Content-Type: multipart/mixed; boundary="-------------0303130516729" This is a multi-part message in MIME format. ---------------0303130516729 Content-Type: text/plain; name="03-112.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-112.keywords" Poincar\'e--Melnikov method, arithmetic properties of frequencies, transverse homoclinic orbits ---------------0303130516729 Content-Type: application/postscript; name="ms2.pss" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ms2.pss" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: ms2.dvi %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Helvetica %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips ms2 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.12.19:1505 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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b(alues)26 b(of)g Fo(")d Fn(!)g Fr(0,)k(since)f(at)g(some)g(sequence)g(of)g(v)-5 b(alues)26 b(of)g Fo(")h Fr(the)f(dominan)n(t)g(harmonics)g(of)g(the)h (splitting)f(function)h(c)n(hange,)-260 615 y(and)h(homo)r(clinic)f (bifurcations)g(could)g(tak)n(e)g(place.)37 b(Some)27 b(examples)g(of)h(suc)n(h)f(bifurcations)g(ha)n(v)n(e)f(b)r(een)i (describ)r(ed)g(in)g([SV01)o(].)-135 814 y(W)-7 b(e)25 b(consider)e(in)i(this)f(pap)r(er)g(some)g(concrete)g(p)r(erturbations) g(in)g(the)h(case)f(of)g(3)g(degrees)f(of)i(freedom)f(\()p Fo(n)f Fr(=)g(2\).)35 b(W)-7 b(e)25 b(c)n(ho)r(ose)e(\(lik)n(e)-260 914 y(other)g(pap)r(ers)f(quoted)i(ab)r(o)n(v)n(e\))e(the)i Fp(golden)i(ve)l(ctor)33 b Fr(of)23 b(frequencies,)h(and)f(a)g(p)r (erturbation)g(with)h(an)f Fp(in\014nite)29 b Fr(n)n(um)n(b)r(er)23 b(of)g(harmonics.)-260 1014 y(In)i(this)g(situation,)g(small)f (divisors)g(app)r(ear)f(in)i(the)g(Melnik)n(o)n(v)f(function)h(but,)h (thanks)e(to)h(the)g(simple)g(arithmetic)f(prop)r(erties)g(of)h(the) -260 1113 y(golden)30 b(v)n(ector,)g(it)h(is)f(p)r(ossible)g(to)h (carry)e(out)h(an)g(accurate)f(analysis)h(of)g(the)h(Melnik)n(o)n(v)e (function)i(and)g(its)f(dominan)n(t)g(harmonics.)-260 1213 y(Applying)j(the)f(accurate)f(b)r(ounds)h(for)g(the)h(size)f(of)g (the)g(error)f(term)h(obtained)g(\(from)g(the)g(use)h(of)f(\015o)n(w-b) r(o)n(x)e(co)r(ordinates\))h(in)i(our)-260 1313 y(pap)r(er)24 b([DGS02],)h(w)n(e)f(sho)n(w)f(that)h(the)h(dominan)n(t)f(harmonics)f (of)h(the)g(splitting)h(function)f(corresp)r(ond)f(to)h(the)h(dominan)n (t)e(harmonics)-260 1412 y(in)39 b(the)f(Melnik)n(o)n(v)g(appro)n (ximation,)h(and)f(from)g(this)h(w)n(e)f(pro)n(vide)f Fp(asymptotic)k(estimates)k Fr(for)38 b(the)h(splitting.)69 b(Suc)n(h)39 b(estimates)-260 1512 y(allo)n(w)30 b(us)h(to)f(sho)n(w)g (the)i(existence)e(of)h(exactly)f(4)h(transv)n(erse)d(homo)r(clinic)j (orbits,)g(and)g(their)f Fp(c)l(ontinuation)38 b Fr(for)30 b(all)h(v)-5 b(alues)30 b(of)h(the)-260 1611 y(p)r(erturbation)c (parameter)f Fo(")d Fn(!)g Fr(0)k(\(with)i(no)e(bifurcations\).)-135 1811 y(Next)e(w)n(e)g(giv)n(e)f(a)g(more)h(precise)f(description)g(of)h (the)h(setting,)f(the)h(results)e(from)h([DGS02)o(])h(to)e(b)r(e)i (applied,)f(and)g(the)h(new)f(results)-260 1910 y(obtained)i(in)h(the)g (presen)n(t)f(pap)r(er.)