492 a Fm(z)1804 4466 y FJ(2)2167 4430 y Fm(dz)2260 4270 y Fn(\014)2260 4325 y(\014)2260 4380 y(\014)2260 4434 y(\014)2260 4489 y(\014)2316 4430 y Fo(\024)25 b FI(2)p Fm(C)2529 4392 y Fp(0)2552 4430 y Fm(")p Fo(j)15 b FI(log)j Fm(")p Fo(j)p Fm(;)71 4753 y FI(obtaining)29 b Fm(b)p FI(\()p Fm(")p FI(\))d(=)f(1)c(+)f Fo(O)s FI(\()p Fm(")p Fo(j)15 b FI(log)i Fm(")p Fo(j)p FI(\),)32 b(and)e(therefore)h(ending)d(the)j(pro)s(of)f(of)g(Prop)s (osition)f(4.8.)3780 4995 y Fe(\003)71 5225 y FI(This)f(concludes)i (the)h(pro)s(of)e(of)i(Theorem)f(A,)h(up)e(to)i(pro)m(ving)f(Prop)s (osition)e(B.)71 5573 y Fu(5)135 b(Pro)t(of)45 b(of)g(Prop)t(osition)g (B)71 5796 y FI(Our)29 b(goal)i(in)e(this)g(section)i(is)e(to)i(pro)m (v)m(e)h(Prop)s(osition)c(B.)p eop %%Page: 25 25 25 24 bop 71 274 a Fs(N.)31 b(F)-8 b(agella,)31 b(T.)f(M.)h(Seara)g (and)f(J.)g(Villan)m(uev)-5 b(a)2025 b FI(25)71 573 y Fg(5.1)112 b(Pro)s(of)37 b(of)h(\(a\))71 752 y FI(Let)31 b(us)e(consider)h(the)g(follo)m(wing)f(one-parameter)j(family)c(of)j (Beltrami)f(forms:)1741 982 y Fm(\026)1796 996 y Fl(t)1851 982 y FI(=)25 b Fm(t)2040 921 y(\026)p 1990 961 156 4 v 1990 1045 a Fo(jj)p Fm(\026)p Fo(jj)2155 982 y Fm(;)71 1247 y FI(where)35 b Fm(t)f Fo(2)f Fq(D)17 b FI(.)63 b(As)35 b Fo(jj)p Fm(\026)892 1261 y Fl(t)922 1247 y Fo(jj)g FI(=)e Fo(j)p Fm(t)p Fo(j)i Fm(<)e FI(1)k(for)e(all)f Fm(t)g Fo(2)g Fq(D)17 b FI(,)43 b(w)m(e)36 b(can)g(apply)e(Theorem)h (2.12)j(and)d(obtain)g(a)h(one-)71 1360 y(parameter)g(family)f(of)h(in) m(tegrating)g(maps)f Fm(h)1667 1374 y Fl(t)1732 1360 y FI(:)g Fq(C)58 b Fo(!)34 b Fq(C)60 b FI(\014xing)34 b(0)j(and)e(1,)j(and)e(suc)m(h)f(that)3352 1324 y Fl(@)t(h)3434 1332 y Fk(t)p 3352 1339 110 4 v 3368 1392 a Fl(@)p 3409 1354 36 3 v 4 w(z)3506 1360 y FI(=)f Fm(\026)3666 1374 y Fl(t)3706 1324 y(@)t(h)3788 1332 y Fk(t)p 3706 1339 110 4 v 3722 1392 a Fl(@)t(z)3825 1360 y FI(.)71 1473 y(Moreo)m(v)m(er,)45 b Fm(h)560 1487 y Fl(t)590 1473 y FI(\()p Fm(z)t FI(\))c(dep)s(ends)d(analytically)h(on)h Fm(t)p FI(.)70 b(Observ)m(e)41 b(that,)i(as)e Fm(\026)2643 1487 y Fl(t)2714 1473 y FI(=)g(0)g(if)e Fm(t)i FI(=)h(0)e(and)g Fm(\026)3519 1487 y Fl(t)3590 1473 y FI(=)h Fm(\026)f FI(if)71 1586 y Fm(t)25 b FI(=)g Fo(k)p Fm(\026)p Fo(k)p FI(,)31 b(w)m(e)g(ha)m(v)m(e)h(that)f Fm(h)1019 1600 y FJ(0)1084 1586 y FI(=)25 b(Id)k(and)h Fm(h)1522 1605 y Fp(jj)p Fl(\026)p Fp(jj)1673 1586 y FI(=)25 b Fm(h)p FI(.)212 1703 y(No)m(w,)31 b(let)g Fm(z)e Fo(2)c Fq(C)44 b Fo(n)21 b(f)p FI(0)p Fm(;)15 b FI(1)p Fo(g)32 b FI(\014xed,)e(and)g (consider)f(the)i(holomorphic)d(map)1476 1917 y Fm(f)1521 1931 y Fl(z)1585 1917 y FI(:)84 b Fq(D)105 b Fo(\000)-15 b(!)83 b Fq(C)44 b Fo(n)20 b(f)p FI(0)p Fm(;)15 b FI(1)p Fo(g)1710 2030 y Fm(t)99 b Fo(7\000)-15 b(!)157 b Fm(h)2198 2044 y Fl(t)2229 2030 y FI(\()p Fm(z)t FI(\))p Fm(:)71 2243 y FI(Since)25 b(b)s(oth)h Fq(D)49 b FI(and)26 b Fq(C)j Fo(n)13 b(f)p FI(0)q Fm(;)i FI(1)q Fo(g)33 b FI(are)26 b(h)m(yp)s(erb)s(olic)e(sets,)k(w)m(e)f(conclude)e(from)h(Big)h(Sc)m(h) m(w)m(artz-Pic)m(k)g(lemma)f(\(see)71 2356 y(Theorem)k(2.23\))i(that)f Fm(f)924 2370 y Fl(z)994 2356 y FI(is)e(a)i(con)m(traction)g(in)f(the)g (P)m(oincar)m(\023)-43 b(e)32 b(metrics,)e(that)h(is)1293 2573 y Fm(\016)1333 2591 y Fh(C)9 b Fp(nf)p FJ(0)q Fl(;)p FJ(1)p Fp(g)1581 2573 y FI(\()p Fm(h)1668 2587 y Fl(t)1693 2596 y Fj(1)1733 2573 y FI(\()p Fm(z)t FI(\))p Fm(;)15 b(h)1941 2587 y Fl(t)1966 2596 y Fj(2)2007 2573 y FI(\()p Fm(z)t FI(\)\))27 b Fo(\024)e Fm(\016)2321 2587 y Fh(D)2373 2573 