-260 2242 y Fm(1.1)112 b(Setup:)51 b(A)37 b(concrete)g(example)g(of)h(singular)e(Hamiltonian)f(with)h(3)i (degrees)g(of)g(freedom)-260 2495 y Fr(W)-7 b(e)25 b(consider)f(a)h (Hamiltonian)g(system,)g(with)g(3)g(degrees)f(of)g(freedom,)i(dep)r (ending)f(on)g(t)n(w)n(o)f(p)r(erturbation)g(parameters)f Fo(")i Fr(and)g Fo(\026)p Fr(.)36 b(In)-260 2594 y(canonical)26 b(co)r(ordinates)g(\()p Fo(x;)14 b(y)s(;)g(';)g(I)7 b Fr(\))24 b Fn(2)f Fl(T)18 b Fn(\002)g Fl(R)24 b Fn(\002)17 b Fl(T)1383 2564 y Fk(2)1438 2594 y Fn(\002)g Fl(R)1574 2564 y Fk(2)1618 2594 y Fr(,)27 b(with)h(the)g(symplectic)f(form)g(d)p Fo(x)19 b Fn(^)f Fr(d)p Fo(y)j Fr(+)d(d)p Fo(')g Fn(^)h Fr(d)p Fo(I)7 b Fr(,)28 b(our)e(Hamiltonian)h(is)-260 2694 y(de\014ned)h(b)n(y)1050 2873 y Fo(H)7 b Fr(\()p Fo(x;)14 b(y)s(;)g(';)g(I)7 b Fr(\))23 b(=)g Fo(H)1669 2885 y Fk(0)1706 2873 y Fr(\()p Fo(x;)14 b(y)s(;)g(I)7 b Fr(\))19 b(+)f Fo(\026H)2199 2885 y Fk(1)2236 2873 y Fr(\()p Fo(x;)c(')p Fr(\))p Fo(;)1425 b Fr(\(1\))1050 3050 y Fo(H)1119 3062 y Fk(0)1156 3050 y Fr(\()p Fo(x;)14 b(y)s(;)g(I)7 b Fr(\))23 b(=)g Fn(h)p Fo(!)1623 3062 y Fj(")1659 3050 y Fo(;)14 b(I)7 b Fn(i)18 b Fr(+)1882 2994 y(1)p 1882 3031 42 4 v 1882 3107 a(2)1947 3050 y Fn(h)q Fr(\003)p Fo(I)7 b(;)14 b(I)7 b Fn(i)18 b Fr(+)2304 2994 y Fo(y)2348 2964 y Fk(2)p 2304 3031 81 4 v 2324 3107 a Fr(2)2413 3050 y(+)g(cos)13 b Fo(x)19 b Fn(\000)f Fr(1)p Fo(;)1051 b Fr(\(2\))1050 3203 y Fo(H)1119 3215 y Fk(1)1156 3203 y Fr(\()p Fo(x;)14 b(')p Fr(\))24 b(=)f Fo(h)p Fr(\()p Fo(x)p Fr(\))p Fo(f)9 b Fr(\()p Fo(')p Fr(\))p Fo(:)2066 b Fr(\(3\))-260 3382 y(The)28 b(concrete)f(v)n(ector) f Fo(!)540 3394 y Fj(")603 3382 y Fr(of)h Fp(fast)j(fr)l(e)l(quencies)f 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Fo(k)2250 5175 y Fk(2)2311 5163 y Fr(=)c(0)p Fo(;)14 b(k)2521 5175 y Fk(1)2581 5163 y Fn(\025)22 b Fr(0\))2742 5096 y Fi(\011)2804 5163 y Fr(;)1059 b(\(8\))-260 5342 y(note)26 b(that)g(the)h(constan)n(t)e Fo(\032)e(>)g Fr(0)j(in)g(\(7\))g(giv)n(es)f(the)h(complex)g(width)h (of)f(analiticit)n(y)f(of)h Fo(f)9 b Fr(\()p Fo(')p Fr(\).)37 b(In)26 b(fact,)h(our)e(results)h(will)g(b)r(e)h(v)-5 b(alid)26 b(for)-260 5441 y(a)h(somewhat)g(more)g(general)f(function)i Fo(f)9 b Fr(\()p Fo(')p Fr(\);)29 b(see)e(Section)g(3.)-135 5641 y(The)j(Hamiltonian)g(de\014ned)h(in)g(\(1{7\))e(is)h(a)g (particular)f(case,)i(with)f(3)g(degrees)f(of)i(freedom,)f(of)h(the)f (mo)r(del)h(considered)e(in)i(our)-260 5740 y(pap)r(er)g([DGS02].)48 b(Our)30 b(c)n(hoice)h(in)g(\(4\))h(of)f(the)g(frequency)g(v)n(ector)f (giv)n(en)h(b)n(y)g(the)g(golden)g(n)n(um)n(b)r(er)g(\012)g(is)g(motiv) -5 b(ated)32 b(b)n(y)f(its)g(simple)1845 5989 y(2)p eop %%Page: 3 3 3 2 bop -260 -82 a Fr(arithmetic)25 b(prop)r(erties.)36 b(On)25 b(the)h(other)e(hand,)i(as)f(in)h([DGS02)o(])g(the)g(t)n(w)n(o) e(parameters)g Fo(")h Fr(and)g Fo(\026)h Fr(will)f(not)h(b)r(e)g(indep) r(enden)n(t.)37 b(On)25 b(the)-260 17 y(con)n(trary)-7 b(,)27 b(they)i(will)g(b)r(e)g(link)n(ed)f(b)n(y)g(a)h(relation)e(of)i (the)g(t)n(yp)r(e)g Fo(\026)24 b Fr(=)h Fo(")1918 -13 y Fj(p)1984 17 y Fr(with)k(a)g(suitable)f Fo(p)c(>)h Fr(0)j(\(the)h(smaller)f Fo(p)g Fr(the)h(b)r(etter\),)h(i.e.)f(w)n(e) -260 117 y(consider)21 b(a)h Fp(singular)h Fr(problem)e(for)h Fo(")g Fn(!)i Fr(0)d(\(also)g(called)h Fp(we)l(akly)k(hyp)l(erb)l(olic) p Fr(,)g(or)21 b Fp(a-pr)l(ori)26 b(stable)6 b Fr(\).)36 b(See)22 b(for)g(instance)f([DGS02,)h(DG01])-260 217 y(for)27 b(a)g(motiv)-5 b(ation)28 b(of)f(this)h(singular)f(setting.) -260 548 y Fm(1.2)112 b(Bac)m(kground:)51 b(The)38 b(P)m(oincar)n(\023) -54 b(e{Melnik)m(o)m(v)35 b(metho)s(d)j(with)e(a)i(b)s(ound)g(for)g (the)f(error)g(term)-260 801 y Fr(As)26 b(in)h([DGS02],)f(our)g 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b(to)i(the)g(exp)r(onen)n(ts)g(app)r(earing) e(in)i([DGS02],)h(easily)e(computed)h(from)f(the)i(general)1845 5989 y(4)p eop %%Page: 5 5 5 4 bop -260 -82 a Fr(expressions)25 b(giv)n(en)g(there.)36 b(W)-7 b(e)27 b(stress)e(that)h(some)g(impro)n(v)n(emen)n(t)f(of)h(the) h(exp)r(onen)n(ts)e(can)h(b)r(e)h(giv)n(en)e(for)h(the)g(case)g(of)g(a) g(\014xed)g(torus.)-260 17 y(Because)h(of)g(this,)h(w)n(e)f(ha)n(v)n(e) g(di\013eren)n(t)h(estimates)f(if)h(w)n(e)f(consider)g Fo(\027)h Fr(=)23 b(0)k(or)g Fo(\027)h Fr(=)23 b(1)k(in)h(\(6\).)-135 217 y(T)-7 b(o)34 b(express)g(the)h(b)r(ounds)g(of)g(functions)g(w)n(e) g(use)g(the)g(notation)f Fn(j)q Fo(f)9 b Fn(j)34 b(\026)h(j)p Fo(g)s Fn(j)p Fr(,)i(whic)n(h)e(means)f(that)h(w)n(e)g(can)f(b)r(ound)i Fn(j)p Fo(f)9 b Fn(j)35 b(\024)g Fo(c)14 b Fn(j)o Fo(g)s Fn(j)p Fr(,)-260 316 y(with)33 b(some)f(constan)n(t)g Fo(c)g Fr(not)h(dep)r(ending)g(on)f(an)n(y)g(of)g(the)h(parameters)e (that)i(will)f(b)r(e)h(relev)-5 b(an)n(t)32 b(to)h(us:)46 b Fo(")p Fr(,)34 b Fo(\026)p Fr(,)g Fo(\016)s Fr(.)51 b(In)33 b(this)g(w)n(a)n(y)-7 b(,)33 b(w)n(e)-260 416 y(do)c(not)h(describ)r(e)f(the)h(\(usually)g(complicated\))f(dep)r (endence)h(on)g(amoun)n(ts)e(lik)n(e)i Fo(n;)14 b(\034)5 b(;)14 b(\032;)g(:)g(:)g(:)29 b Fr(and)g(include)h(this)g(dep)r (endence)g(in)g(the)-260 516 y(`constan)n(ts'.)-260 788 y Fs(Theorem)h(0)41 b Fp(F)-6 b(or)30 b(a)g(given)g Fo(\016)c(>)d Fr(0)p Fp(,)30 b(assuming)g(the)f(c)l(onditions)1189 962 y Fo(")23 b Fn(\026)f Fr(1)p Fo(;)183 b(\026)23 b Fn(\026)g Fo(\016)1787 927 y Fj(q)1817 935 y Fg(1)1854 962 y Fo(;)184 b(\026)23 b Fn(\026)f Fo(\016)2261 927 y Fj(q)2291 935 y Fg(2)2324 927 y Fk(+1)2413 898 y Fn(p)p 2482 898 39 4 v 64 x Fo(")o(;)1302 b Fr(\(19\))-260 1135 y Fp(the)25 b(splitting)f(function)h Fn(M)14 b Fr(\()p Fo(s;)g(\022)r Fr(\))23 b(=)g Fo(@)957 1147 y Fj(\022)994 1135 y Fn(L)14 b Fr(\()q Fo(s;)g(\022)r Fr(\))24 b Fp(is)h(analytic)h (on)e Fn(P)1835 1150 y Fj(\024;\031)r(=)p Fk(2)p Fh(\000)p Fj(\016)o(;\032)p Fh(\000)p Fj(\016)2249 1135 y Fp(\()p Fo(\024)f(>)f Fr(0)p Fp(\),)k(and)31 b Fr(^)-49 b Fo(!)2775 1147 y Fj(")2811 1135 y Fp(-quasip)l(erio)l(dic)26 b(with)32 b Fr(^)-49 b Fo(!)3557 1147 y Fj(")3617 1135 y Fp(as)24 b(in)h(\(16\).)-260 1234 y(F)-6 b(or)30 b(the)g(amounts)f Fo(b)h Fp(and)g Fo(b)631 1204 y Fh(0)683 1234 y Fp(in)g(\(12\))h(and)f (\(16\),)h(one)f(has)g(the)g(b)l(ounds)1474 1429 y Fn(j)p Fo(b)18 b Fn(\000)g Fr(1)p Fn(j)c Fo(;)g Fn(j)p Fo(b)1809 1395 y Fh(0)1850 1429 y Fn(\000)k Fr(1)p Fn(j)23 b(\026)2147 1373 y Fo(\026)p 2118 1410 108 4 v 2118 1486 a(\016)2158 1462 y Fj(q)2188 1470 y Fg(2)2235 1429 y Fo(:)-260 1630 y Fp(The)31 b(err)l(or)f(term)f Fn(R)p Fr(\()p Fo(s;)14 b(\022)r Fr(\))31 b Fp(de\014ne)l(d)f(in)g(\(18\))g(is)h(also)37 b Fr(^)-49 b Fo(!)1476 1642 y Fj(")1512 1630 y Fp(-quasip)l(erio)l (dic,)32 b(and)e(satis\014es)g(the)g(fol)t(lowing)i(b)l(ound:)1315 1856 y Fn(jRj)1431 1881 