y FI(\()p Fm(t)2441 2587 y FJ(1)2481 2573 y Fm(;)15 b(t)2554 2587 y FJ(2)2593 2573 y FI(\))71 2789 y(for)30 b(all)f Fm(t)369 2803 y FJ(1)409 2789 y Fm(;)15 b(t)482 2803 y FJ(2)546 2789 y Fo(2)25 b Fq(D)17 b FI(.)47 b(If)30 b(w)m(e)g(tak)m(e)i Fm(t)1218 2803 y FJ(1)1283 2789 y FI(=)25 b(0)31 b(and)e Fm(t)1664 2803 y FJ(2)1729 2789 y FI(=)c Fo(jj)p Fm(\026)p Fo(jj)p FI(,)31 b(the)g(statemen)m(t)h (follo)m(ws.)71 3055 y Fg(5.2)112 b(Pro)s(of)37 b(of)h(\(b\))71 3235 y FI(The)25 b(second)g(part)h(of)f(Prop)s(osition)f(B)i(will)d(b)s (e)h(deduced)h(from)g(the)h(\014rst)f(one)g(b)m(y)h(comparing)f(the)g (Euclidean)71 3348 y(distance,)39 b(b)s(et)m(w)m(een)e(p)s(oin)m(ts)f (close)i(to)f(0,)j(with)35 b(the)j(h)m(yp)s(erb)s(olic)c(distance)j(in) f Fq(C)48 b Fo(n)25 b(f)p FI(0)p Fm(;)15 b FI(1)p Fo(g)p FI(,)41 b(and)c(using)e(the)71 3461 y(explicit)30 b(form)m(ula)i(for)g Fm(\016)910 3475 y Fh(D)962 3461 y FI(\(0)p Fm(;)15 b Fo(k)p Fm(\026)p Fo(k)p FI(\))35 b(giv)m(en)d(in)f(Prop)s(osition)f (2.18.)48 b(W)-8 b(e)33 b(notice)f(that)h(it)f(is)f(not)i(easy)f(to)h (w)m(ork)71 3574 y(directly)25 b(with)g Fm(\016)641 3592 y Fh(C)9 b Fp(nf)q FJ(0)p Fl(;)p FJ(1)p Fp(g)889 3574 y FI(,)28 b(since)d(there)i(is)e(no)h(explicit)f(form)m(ula)g(for)h (this)f(h)m(yp)s(erb)s(olic)f(distance.)39 b(Ho)m(w)m(ev)m(er,)29 b(as)71 3687 y(stated)g(in)d(the)i(follo)m(wing)e(prop)s(osition,)h Fm(\016)1522 3705 y Fh(C)9 b Fp(nf)q FJ(0)p Fl(;)p FJ(1)p Fp(g)1798 3687 y FI(is)27 b(comparable)g({close)i(to)g(the)f(origin{)e (to)j Fm(\016)3342 3701 y Fh(D)3384 3682 y Ff(\003)3430 3687 y FI(,)g(for)f(whic)m(h)71 3799 y(w)m(e)j(ha)m(v)m(e)g(an)g (explicit)d(expression)h(\(see)j(Prop)s(osition)c(2.22\).)71 4003 y Ft(Prop)s(osition)36 b(5.1.)42 b Fr(Ther)-5 b(e)34 b(exist)e(c)-5 b(onstants)35 b FI(0)26 b Fm(<)f(c)g(<)g FI(1)p Fm(=)p FI(2)p Fr(,)34 b Fm(M)h(>)25 b FI(0)33 b Fr(and)h Fm(\033)28 b(>)d FI(0)33 b Fr(such)g(that:)241 4120 y FI(\(a\))47 b Fr(F)-7 b(or)34 b(al)5 b(l)33 b Fm(z)745 4134 y FJ(1)784 4120 y Fm(;)15 b(z)866 4134 y FJ(2)932 4120 y Fo(2)25 b Fq(D)1077 4087 y Fp(\003)1077 4143 y Fl(c)1148 4120 y FI(=)g Fq(D)1304 4134 y Fl(c)1364 4120 y Fo(n)c(f)p FI(0)p Fo(g)1496 4384 y FI(1)k Fo(\024)1752 4323 y Fm(\016)1792 4337 y Fh(D)1835 4318 y Ff(\003)1881 4323 y FI(\()p Fm(z)1958 4337 y FJ(1)1998 4323 y Fm(;)15 b(z)2080 4337 y FJ(2)2120 4323 y FI(\))p 1672 4363 564 4 v 1672 4447 a Fm(\016)1712 4465 y Fh(C)10 b Fp(nf)p FJ(0)q Fl(;)p FJ(1)p Fp(g)1961 4447 y FI(\()p Fm(z)2038 4461 y FJ(1)2078 4447 y Fm(;)15 b(z)2160 4461 y FJ(2)2200 4447 y FI(\))2271 4384 y Fo(\024)25 b FI(1)20 b(+)g Fm(M)5 b(:)241 4707 y FI(\(b\))46 b Fr(If)32 b Fm(z)545 4721 y FJ(1)585 4707 y Fm(;)15 b(z)667 4721 y FJ(2)732 4707 y Fo(2)25 b Fq(D)878 4674 y Fp(\003)923 4707 y Fr(,)33 b(with)g Fo(j)p Fm(z)1248 4721 y FJ(1)1288 4707 y Fo(j)26 b(\024)f Fm(c=)p FI(2)34 b Fr(and)f Fm(\016)1814 4726 y Fh(C)9 b Fp(n)q(f)p FJ(0)p Fl(;)p FJ(1)p Fp(g)2063 4707 y FI(\()p Fm(z)2140 4721 y FJ(1)2180 4707 y Fm(;)15 b(z)2262 4721 y FJ(2)2302 4707 y FI(\))25 b Fo(\024)g Fm(\033)s Fr(,)33 b(then)g Fo(j)p Fm(z)2843 4721 y FJ(2)2883 4707 y Fo(j)26 b(\024)f Fm(c)p Fr(.)212 4911 y FI(The)30 b(comparison)f(b)s(et)m(w)m(een)i(the)g(Euclidean)d(distance)j(and)e Fm(\016)2378 4925 y Fh(D)2421 4907 y Ff(\003)2497 4911 y FI(is)g(giv)m(en)i(b)m(y)f(the)h(follo)m(wing)e(lemma.)71 5115 y Ft(Lemma)k(5.2.)42 b Fr(L)-5 b(et)33 b Fm(z)845 5129 y FJ(1)884 5115 y Fm(;)15 b(z)966 5129 y FJ(2)1032 5115 y Fo(2)25 b Fq(D)1177 5082 y