y Fj(\024;\031)r(=)p Fk(2)p Fh(\000)p Fj(\016)o(;\032)p Fh(\000)p Fj(\016)1844 1856 y Fn(\026)1951 1800 y Fo(\026)2001 1770 y Fk(2)p 1941 1837 V 1941 1913 a Fo(\016)1981 1889 y Fj(q)2011 1897 y Fg(3)2077 1856 y Fr(+)2234 1800 y Fo(\026)2284 1770 y Fk(2)p 2170 1837 215 4 v 2170 1914 a Fo(\016)2210 1890 y Fj(q)2240 1898 y Fg(4)2277 1854 y Fn(p)p 2346 1854 39 4 v 60 x Fo(")2395 1856 y(:)1427 b Fr(\(20\))-260 2073 y Fp(The)31 b(exp)l(onents)e Fo(q)327 2085 y Fk(1)364 2073 y Fo(;)14 b(:)g(:)g(:)g(;)g(q)586 2085 y Fk(4)623 2073 y Fp(,)30 b(in)g(the)g(c)l(ase)g Fo(\027)f Fr(=)22 b(0)p Fp(,)30 b(ar)l(e)g(given)g(by)1062 2246 y Fo(q)1099 2258 y Fk(1)1159 2246 y Fr(=)23 b(8)p Fo(;)183 b(q)1532 2258 y Fk(2)1592 2246 y Fr(=)23 b(4)p Fo(;)183 b(q)1965 2258 y Fk(3)2025 2246 y Fr(=)23 b(14)p Fo(;)183 b(q)2440 2258 y Fk(4)2500 2246 y Fr(=)23 b(12)-260 2420 y Fp(and,)31 b(in)f(the)g(c)l(ase)g Fo(\027)e Fr(=)23 b(1)p Fp(,)29 b(the)h(exp)l(onents)g(ar)l(e)g(given)g(by)1071 2593 y Fo(q)1108 2605 y Fk(1)1168 2593 y Fr(=)23 b(6)p Fo(;)183 b(q)1541 2605 y Fk(2)1601 2593 y Fr(=)23 b(2)p Fo(;)183 b(q)1974 2605 y Fk(3)2035 2593 y Fr(=)22 b(10)p Fo(;)183 b(q)2449 2605 y Fk(4)2509 2593 y Fr(=)23 b(8)p Fo(:)-135 2866 y Fr(W)-7 b(e)30 b(remark)e(that,)i(in)g(the)g(quoted)f (result)g([DGS02,)h(Th.)g(10],)f(the)h(splitting)g(function)g(is)g (de\014ned)g(in)f(a)g(complex)h(strip)f(whose)-260 2965 y(widths)h(in)f Fn(j)p Fr(Im)14 b Fo(s)p Fn(j)29 b Fr(and)g Fn(j)p Fr(Im)14 b Fo(\022)r Fn(j)30 b Fr(are)e(written)h(as)f Fo(\027)1304 2977 y Fk(3)1367 2965 y Fr(=)d Fo(\031)s(=)p Fr(2)19 b Fn(\000)g Fr(3)p Fo(\016)32 b Fr(and)d Fo(\032)2011 2977 y Fk(6)2074 2965 y Fr(=)c Fo(\032)19 b Fn(\000)g Fr(6)p Fo(\016)s Fr(.)41 b(This)29 b(comes)f(from)h(the)h(n)n(um)n(b)r (er)f(of)g(successiv)n(e)-260 3065 y(reductions)h(carried)f(out)h(un)n (til)h(the)g(splitting)g(function)g(can)f(b)r(e)h(de\014ned.)46 b(Of)30 b(course,)g(the)h(v)-5 b(alue)30 b(of)h Fo(\016)i Fr(can)e(b)r(e)f(rescaled.)45 b(On)30 b(the)-260 3164 y(other)22 b(hand,)h(the)f(width)h(in)g Fn(j)p Fr(Re)14 b Fo(s)p Fn(j)22 b Fr(is)g(some)f Fo(\024)1184 3176 y Fk(3)1244 3164 y Fr(that)h(w)n(e)g(ha)n(v)n(e)f(renamed)g(as)h Fo(\024)g Fr(\(related)g(to)g(the)g(reac)n(h)f(of)h(the)h(\015o)n(w-b)r (o)n(x)d(co)r(ordinates\).)-260 3495 y Fm(1.3)112 b(Description)36 b(of)i(the)f(results)-260 3748 y Fr(The)28 b(main)g(goal)e(of)i(this)g (pap)r(er)f(is)h(to)f(sho)n(w)g(that,)h(for)f(a)h(singular)e (Hamiltonian)i(with)g(the)g(t)n(w)n(o)f(parameters)f(link)n(ed)h(b)n(y) h(a)f(relation)-260 3847 y(of)33 b(the)h(t)n(yp)r(e)f Fo(\026)f Fr(=)g Fo(")399 3817 y Fj(p)437 3847 y Fr(,)j(the)e (splitting)h(can)f(b)r(e)g(appro)n(ximated)f(b)n(y)g(the)i(Melnik)n(o)n (v)e(function.)54 b(This)33 b(w)n(as)f(sho)n(wn)g(in)i([DG00a)o(],)h (but)-260 3947 y(assuming)i(that)i Fo(\026)f Fr(is)g(exp)r(onen)n (tially)g(small)g(in)g Fo(")p Fr(.)68 b(In)39 b(the)f(presen)n(t)g (singular)f(case)g Fo(\026)k Fr(=)f Fo(")2786 3917 y Fj(p)2824 3947 y Fr(,)h(w)n(e)d(need)g(to)g(sho)n(w)f(that,)k(in)e(the) -260 4047 y(P)n(oincar)n(\023)-39 b(e{Melnik)n(o)n(v)29 b(appro)n(ximation)j(\(18\))i(for)f(whole)h(splitting)g(function)g Fn(M)p Fr(\()p Fo(s;)14 b(\022)r Fr(\),)36 b(the)e(term)g Fo(\026M)9 b Fr(\()p Fo(s)23 b Fr(+)f Fo(s)3294 4059 y Fk(0)3331 4047 y Fo(;)14 b(\022)r Fr(\))35 b(\(exp)r(onen)n(tially) -260 4146 y(small)f(in)g Fo(")p Fr(\))g(dominates,)i(in)e(some)f (sense,)j(the)e(error)e(term)i Fn(R)p Fr(\()p Fo(s;)14 b(\022)r Fr(\).)58 b(A)34 b(natural)f(approac)n(h)g(to)h(this)g(is)g (to)g(pro)n(vide)f(asymptotic)-260 4246 y(estimates)28 b(\(or)g(at)h(least)f(lo)n(w)n(er)f(b)r(ounds\))i(of)g(the)g(dominan)n (t)f(harmonics)f(of)i(the)g(Melnik)n(o)n(v)e(p)r(oten)n(tial)i Fo(L)p Fr(.)39 b(Suc)n(h)29 b(estimates)f(ha)n(v)n(e)f(to)-260 4345 y(b)r(e)32 b(big)f(enough)g(in)g(order)f(to)i(b)r(e)f(v)-5 b(alid)32 b(also)e(for)h(the)h(dominan)n(t)f(harmonics)f(of)h (splitting)h(p)r(oten)n(tial)f Fn(L)h Fr(\(recall)e(that)i Fn(M)d Fr(=)g Fo(@)3842 4357 y Fj(\022)3880 4345 y Fn(L)p Fr(\),)-260 4445 y(sho)n(wing)e(that)i(they)f(dominate)h(the)f(part)g (coming)g(from)g Fn(R)p Fr(.)39 b(It)29 b(is)f(then)h(p)r(ossible)f(to) g(pro)n(v)n(e)f(the)h(existence)g(of)h(simple)f(zeros)f(of)h Fn(M)-260 4545 y Fr(\(with)h(asymptotic)e(estimates)g(of)h(the)f(asso)r (ciated)g(eigen)n(v)-5 b(alues)26 b(of)i Fo(@)1955 4557 y Fj(\022)1992 4545 y Fn(M)p Fr(\),)g(and)g(hence)f(transv)n(erse)f (homo)r(clinic)h(orbits.)-135 4744 y(W)-7 b(e)37 b(carry)f(out)i(this)f (sc)n(heme)g(for)g(the)h(example)f(with)h(3)f(degrees)f(of)h(freedom)g (de\014ned)h(in)g(\(1{7\).)65 b(W)-7 b(e)38 b(ha)n(v)n(e)e(considered)h (a)-260 4844 y(v)n(ery)d(concrete)g(frequency)h(v)n(ector)f(in)h (\(4\),)j(the)d(golden)g(one,)h(for)f(whic)n(h)g(a)g(careful)f (analysis)g(of)h(the)h(asso)r(ciated)e(small)h(divisors)-260 4943 y(can)29 b(b)r(e)h(done,)f(making)f(it)i(p)r(ossible)f(to)g(giv)n (e)f(the)i(ab)r(o)n(v)n(e)e(men)n(tioned)h(asymptotic)g(estimates.)41 b(Then,)30 b(applying)f(previous)f(results,)-260 5043 y(summarized)f(in)h(Section)g(1.2,)e(w)n(e)i(pro)n(v)n(e)d(the)j (existence)g(of)f(exactly)g(4)g(transv)n(erse)f(homo)r(clinic)h (orbits.)-135 5242 y(As)19 b(a)h(di\013erence)f(with)i(resp)r(ect)e(to) h(previous)e(w)n(orks)g([Sau01)o(,)i(LMS99)o(,)g(DG00a)o(],)i(w)n(e)d (sho)n(w)g(that)h(there)f(exist)h(exactly)f(4)g(transv)n(erse)-260 5342 y(homo)r(clinic)27 b(orbits)f Fp(for)j(any)e Fo(")c(>)g Fr(0)j(small)g(enough.)36 b(So)26 b(w)n(e)h(ha)n(v)n(e)e(a)h(con)n(tin) n(uation)g(of)g(the)h(homo)r(clinic)g(orbits)f(as)g Fo(")g Fr(go)r(es)g(to)g(0,)h(with)-260 5441 y(no)g(bifurcations.)-135 5641 y(In)h(fact,)f(w)n(e)h(pro)n(v)n(e)e(this)i(result)f(for)g(a)g (sligh)n(t)h(generalization)d(of)j(the)g(example)f(\(1{7\).)36 b(Indeed,)28 b(w)n(e)g(replace)e(the)i(function)h Fo(f)9 b Fr(\()p Fo(')p Fr(\))-260 5740 y(in)34 b(\(7\))h(b)n(y)e(a)h (function)h(ha)n(ving)e(a)g(more)h(general)e(F)-7 b(ourier)33 b(expansion)g(\(33\),)j(in)n(tro)r(duced)d(in)i(Section)f(3.)56 b(In)34 b(order)f(to)g(pro)n(v)n(e)g(the)1845 5989 y(5)p eop %%Page: 6 6 6 5 bop -260 -82 a Fr(con)n(tin)n(uation)27 b(of)g(the)h(transv)n(erse) d(homo)r(clinic)j(orbits,)f(w)n(e)g(assume)g(a)g(quite)g(general)g (condition)g(\(explicited)h(in)g(Section)f(6\))h(on)f(the)-260 17 y(phases)f(of)g(the)h(F)-7 b(ourier)26 b(expansion)f(of)i(the)g (function)g Fo(f)9 b Fr(\()p Fo(')p Fr(\).)37 b(This)27 b(assumption)f(on)g(the)h(phases)e(excludes)i(the)g(case)e(of)i(a)f (rev)n(ersible)-260 117 y(p)r(erturbation)31 