Fp(\003)1255 5115 y Fr(satisfying)776 5331 y Fo(j)15 b FI(log)i Fo(j)p Fm(z)1016 5345 y FJ(1)1056 5331 y Fo(jj)p Fm(\016)1146 5345 y Fh(D)1189 5327 y Ff(\003)1235 5331 y FI(\()p Fm(z)1312 5345 y FJ(1)1352 5331 y Fm(;)e(z)1434 5345 y FJ(2)1474 5331 y FI(\))26 b Fo(\024)f FI(1)p Fm(=)1721 5252 y Fo(p)p 1797 5252 46 4 v 79 x FI(2)q Fm(;)201 b(\016)2109 5345 y Fh(D)2152 5327 y Ff(\003)2198 5331 y FI(\()p Fm(z)2275 5345 y FJ(1)2315 5331 y Fm(;)15 b(z)2397 5345 y FJ(2)2437 5331 y FI(\))25 b Fo(\024)g FI(2)15 b(log)r(\(1)21 b(+)2963 5252 y Fo(p)p 3039 5252 V 79 x FI(2)q(\))p Fm(;)71 5548 y Fr(then:)1143 5712 y Fo(j)p Fm(z)1210 5726 y FJ(1)1270 5712 y Fo(\000)f Fm(z)1403 5726 y FJ(2)1443 5712 y Fo(j)25 b(\024)1678 5575 y(p)p 1753 5575 V 1753 5650 a FI(2)p 1599 5691 278 4 v 1599 5709 a Fo(p)p 1675 5709 46 4 v 75 x FI(2)c Fo(\000)f FI(1)1887 5712 y Fo(j)p Fm(z)1954 5726 y FJ(1)1994 5712 y Fo(jj)15 b FI(log)i Fo(j)p Fm(z)2259 5726 y FJ(1)2299 5712 y Fo(jj)p Fm(\016)2389 5726 y Fh(D)2432 5707 y Ff(\003)2478 5712 y FI(\()p Fm(z)2555 5726 y FJ(1)2595 5712 y Fm(;)e(z)2677 5726 y FJ(2)2717 5712 y FI(\))p Fm(:)p eop %%Page: 26 26 26 25 bop 71 274 a FI(26)1001 b Fs(Asymptotic)30 b(Size)g(of)h(Herman)f (Rings)f(of)i(the)f(Complex)g(Standard)f(F)-8 b(amily)71 573 y Ft(Remark)42 b(5.3.)k FI(W)-8 b(e)38 b(p)s(oin)m(t)e(out)i(that)g (part)f Ft(\(a\))f FI(of)h(Prop)s(osition)e(B)j Fr(only)46 b FI(pro)m(vides)36 b(information)g(ab)s(out)71 686 y Fm(\016)111 704 y Fh(C)9 b Fp(n)q(f)p FJ(0)p Fl(;)p FJ(1)p Fp(g)359 686 y FI(\()p Fm(z)t(;)15 b(h)p FI(\()p Fm(z)t FI(\)\),)40 b(and)35 b(th)m(us)g(on)g Fm(\016)1306 700 y Fh(D)1349 681 y Ff(\003)1395 686 y FI(\()p Fm(z)t(;)15 b(h)p FI(\()p Fm(z)t FI(\)\))38 b(\(after)e(using)e(Prop)s(osition)g (5.1\).)57 b(This)34 b(is)g(the)i(reason)g(for)71 799 y(whic)m(h)31 b(the)i(h)m(yp)s(otheses)f(of)g(Lemma)h(5.2)g(are)g(form) m(ulated)f(in)f(terms)h(of)h(the)g(h)m(yp)s(erb)s(olic)c(distance.)47 b(If,)33 b(for)71 912 y(instance,)c Fr(a-priori)38 b FI(estimates)29 b(on)g Fo(j)p Fm(z)1380 926 y FJ(1)1436 912 y Fo(\000)16 b Fm(z)1565 926 y FJ(2)1604 912 y Fo(j)p Fm(=)p Fo(j)p Fm(z)1741 926 y FJ(1)1782 912 y Fo(j)28 b FI(w)m(ere)h(kno)m(wn,)g(then)f(the)g(statemen)m(t)j(\(and)d(the)g (pro)s(of)7 b(\))28 b(of)71 1024 y(the)i(lemma)g(could)g(b)s(e)g (simpli\014ed.)212 1200 y(The)j(pro)s(of)f(of)h(Prop)s(osition)e(5.1)j (and)f(Lemma)g(5.2)h(are)f(p)s(ostp)s(oned)f(to)i(the)f(follo)m(wing)e (section.)49 b(No)m(w,)71 1313 y(w)m(e)31 b(pro)m(v)m(e)g(the)f(second) h(part)f(of)h(Prop)s(osition)d(B.)212 1426 y(The)i(k)m(ey)h(p)s(oin)m (t)e(to)i(pro)m(v)m(e)h(\()p Ft(b)p FI(\))f(is)e(the)i(estimate)g(pro)m (vided)e(b)m(y)h(\()p Ft(a)p FI(\))q(:)1382 1616 y Fm(\016)1422 1635 y Fh(C)10 b Fp(nf)p FJ(0)q Fl(;)p FJ(1)p Fp(g)1671 1616 y FI(\()p Fm(z)t(;)15 b(h)p FI(\()p Fm(z)t FI(\)\))28 b Fo(\024)d Fm(\016)2159 1630 y Fh(D)2211 1616 y FI(\(0)p Fm(;)15 b Fo(k)p Fm(\026)p Fo(k)p FI(\))p Fm(:)1153 b FI(\(22\))71 1841 y(Using)19 b(this)g(inequalit)m(y)-8 b(,)22 b(w)m(e)f(will)c(c)m(hec)m(k)22 b(that)f(the)f(desired)f(result) g(follo)m(ws)h(b)m(y)g(taking)g Fm(\032)25 b 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2677 y Fo(jj)p Fm(z)1897 2691 y FJ(1)1937 2677 y Fo(j)21 b(\000)f(j)p Fm(z)2141 2691 y FJ(2)2181 2677 y Fo(jj)p Fm(;)197 b(\030)30 b Fo(2)24 b(hj)p Fm(z)2710 2691 y FJ(1)2751 2677 y Fo(j)p Fm(;)15 b Fo(j)p Fm(z)2883 2691 y FJ(2)2923 2677 y Fo(ji)p Fm(;)71 2925 y FI(and)30 b(the)g(b)s(ound)1331 3076 y(1)20 b Fo(\000)g FI(cos)c