b(\(giv)n(en)g(b)n(y)g(an) g(ev)n(en)g(function)h Fo(f)9 b Fr(\()p Fo(')p Fr(\)\),)33 b(often)f(considered)f(in)g(related)g(pap)r(ers)g([Gal94)o(,)g(GGM99,)g (R)-9 b(W98)o(].)49 b(In)32 b(suc)n(h)-260 217 y(a)d(case,)g (bifurcations)g(of)g(some)g(of)h(the)f(homo)r(clinic)h(orbits)e(can)h (o)r(ccur)g(when)h Fo(")f Fr(go)r(es)f(across)g(some)h(critical)f(v)-5 b(alues.)42 b(This)30 b(kind)f(of)-260 316 y(bifurcations)e(has)g(b)r (een)h(describ)r(ed)g(in)f([SV01],)h(where)f(the)h(Arnold)f(example)g (\(sligh)n(tly)h(mo)r(di\014ed\))g(is)f(considered.)-135 516 y(Let)g(us)f(giv)n(e)g(no)n(w)g(a)g(short)g(summary)g(of)h(the)g (results)f(presen)n(ted.)36 b(As)26 b(said)h(ab)r(o)n(v)n(e,)e(one)i (of)f(the)h(main)g(features)f(of)h(the)g(example)-260 615 y(considered)h(is)h(that)h(the)f(frequency)g(v)n(ector)f Fo(!)s Fr(,)h(giv)n(en)f(b)n(y)h(the)h(golden)e(n)n(um)n(b)r(er,)h(has) g(nice)g(arithmetic)g(prop)r(erties)f(\(in)i(fact,)g(other)-260 715 y(quadratic)25 b(irrationals)f(could)i(also)f(b)r(e)h (considered\).)36 b(T)-7 b(o)25 b(start,)h(in)g(Section)g(2)g(w)n(e)g (review)f(these)h Fp(arithmetic)j(pr)l(op)l(erties)p Fr(,)f(putting)f(a)-260 814 y(sp)r(ecial)g(emphasis)h(on)f(a)g (classi\014cation)f(of)i(the)g(asso)r(ciated)e(resonances.)-135 1014 y(In)35 b(Section)h(3,)h(w)n(e)e(de\014ne)g(the)h(generalization)e (of)h(the)h(function)g Fo(f)9 b Fr(\()p Fo(')p Fr(\),)38 b(in)e(terms)f(of)g(arbitrary)e(phases,)k(and)e(compute)h(the)-260 1113 y(F)-7 b(ourier)24 b(co)r(e\016cien)n(ts)h(of)g(its)g(Melnik)n(o)n (v)e(p)r(oten)n(tial)i Fo(L)p Fr(.)36 b(Next,)26 b(in)f(Section)g(4)g (w)n(e)f(carry)f(out)i(an)g(accurate)f(analysis)f(of)i(the)h(co)r (e\016cien)n(ts)-260 1213 y(and)j(\014nd)g(the)g Fp(dominant)j (harmonics)e Fr(of)f(the)g(F)-7 b(ourier)28 b(expansion)g(in)h Fo(\022)i Fr(of)e Fo(L)p Fr(.)41 b(Since)29 b(w)n(e)f(lo)r(ok)g(for)h (nondegenerate)e(critical)h(p)r(oin)n(ts)-260 1313 y(on)33 b Fl(T)-83 1282 y Fk(2)-13 1313 y Fr(of)g(the)h(p)r(oten)n(tial,)h(w)n (e)e(need)g(at)g(least)g(the)h(2)f(most)g(dominan)n(t)g(harmonics.)52 b(W)-7 b(e)34 b(sho)n(w)f(that,)i(when)e Fo(")g Fr(go)r(es)f(across)g (some)-260 1412 y(critical)25 b(v)-5 b(alues)24 b Fo(")298 1424 y Fj(n)368 1412 y Fr(\(de\014ned)i(b)r(elo)n(w\),)g(some)e(c)n (hanges)g(in)h(the)h(dominance)f(o)r(ccur.)35 b(In)25 b(fact,)h(for)f Fo(")g Fr(close)f(to)h Fo(")3195 1424 y Fj(n)3240 1412 y Fr(,)h(w)n(e)e(ha)n(v)n(e)g(to)h(consider)-260 1512 y(the)k(3)f(most)h(dominan)n(t)f(harmonics)f(b)r(ecause)h(the)h (second)f(and)h(third)f(ones)g(are)g(of)g(the)h(same)f(magnitude.)40 b(W)-7 b(e)29 b(estimate)g(the)g(size)-260 1611 y(of)36 b(these)f(dominan)n(t)g(harmonics,)h(and)f(sho)n(w)g(that)h(the)f(sum)h (of)f(all)g(other)g(harmonics)f(is)i(m)n(uc)n(h)f(smaller.)59 b(In)36 b(Section)f(5,)i(these)-260 1711 y(estimates)24 b(are)g(translated)g(from)g(the)h(Melnik)n(o)n(v)f(p)r(oten)n(tial)g Fo(L)g Fr(to)h(the)g(splitting)g(p)r(oten)n(tial)f Fn(L)p Fr(,)i(using)e(the)h(b)r(ound)g(on)g(the)g(error)d(term)-260 1811 y(giv)n(en)27 b(ab)r(o)n(v)n(e)f(in)i(Theorem)f(0.)-135 2010 y(Finally)-7 b(,)26 b(in)h(Section)f(6)g(w)n(e)g(study)g(the)g (nondegenerate)f(critical)h(p)r(oin)n(ts)g(of)g Fn(L)h Fr(\(whic)n(h)f(corresp)r(ond)f(to)h(simple)g(zeros)f(of)h Fn(M)p Fr(\))g(and)-260 2110 y(obtain)g(the)g Fp(main)i(r)l(esult)e Fr(of)f(this)i(pap)r(er,)e(whic)n(h)h(implies)g(the)g(existence)g(of)g 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b(in)h(\(6\),)g(i.e.)f(whether)h(the)g(whisk)n(ered)e(torus)h(remains)g (\014xed)h(or)e(not.)1845 5989 y(6)p eop %%Page: 7 7 7 6 bop -135 -82 a Fr(W)-7 b(e)26 b(use)f(the)i(notation)e Fo(f)31 b Fn(\030)23 b Fo(g)28 b Fr(if)f(w)n(e)e(can)g(b)r(ound)h Fo(c)1482 -70 y Fk(1)1533 -82 y Fn(j)q Fo(g)s Fn(j)c(\024)h(j)p Fo(f)9 b Fn(j)23 b(\024)f Fo(c)1975 -70 y Fk(2)2026 -82 y Fn(j)p Fo(g)s Fn(j)k Fr(with)g(p)r(ositiv)n(e)f(constan)n(ts)g Fo(c)3035 -70 y Fk(1)3072 -82 y Fr(,)i Fo(c)3158 -70 y Fk(2)3221 -82 y Fr(not)e(dep)r(ending)h(on)g(the)-260 17 y(parameters)g Fo(")p Fr(,)h Fo(\026)p Fr(,)h Fo(\016)s Fr(.)-260 296 y Fs(Theorem)j(1)41 b Fp(In)30 b(the)g(example)h (\(1{7\),)h(assume)f Fo(")f Fp(smal)t(l)h(enough)g(and)f Fo(\026)24 b Fr(=)g Fo(")2280 266 y Fj(p)2348 296 y Fp(with)31 b Fo(p)24 b(>)g(p)2726 266 y Fh(\003)2764 296 y Fp(,)31 b(wher)l(e)g(we)f(de\014ne)h Fo(p)3463 266 y Fh(\003)3525 296 y Fr(=)23 b(2)30 b Fp(if)h Fo(\027)f Fr(=)23 b(1)p Fp(,)-260 396 y(and)30 b Fo(p)-57 366 y 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h(clearly)f(ful\014lled)h(in)g(the)g(concrete)f(case)g(\(7\).)-260 2148 y Fq(2)135 b(The)44 b(golden)i(v)l(ector)-260 2429 y Fr(In)28 b(order)e(to)i(carry)e(out)h(the)h(analysis)e(of)i(the)g (small)f(divisors,)g(the)g(simplest)h(case)f(is)g(that)h(of)g(the)g Fp(golden)j(ve)l(ctor)9 b Fr(:)1146 2664 y Fo(!)26 b Fr(=)d(\(1)p Fo(;)14 b Fr(\012\))p Fo(;)179 b Fr(\012)23 b(=)1898 2540 y Fn(p)p 1967 2540 42 4 v 68 x Fr(5)18 b Fn(\000)g Fr(1)p 1898 2645 254 4 v 2004 2721 a(2)2185 2664 y(=)k(0)p Fo(:)p Fr(618034)1256 b(\(25\))-260 2877 y(\(the)33 b(n)n(um)n(b)r(er)e(considered)g(in)h([DG00a)o(])g(w)n(as)f (\012)1293 2847 y Fh(\000)p Fk(1)1382 2877 y Fr(,)i(but)g(this)f(mak)n (es)e(no)i(imp)r(ortan)n(t)f(di\013erence\).)50 b(Note)32 b(that)g(\012)3428 2847 y Fk(2)3496 2877 y Fr(=)d(1)21 b Fn(\000)g Fr(\012)32 b(with)-260 2976 y(0)39 b Fo(<)g Fr(\012)g Fo(<)g Fr(1.)65 b(This)38 b(frequency)e(v)n(ector)g(allo)n (ws)g(us)i(to)f(tak)n(e)f(adv)-5 b(an)n(tage)36 b(of)h(the)h(nice)f 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5277 y Fo(k)r Fr(\()p Fo(j)5 b Fr(\))29 b(as)e(the)h(follo)n(wing)e (linear)h(com)n(bination)g(of)g(the)h(eigen)n(v)n(ectors)d(of)j Fo(U)9 b Fr(:)1038 5465 y(^)1036 5487 y Fo(k)r Fr(\()p Fo(j)c Fr(\))24 b(=)1296 5370 y Fi(\022)1398 5437 y Fn(\000)p Fo(m)p Fr(\()p Fo(j)5 b Fr(\))1499 5536 y Fo(j)1681 5370 y Fi(\023)1765 5487 y Fr(=)22 b Fo(c)1888 5499 y Fk(1)1939 5370 y Fi(\022)2051 5437 y Fr(1)2042 5536 y(\012)2144 5370 y Fi(\023)2223 5487 y Fn(\000)c Fo(c)2342 5499 y Fk(2)2393 5370 y Fi(\022)2582 5437 y Fr(1)2496 5536 y Fn(\000)p Fr(\012)2621 5506 y Fh(\000)p Fk(1)2751 5370 y Fi(\023)2826 5487 y Fo(;)1036 5718 y(c)1072 5730 y Fk(1)1132 5718 y Fr(=)1257 5662 y Fo(d)p Fr(\()p Fo(j)5 b Fr(\))p 1229 5699 203 4 v 1229 5775 a(2)18 b Fn(\000)g Fr(\012)1442 5718 y Fo(;)97 b(c)1598 5730 y Fk(2)1658 5718 y Fr(=)23 b Fo(m)p Fr(\()p Fo(j)5 b Fr(\))19 b(+)f Fo(c)2060 5730 y Fk(1)1845 5989 y Fr(8)p eop %%Page: 9 9 9 8 bop -260 -82 a Fr(\(note)28 b(that)g Fn(j)p Fo(c)196 -70 y Fk(1)233 -82 y Fn(j)23 b Fo(<)g Fr(1)p