Fm(x)25 b Fo(\024)1807 3015 y Fm(x)1859 2982 y FJ(2)p 1807 3056 92 4 v 1830 3139 a FI(2)1908 3076 y Fm(;)198 b(x)25 b Fo(2)g FI([)p Fo(\000)p Fm(\031)s(;)15 b(\031)s FI(])p Fm(:)3780 3292 y Fe(\003)71 3487 y FI(The)25 b(third)e(part)i(of)h(Lemma)f(5.5)i(is)d(v)m(ery)i(close)f(to)h(what)g (it)e(is)h(stated)h(in)e(Lemma)h(5.2,)j(but)d(it)g(is)f(not)i(exactly) 71 3600 y(what)h(w)m(e)h(need:)39 b(w)m(e)28 b(w)m(an)m(t)g(a)g(b)s (ound)d(for)j Fo(j)p Fm(z)1576 3614 y FJ(1)1630 3600 y Fo(\000)14 b Fm(z)1757 3614 y FJ(2)1797 3600 y Fo(j)27 b FI(dep)s(ending)e(just)i(on)h Fm(z)2626 3614 y FJ(1)2693 3600 y FI(and)e Fm(\016)2906 3614 y Fh(D)2949 3596 y Ff(\003)2995 3600 y FI(\()p Fm(z)3072 3614 y FJ(1)3112 3600 y Fm(;)15 b(z)3194 3614 y FJ(2)3234 3600 y FI(\).)40 b(This)26 b(requires)71 3713 y(the)k(follo)m(wing)f Fr(a-priori)41 b FI(estimate.)71 3908 y Ft(Corollary)36 b(5.6.)43 b Fr(Given)32 b Fm(z)1049 3922 y FJ(1)1089 3908 y Fm(;)15 b(z)1171 3922 y FJ(2)1236 3908 y Fo(2)25 b Fq(D)1382 3875 y Fp(\003)1460 3908 y Fr(such)32 b(that)i Fm(\016)1892 3922 y Fh(D)1935 3903 y Ff(\003)1981 3908 y FI(\()p Fm(z)2058 3922 y FJ(1)2098 3908 y Fm(;)15 b(z)2180 3922 y FJ(2)2220 3908 y FI(\))25 b Fo(\024)g FI(2)15 b(log)r(\(1)21 b(+)2746 3833 y Fo(p)p 2822 3833 46 4 v 75 x FI(2)q(\))p Fr(,)32 b(we)h(have:)1408 4180 y Fo(j)15 b FI(log)i Fo(j)p Fm(z)1648 4194 y FJ(2)1688 4180 y Fo(jj)26 b(\024)1870 4118 y FI(3)20 b(+)2026 4043 y Fo(p)p 2102 4043 V 75 x FI(5)p 1870 4159 278 4 v 1986 4242 a(2)2157 4180 y Fo(j)15 b FI(log)i Fo(j)p Fm(z)2397 4194 y FJ(1)2437 4180 y Fo(jj)p Fm(:)71 4429 y Ft(Pro)s(of)40 b(:)48 b FI(If)34 b(w)m(e)g(set)h Fm(x)d FI(=)f Fo(\000)15 b FI(log)h Fo(j)p Fm(z)1290 4443 y FJ(2)1330 4429 y Fo(j)32 b Fm(>)g FI(0)i(and)g Fm(d)e FI(=)f Fm(\016)1971 4443 y Fh(D)2013 4424 y Ff(\003)2059 4429 y FI(\()p Fm(z)2136 4443 y FJ(1)2176 4429 y Fm(;)15 b(z)2258 4443 y FJ(2)2299 4429 y FI(\),)35 b(then)f(from)g(part)g(\()p Fm(i)p FI(\))h(of)f(Lemma)h(5.5)g(w)m(e)71 4542 y(ha)m(v)m(e)c(that) 1078 4660 y Fm(x)1130 4623 y FJ(2)1190 4660 y FI(+)20 b(log)d Fo(j)p Fm(z)1481 4674 y FJ(1)1521 4660 y Fo(j)p FI(\(2)k(+)f(sinh)1900 4623 y FJ(2)1939 4660 y FI(\()p Fm(d=)p FI(2\)\))p Fm(x)j FI(+)d(log)2464 4623 y FJ(2)2519 4660 y Fo(j)p Fm(z)2586 4674 y FJ(1)2626 4660 y Fo(j)25 b(\024)g FI(0)p Fm(:)847 b FI(\(24\))71 4833 y(Th)m(us,)38 b Fm(x)387 4847 y Fp(\000)483 4833 y Fo(\024)f(\000)15 b FI(log)h Fo(j)p Fm(z)876 4847 y FJ(2)916 4833 y Fo(j)37 b(\024)g Fm(x)1138 4847 y FJ(+)1197 4833 y FI(,)i(where)e Fm(x)1583 4847 y Fp(\006)1679 4833 y FI(are,)j(resp)s(ectiv)m(ely)-8 b(,)39 b(the)f(t)m(w)m(o)g(\(p)s(ositiv)m(e\))f(zeros)h(for)f Fm(x)h FI(of)f(the)71 4946 y(left-hand)28 b(side)g(of)i(equation)f (\(24\).)42 b(This)27 b(pro)m(vides)h(lo)m(w)m(er)i(and)e(upp)s(er)f(b) s(ounds)g(for)i Fo(\000)15 b FI(log)i Fo(j)p Fm(z)3311 4960 y FJ(2)3351 4946 y Fo(j)p FI(,)30 b(but)e(w)m(e)i(are)71 5059 y(only)f(in)m(terested)i(in)e(the)h(upp)s(er)f(one:)354 5326 y Fm(x)406 5340 y FJ(+)490 5326 y FI(=)596 5264 y Fo(j)15 b FI(log)i Fo(j)p Fm(z)836 5278 y FJ(1)876 5264 y Fo(jj)p 596 5305 331 4 v 739 5388 a FI(2)952 5198 y Fn(\022)1019 5326 y FI(2)j(+)g(sinh)1337 5288 y FJ(2)1377 5326 y FI(\()p Fm(d=)p FI(2\))i(+)e(sinh)n(\()p Fm(d=)p FI(2\))2066 5211 y Fn(q)p 2159 5211 568 4 v 2159 5326 a FI(sinh)2321 5289 y FJ(2)2360 5326 y FI(\()p Fm(d=)p FI(2\))i(+)e(4)2726 5198 y Fn(\023)2818 5326 y Fo(\024)2924 5264 y FI(3)h(+)3080 5189 y Fo(p)p 3156 5189 46 4 v 75 x FI(5)p 2924 5305 278 4 v 3040 5388 a(2)3212 5326 y Fo(j)15 b FI(log)i Fo(j)p Fm(z)3452 5340 y FJ(1)3492 5326 y Fo(jj)p Fm(;)71 5600 y FI(where)30 b(w)m(e)h(ha)m(v)m(e)g(used)f (that)h Fm(d)25 b Fo(\024)g FI(2)15 b(log)r(\(1)21 b(+)1620 5525 y Fo(p)p 1696 5525 46 4 v 75 x FI(2\))31 b(implies)d(sinh)n(\()p Fm(d=)p FI(2\))f Fo(\024)e FI(1.)3780 5796 y Fe(\003)p eop %%Page: 31 31 31 30 bop 71 274 a Fs(N.)31 b(F)-8 b(agella,)31 b(T.)f(M.)h(Seara)g (and)f(J.)g(Villan)m(uev)-5 b(a)2025 b FI(31)71 573 y Ft(Pro)s(of)32 b(of)g(Lemma)d(5.2:)40 b FI(In)26 b(the)i(pro)s(of)f(w)m (e)g(set)h Fm(d)e FI(=)f Fm(\016)2018 587 y Fh(D)2060 568 y Ff(\003)2106 573 y FI(\()p Fm(z)2183 587 y FJ(1)2223 573 y Fm(;)15 b(z)2305 587 y FJ(2)2346 573 y FI(\).)40 b(T)-8 b(o)27 b(pro)m(v)m(e)i(this)d(result)g(w)m(e)i(ha)m(v)m(e)h(to)f (deal)71 686 y(with)h(t)m(w)m(o)j(di\013eren)m(t)d(cases:)71 799 y(\()p Fm(i)p FI(\))i(If)f Fo(j)p Fm(z)361 813 y FJ(2)401 799 y Fo(j)c(\025)f(j)p Fm(z)615 813 y FJ(1)655 799 y Fo(j)p FI(,)30 b(then)g(from)g(part)h(\()p Fm(iii)p FI(\))g(of)g(Lemma)f(5.5)i(w)m(e)f(ha)m(v)m(e:)1240 1001 y Fo(j)p Fm(z)1307 1015 y FJ(1)1367 1001 y Fo(\000)20 b Fm(z)1500 1015 y FJ(2)1540 1001 y Fo(j)25 b(\024)1686 922 y(p)p 1762 922 46 4 v 79 x FI(2)p Fo(j)p Fm(z)1874 1015 y FJ(2)1914 1001 y Fo(jj)15 b FI(log)i Fo(j)p Fm(z)2179 1015 y FJ(1)2219 1001 y Fo(jj)e FI(sinh)o(\()p Fm(d=)p FI(2\))p Fm(:)71 1204 y FI(From)30 b(here,)h(it)f(follo)m(ws)f(the)i (expression)681 1425 y Fo(j)p Fm(z)748 1439 y FJ(1)808 1425 y Fo(\000)20 b Fm(z)941 1439 y FJ(2)981 1425 y Fo(j)1021 1324 y Fn(\020)1075 1425 y FI(1)h Fo(\000)1232 1345 y(p)p 1308 1345 V 80 x FI(2)p Fo(j)15 b FI(log)i Fo(j)p Fm(z)1593 1439 y FJ(1)1633 1425 y Fo(jj)e FI(sinh)o(\()p Fm(d=)p FI(2\))2068 1324 y Fn(\021)2149 1425 y Fo(\024)2245 1345 y(p)p 2321 1345 V 80 x FI(2)p Fo(j)p Fm(z)2433 1439 y FJ(1)2473 1425 y Fo(jj)g FI(log)j Fo(j)p Fm(z)2739 1439 y FJ(1)2779 1425 y Fo(jj)d FI(sinh)n(\()p Fm(d=)p FI(2\))p Fm(:)71 1664 y FI(W)-8 b(e)33 b(p)s(oin)m(t)e(out)h(that)h(from)f(the)g (h)m(yp)s(otheses)g(on)g(the)g(statemen)m(t)i(w)m(e)e(ha)m(v)m(e)i (that)e(cosh\()p Fm(d=)p FI(2\))e Fo(\024)3433 1589 y(p)p 3509 1589 V 75 x FI(2.)46 b(Th)m(us,)71 1777 y(using)29 b(the)h(main)g(v)-5 b(alue)29 b(theorem)i(w)m(e)g(deduce:)559 1901 y Fo(p)p 635 1901 V 79 x FI(2)p Fo(j)15 b FI(log)i Fo(j)p Fm(z)920 1994 y FJ(1)960 1980 y Fo(jj)e FI(sinh)o(\()p Fm(d=)p FI(2\))27 b Fo(\024)1518 1901 y(p)p 1594 1901 V 79 x FI(2)q Fo(j)15 b FI(log)h Fo(j)p Fm(z)1879 1994 y FJ(1)1919 1980 y Fo(jj)f FI(cosh)q(\()p Fm(d=)p FI(2\))p Fm(d=)p FI(2)29 b Fo(\024)c(j)15 b FI(log)i Fo(j)p Fm(z)2866 1994 y FJ(1)2906 1980 y Fo(jj)p Fm(d)26 b Fo(\024)f FI(1)p Fm(=)3215 1901 y Fo(p)p 3291 1901 V 79 x FI(2)q Fm(:)71 2183 y FI(From)30 b(here,)h(w)m(e)g(obtain)1152 2329 y Fo(j)p Fm(z)1219 2343 y FJ(1)1279 2329 y Fo(\000)20 b Fm(z)1412 2343 y FJ(2)1451 2329 y Fo(j)26 b(\024)1724 2268 y FI(2)p 1608 2308 278 4 v 1608 2326 a Fo(p)p 1684 2326 46 4 v 76 x FI(2)20 b Fo(\000)g FI(1)1896 2329 y Fo(j)p Fm(z)1963 2343 y FJ(1)2003 2329 y Fo(jj)15 b FI(log)i Fo(j)p Fm(z)2268 2343 y FJ(1)2308 2329 y Fo(jj)e FI(sinh)n(\()p Fm(d=)p FI(2\))p Fm(:)71 2543 y FI(\()p Fm(i)p FI(\))31 b(If)f Fo(j)p Fm(z)361 2557 y FJ(1)401 2543 y Fo(j)c(\025)f(j)p Fm(z)615 2557 y FJ(2)655 2543 y Fo(j)p FI(,)30 b(applying)f(again)h (part)g(\()p Fm(iii)p FI(\))i(of)f(Lemma)f(5.5)i(and)d(Corollary)g (5.6,)j(w)m(e)f(obtain)606 2808 y Fo(j)p Fm(z)673 2822 y FJ(1)733 2808 y Fo(\000)20 b Fm(z)866 2822 y FJ(2)906 2808 y Fo(j)25 b(\024)1052 2729 y(p)p 1128 2729 V 79 x FI(2)p Fo(j)p Fm(z)1240 2822 y FJ(1)1280 2808 y Fo(jj)15 b FI(log)j Fo(j)p Fm(z)1546 2822 y FJ(2)1586 2808 y Fo(jj)d FI(sinh)n(\()p Fm(d=)p FI(2\))28 b Fo(\024)2153 2747 y FI(3)21 b(+)2310 2672 y Fo(p)p 2386 2672 V 75 x FI(5)p 2153 2787 278 4 v 2232 2806 a Fo(p)p 2307 2806 46 4 v 2307 2881 a FI(2)2441 2808 y Fo(j)p Fm(z)2508 2822 y FJ(1)2548 2808 y Fo(jj)15 b FI(log)i Fo(j)p Fm(z)2813 2822 y FJ(1)2853 2808 y Fo(jj)e FI(sinh)o(\()p Fm(d=)p FI(2\))p Fm(:)71 3054 y FI(The)30 b(pro)s(of)f(ends)h(b)m(y)g(applying) e(again)j(the)f(main)g(v)-5 b(alue)30 b(theorem.)3780 3241 y Fe(\003)71 3527 y Fu(Ac)l(kno)l(wledgemen)l(ts)71 3730 y FI(W)-8 b(e)27 b(wish)d(to)i(thank)g(Adrien)e(Douady)i(and)f (Ricardo)g(P)m(\023)-43 b(erez-Marco)29 b(for)d(v)-5 b(aluable)24 b(discussions)f(and)j(sugges-)71 3843 y(tions.)39 b(The)28 b(authors)f(ha)m(v)m(e)j(b)s(een)d(partially)f(supp)s(orted)g (b)m(y)i(the)g(Catalan)g(CIRIT)f(gran)m(t)i(2001SGR-00070,)71 3956 y(Spanish)h(DGES)j(gran)m(t)h(BFM2002-01344)k(\(NF\),)d(Spanish)30 b(DGICYT)j(gran)m(t)h(BFM2000-0805-C02)39 b(\(NF)71 4068 y(and)h(TMS\),)i(Spanish)c(DGICYT)k(gran)m(t)f(BFM2000-0623)47 b(\(JV\),)42 b(INT)-8 b(AS)40 b(gran)m(t)i(2000-221)j(\(TMS)c(and)71 4181 y(JV\),)31 b(Catalan)f(gran)m(t)h(2002/XT/00094)k(and)30 b(Spanish)e(DGICYT)i(gran)m(t)i(BFM2001-5237-E.)71 4468 y Fu(References)71 4670 y FI([Ah])171 b(Ahlfors,)31 b(L.,)i(Lectures)f (on)g(quasiconformal)f(mappings,)g(W)-8 b(adsw)m(orth)33 b(&)f(Bro)s(oks/Cole)h(mathe-)411 4783 y(matics)d(series,)g(1966.)71 4971 y([Ar])186 b(Arnold,)41 b(V.I.,)j(Small)39 b(denominators)h(I.)h (Mapping)e(the)i(circle)f(on)m(to)i(itself,)g Fr(Izv.)g(A)n(kad.)g (Nauk)411 5083 y(SSSR)e(Ser.)g(Mat.)f Ft(25)g FI(\(1961\))i(21{86.)g (English)36 b(translation:)57 b Fr(A)n(mer.)39 b(Math.)i(So)-5 b(c.)41 b(T)-7 b(r)i(ans.)40 b Ft(46)p FI(,)411 5196 y(2nd)29 b(series)h(\(1965\),)j(213{284.)71 5384 y([Ba])181 b(Bak)m(er,)25 b(I.N.,)f(W)-8 b(andering)22 b(domains)f(for)h(maps)g (of)g(the)h(punctured)d(plane,)j Fr(A)n(nn.)h(A)-5 b(c)g(ad.)26 b(Sci.)f(F)-7 b(enn.)411 5496 y(Ser.)32 b(A)g(I)h(Math.)e Ft(12)g FI(\(1987\),)i(191{198.)71 5684 y([Be1])141 b(Beardon,)42 b(A.F.,)h(Iteration)c(of)h(rational)f(functions.)f(Complex)g(analytic)h (dynamical)f(systems.)411 5796 y(Graduate)30 b(T)-8 b(exts)31 b(in)e(Mathematics)j Ft(132)p FI(,)f(Springer-V)-8 b(erlag,)30 b(New)g(Y)-8 b(ork,)32 b(1991.)p eop %%Page: 32 32 32 31 bop 71 274 a FI(32)1001 b Fs(Asymptotic)30 b(Size)g(of)h(Herman)f (Rings)f(of)i(the)f(Complex)g(Standard)f(F)-8 b(amily)71 573 y FI([Be2])141 b(Beardon,)35 b(A.F.,)i(The)d(geometry)i(of)e (discrete)g(groups.)g(Graduate)h(T)-8 b(exts)35 b(in)e(Mathematics)j Ft(91)p FI(,)411 686 y(Springer-V)-8 b(erlag,)29 b(New)i(Y)-8 b(ork,)31 b(1983.)71 871 y([Br1])145 b(Brjuno,)27 b(A.D.,)i(Analytic)d (form)h(of)g(di\013eren)m(tial)e(equations)i(\(I\),)h Fr(T)-7 b(rudy)30 b(Moskov.)g(Mat.)f(Ob)-5 b(\024)-42 b(s)n(\024)e(c.)28 b Ft(25)411 984 y FI(\(1971\),)33 b(119{262.)g(English)28 b(translation:)40 b Fr(T)-7 b(r)i(ans.)34 b(Mosc)-5 b(ow)34 b(Math.)f(So)-5 b(c.)31 b Ft(25)f FI(\(1973\),)k (131{288.)71 1169 y([Br2])145 b(Brjuno,)24 b(A.D.,)j(Analytic)c(form)h (of)g(di\013eren)m(tial)e(equations)i(\(I)s(I\),)g Fr(T)-7 b(rudy)27 b(Moskov.)g(Mat.)g(Ob)-5 b(\024)-42 b(s)n(\024)e(c.)24 b Ft(26)411 1282 y FI(\(1972\),)33 b(119{239.)g(English)28 b(translation:)40 b Fr(T)-7 b(r)i(ans.)34 b(Mosc)-5 b(ow)34 b(Math.)f(So)-5 b(c.)31 b Ft(26)f FI(\(1974\),)k(119{239.)71 1468 y([BD])157 b(Bu\013,)27 b(X.)g(and)f(Douady)-8 b(,)29 b(A.,)f(Le)f(th)m(\023)-43 b(eor)m(\022)g(eme)30 b(d'in)m(t)m(\023)-43 b(egrabilit)m(\023)g(e)27 b(des)f(structures)g(presque)g(complexes.)411 1581 y(The)31 b(Mandelbrot)f(set,)j(theme)f(and)f(v)-5 b(ariations,)31 b(307{324,)k(London)c(Math.)h(So)s(c.)g(Lecture)g(Note) 411 1693 y(Ser.,)e(274,)i(Cam)m(bridge)d(Univ.)h(Press,)g(Cam)m (bridge,)f(2000.)71 1879 y([C])224 b(Con)m(w)m(a)m(y)-8 b(,)38 b(J.B.,)f(F)-8 b(unctions)35 b(of)h(one)f(complex)g(v)-5 b(ariable.)35 b(I)s(I.)f(Graduate)i(T)-8 b(exts)36 b(in)e(Mathematics) 411 1992 y Ft(159)p FI(,)d(Springer-V)-8 b(erlag,)29 b(New)i(Y)-8 b(ork,)31 b(1995.)71 2177 y([D])221 b(Da)m(v)m(enp)s(ort,) 32 b(H.,)g(Multiplicativ)m(e)e(n)m(um)m(b)s(er)g(theory)-8 b(.)32 b(Third)d(edition.)h(Graduate)i(T)-8 b(exts)31 b(in)f(Math-)411 2290 y(ematics)g Ft(74)p FI(,)h(Springer-V)-8 b(erlag,)30 b(New)g(Y)-8 b(ork,)32 b(2000.)71 2475 y([dMvS])57 b(de)38 b(Melo,)j(W.)e(and)f(v)-5 b(an)38 b(Strien,)h(S.,)h (One-dimensional)c(dynamics,)j(Springer-V)-8 b(erlag,)40 b(Berlin,)411 2588 y(1993.)71 2774 y([F])231 b(F)-8 b(agella,)46 b(N.,)f(Dynamics)c(of)i(the)f(complex)f(standard)g(family)-8 b(,)44 b Fr(J.)f(Math.)h(A)n(nal.)f(Appl.)f Ft(229:1)411 2886 y FI(\(1999\),)33 b(1{31.)71 3072 y([F)m(G])163 b(F)-8 b(agella,)36 b(N.)f(and)e(Gey)m(er,)k(L.,)f(Surgery)d(on)h (Herman)g(Rings)g(of)g(the)h(complex)f(standard)f(family)-8 b(,)411 3185 y Fr(Er)j(go)g(dic)33 b(The)-5 b(ory)34 b(Dynam.)f(Systems)f Ft(23:2)f FI(\(2003\),)i(493-508.)71 3370 y([G])219 b(Gey)m(er,)33 b(L.,)e(Siegel)g(discs,)g(Herman)g(rings) f(and)h(the)g(Arnold)f(family)-8 b(,)31 b Fr(T)-7 b(r)i(ans.)35 b(A)n(mer.)e(Math.)h(So)-5 b(c.)411 3483 y Ft(353:9)31 b FI(\(2001\),)i(3661{3683)h(\(electronic\).)71 3668 y([Her1])101 b(Herman,)45 b(M.,)h(Sur)41 b(la)h(conjugaison)g(di\013)m (\023)-43 b(eren)m(tiable)42 b(des)g(di\013)m(\023)-43 b(eomorphismes)41 b(du)g(cercle)i(\022)-45 b(a)43 b(des)411 3781 y(rotations,)30 b Fr(Inst.)j(Hautes)1352 3758 y(\023)1339 3781 y(Etudes)f(Sci.)h(Publ.)f(Math.)e Ft(49)h FI(\(1979\),)i(5{233.)71 3967 y([Her2])101 b(Herman,)24 b(M.,)i(Simple)21 b(pro)s(ofs)h(of)i(lo) s(cal)f(conjugacy)h(theorems)f(for)g(di\013eomorphisms)e(of)j(the)f (circle)411 4079 y(with)29 b(almost)h(ev)m(ery)h(rotation)g(n)m(um)m(b) s(er,)e Fr(Bol.)k(So)-5 b(c.)33 b(Br)-5 b(asil.)34 b(Mat.)d Ft(16)f FI(\(1985\),)k(45{83.)71 4265 y([Ke])179 b(Keen,)24 b(L.,)h(Dynamics)e(of)h(holomorphic)d(self-maps)i(of)g Ft(C)2381 4232 y Fp(\003)2420 4265 y FI(,)i Fr(Holomorphic)j(functions) f(and)g(mo)-5 b(duli,)411 4378 y(V)e(ol.)30 b(I)h(\(Berkeley,)g(CA,)f (1986\))p FI(,)h(9{30,)g(Math.)e(Sci.)f(Res.)h(Inst.)f(Publ.)f Ft(10)p FI(,)i(Springer,)e(New)h(Y)-8 b(ork,)411 4491 y(1988.)71 4676 y([Ko1])129 b(Kotus,)28 b(J.,)h(Iterated)f(holomorphic) f(maps)g(on)h(the)g(punctured)e(plane,)i Fr(Dynamic)-5 b(al)32 b(systems)f(\(So-)411 4789 y(pr)-5 b(on,)27 b(1985\))p FI(,)f(10{28,)g(Lecture)c(Notes)h(in)e(Econom.)h(and)g(Math.)g(Systems) g Ft(287)p FI(,)i(Springer,)e(Berlin,)411 4902 y(1987.)71 5087 y([Ko2])129 b(Kotus,)33 b(J.,)g(The)f(domains)f(of)i(normalit)m(y) e(of)i(holomorphic)e(self-maps)g(of)i Ft(C)3119 5054 y Fp(\003)3158 5087 y FI(,)g Fr(A)n(nn.)h(A)-5 b(c)g(ad.)36 b(Sci.)411 5200 y(F)-7 b(enn.)32 b(Ser.)h(A)f(I)h(Math.)d Ft(15:2)i FI(\(1990\),)h(329{340.)71 5385 y([Mak])114 b(Makienk)m(o,)31 b(P)-8 b(.M.,)32 b(Iterations)e(of)h(analytic)f (functions)f(in)g Ft(C)2530 5352 y Fp(\003)2569 5385 y FI(,)h Fr(Dokl.)j(A)n(kad.)f(Nauk)g(SSSR)f Ft(297:1)411 5498 y FI(\(1987\),)i(35{37.)f(English)c(translation:)40 b Fr(Soviet)33 b(Math.)g(Dokl.)d Ft(36:3)h FI(\(1988\),)j(418{420.)71 5684 y([MMY])56 b(Marmi,)40 b(S.,)g(Moussa,)h(P)-8 b(.)39 b(and)f(Y)-8 b(o)s(ccoz,)43 b(J.-C.,)e(The)d(Brjuno)g(functions)f(and)h (their)f(regularit)m(y)411 5796 y(prop)s(erties,)29 b Fr(Comm.)k(Math.)g(Phys.)e Ft(186)g FI(\(1997\),)i(265{293.)p eop %%Page: 33 33 33 32 bop 71 274 a Fs(N.)31 b(F)-8 b(agella,)31 b(T.)f(M.)h(Seara)g (and)f(J.)g(Villan)m(uev)-5 b(a)2025 b FI(33)71 573 y([PM])145 b(P)m(\023)-43 b(erez-Marco,)43 b(R.,)c(Solution)d(compl)m(\022)-43 b(ete)40 b(au)d(probl)m(\022)-43 b(eme)38 b(de)f(Siegel)g(de)h(lin)m (\023)-43 b(earisation)36 b(d'une)h(ap-)411 686 y(plication)i (holomorphe)g(au)i(v)m(oisinage)g(d'un)e(p)s(oin)m(t)h(\014xe)h (\(d'apr)m(\023)-43 b(es)41 b(J.-C.)g(Y)-8 b(o)s(ccoz\),)46 b Fr(S)n(\023)-44 b(eminair)-5 b(e)411 799 y(Bourb)g(aki,)33 b(V)-7 b(ol.)33 b(1991/92)p FI(,)g(Ast)m(\023)-43 b(erisque)31 b Ft(206)g FI(\(1992\),)i(Exp.)d(No.)h(753,)h(4,)f(273{310.)71 986 y([P)m(et])156 b(P)m(etersen,)31 b(C.L.,)f(Geometry)i(of)f(Julia)d (sets,)j(lecture)f(notes,)h(Gottingen,)h(1998.)71 1174 y([P)m(om])110 b(P)m(ommerenk)m(e,)46 b(Ch.,)e(Boundary)e(b)s(eha)m (viour)e(of)j(conformal)e(maps,)k(Grundlehren)39 b(der)i(Math-)411 1287 y(ematisc)m(hen)j(Wissensc)m(haften)h([F)-8 b(undamen)m(tal)44 b(Principles)d(of)k(Mathematical)g(Sciences])f Ft(299)p FI(,)411 1400 y(Springer-V)-8 b(erlag,)29 b(Berlin,)g(1992.)71 1587 y([Ri])197 b(Risler,)36 b(E.,)j(Lin)m(\023)-43 b(earisation)35 b(des)i(p)s(erturbations)d(holomorphes)h(des)i(rotations)g(et)g (applications,)411 1700 y Fr(M)n(\023)-44 b(em.)32 b(So)-5 b(c.)33 b(Math.)g(F)-7 b(r.)31 b Ft(77)p FI(,)g(1999.)71 1888 y([Ru1])127 b(R)s(\177)-48 b(ussmann,)44 b(H.,)j(Kleine)42 b(Nenner.)g(I)s(I.)h(Bemerkungen)g(zur)f(Newtonsc)m(hen)i(Metho)s(de,)j Fr(Nachr.)411 2001 y(A)n(kad.)32 b(Wiss.)g(G\177)-46 b(ottingen)34 b(Math.-Phys.)f(Kl.)f(II)f FI(\(1972\),)i(1{20.)71 2188 y([Ru2])127 b(R)s(\177)-48 b(ussmann,)37 b(H.,)j(On)c(the)h (frequencies)g(of)g(quasi-p)s(erio)s(dic)d(solutions)i(of)h(analytic)g (nearly)f(in)m(te-)411 2301 y(grable)j(Hamiltonian)f(systems,)k Fr(Seminar)g(on)f(Dynamic)-5 b(al)43 b(Systems)f(\(St.)f(Petersbur)-5 b(g,)43 b(1991\))p FI(,)411 2414 y(Progr.)31 b(Nonlinear)g(Di\013eren)m (tial)g(Equations)g(Appl.,)g(v)m(ol.)h Ft(12)p FI(,)h(Birkh\177)-45 b(auser,)31 b(Basel)h(\(1994\),)j(160{)411 2527 y(183.)71 2715 y([SV])171 b(Seara,)34 b(T.M.)g(and)e(Villan)m(uev)-5 b(a,)33 b(J.,)h(Asymptotic)f(b)s(eha)m(viour)f(of)h(the)h(domain)e(of)h (analyticit)m(y)g(of)411 2828 y(in)m(v)-5 b(arian)m(t)29 b(curv)m(es)i(of)f(the)h(standard)e(map,)i Fr(Nonline)-5 b(arity)31 b Ft(13)g FI(\(2000\),)i(1699{1744.)71 3015 y([Y1])177 b(Y)-8 b(o)s(ccoz,)57 b(J.C.,)e(Th)m(\023)-43 b(eor)m(\022)g(eme)52 b(de)d(Siegel,)55 b(nom)m(bres)49 b(de)h(Bruno)f(et)i(p)s(olyn^)-45 b(omes)48 b(quadratiques,)411 3128 y Fr(Ast)n(\023)-44 b(erisque)29 b Ft(231)j FI(\(1995\),)h(3{88.) 71 3316 y([Y2])177 b(Y)-8 b(o)s(ccoz,)36 b(J.C.,)e(Analytic)f (linearization)f(of)h(circle)g(di\013eomorphisms,)e Fr(Dynamic)-5 b(al)37 b(systems)f(and)411 3429 y(smal)5 b(l)27 b(divisors)g(\(Cetr)-5 b(ar)g(o,)30 b(1998\))p FI(,)d(125{173,)h(Lecture)c(Notes)h(in)d(Math.) j Ft(1784)p FI(,)h(Springer,)d(Berlin,)411 3542 y(2002.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0306030940691--