Fo(=)p Fr(2)j(and)h Fo(c)716 -70 y Fk(2)777 -82 y Fo(>)22 b Fr(1\).)37 b(The)28 b(follo)n(wing)e(equalities)h(are)g(easily)g(deduced)h(from)f (de\014nition)h(\(29\):)970 172 y Fo(s)p Fr(\()p Fo(j;)14 b(n)p Fr(\))23 b(=)g(\()p Fn(\000)p Fr(\012\))1494 137 y Fj(n)p Fh(\000)p Fk(1)1624 172 y Fo(c)1660 184 y Fk(1)1711 54 y Fi(\022)1823 121 y Fr(1)1814 221 y(\012)1915 54 y Fi(\023)1995 172 y Fn(\000)18 b Fr(\012)2138 137 y Fh(\000)p Fk(\()p Fj(n)p Fh(\000)p Fk(1\))2372 172 y Fo(c)2408 184 y Fk(2)2459 54 y Fi(\022)2648 121 y Fr(1)2562 221 y Fn(\000)p Fr(\012)2687 190 y Fh(\000)p Fk(1)2817 54 y Fi(\023)2892 172 y Fo(;)970 361 y Fn(j)p Fo(s)p Fr(\()p Fo(j;)c(n)p Fr(\))p Fn(j)24 b Fr(=)e Fo(f)9 b Fr(\()p Fo(j;)14 b(n)19 b Fr(+)f(1\))k(=)h(\012)1900 327 y Fh(\000)p Fk(\()p Fj(n)p Fk(+1\))2133 361 y Fo(c)2169 373 y Fk(2)2225 361 y Fn(\000)18 b Fr(\()p Fn(\000)p Fr(\012\))2497 327 y Fj(n)p Fk(+1)2626 361 y Fo(c)2662 373 y Fk(1)2699 361 y Fo(;)970 486 y Fn(jh)p Fo(s)p Fr(\()p 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1388 y Fr(=)23 b Fn(j)p Fo(d)p Fr(\()p Fo(j)5 b Fr(\))p Fn(j)15 b Fr(\012)1002 1353 y Fh(\000)p Fk(2)1091 1388 y Fo(c)1127 1400 y Fk(2)1187 1388 y Fr(=)23 b(\012)1335 1353 y Fh(\000)p Fk(2)1438 1267 y Fi(\014)1438 1317 y(\014)1438 1367 y(\014)1438 1417 y(\014)1465 1388 y Fo(d)p Fr(\()p Fo(j)5 b Fr(\))1625 1271 y Fi(\022)1687 1388 y Fo(m)p Fr(\()p Fo(j)g Fr(\))19 b(+)2003 1331 y Fo(d)p Fr(\()p Fo(j)5 b Fr(\))p 1975 1369 203 4 v 1975 1445 a(2)18 b Fn(\000)g Fr(\012)2188 1271 y Fi(\023)2249 1267 y(\014)2249 1317 y(\014)2249 1367 y(\014)2249 1417 y(\014)2300 1388 y Fr(=)2397 1256 y Fi(\014)2397 1305 y(\014)2425 1326 y Fo(j)2464 1296 y Fk(2)2519 1326 y Fn(\000)h Fo(j)5 b(m)p Fr(\()p Fo(j)g Fr(\))18 b Fn(\000)g Fo(m)p Fr(\()p Fo(j)5 b Fr(\))3095 1296 y Fk(2)3133 1256 y Fi(\014)3133 1305 y(\014)p 2397 1369 764 4 v 2677 1445 a Fr(2)18 b Fn(\000)g Fr(\012)3171 1388 y Fo(;)-260 1641 y Fr(and)28 b(w)n(e)f(also)f(deduce)i(part)f(\(b\).)-135 1841 y(Finally)-7 b(,)27 b(part)g(\(c\))i(is)e(easily)g(pro)n(v)n(ed)f(in)i(the)g(follo)n (wing)e(rough)h(w)n(a)n(y:)660 2024 y Fi(\014)660 2074 y(\014)687 2094 y Fo(j)726 2060 y Fk(2)782 2094 y Fn(\000)18 b Fo(j)5 b(m)p Fr(\()p Fo(j)g Fr(\))19 b Fn(\000)f Fo(m)p Fr(\()p Fo(j)5 b Fr(\))1358 2060 y Fk(2)1395 2024 y Fi(\014)1395 2074 y(\014)1446 2094 y Fr(=)23 b Fn(j)p Fo(d)p Fr(\()p Fo(j)5 b Fr(\)\(\(1)19 b(+)f(2\012\))p Fo(j)23 b Fn(\000)18 b Fo(d)p Fr(\()p Fo(j)5 b Fr(\)\))p Fn(j)24 b Fo(>)2508 2038 y Fr(\012)p 2508 2075 60 4 v 2517 2151 a(2)2592 1977 y Fi(\022)2653 2094 y Fr(\(1)18 b(+)g(2\012\))p Fo(j)23 b Fn(\000)3112 2038 y Fr(1)p 3112 2075 42 4 v 3112 2151 a(2)3164 1977 y Fi(\023)826 2327 y Fr(=)913 2210 y Fi(\022)974 2327 y Fr(1)18 b Fn(\000)1127 2271 y Fr(\012)p 1127 2308 60 4 v 1136 2384 a(2)1197 2210 y Fi(\023)1272 2327 y Fo(j)24 b Fn(\000)1423 2271 y Fr(\012)p 1423 2308 V 1432 2384 a(4)1516 2327 y Fn(\025)1603 2210 y Fi(\022)1664 2327 y Fr(1)18 b Fn(\000)1817 2271 y Fr(3\012)p 1817 2308 102 4 v 1847 2384 a(4)1929 2210 y Fi(\023)2004 2327 y Fo(j)28 b(>)2165 2271 y(j)p 2163 2308 42 4 v 2163 2384 a Fr(2)2215 2327 y Fo(;)-260 2581 y Fr(where)j(w)n(e)f(ha)n(v)n(e) g(used)h(that)g(\012)p Fo(=)p Fr(2)c Fo(<)i Fn(j)p Fo(d)p Fr(\()p Fo(j)5 b Fr(\))p Fn(j)29 b Fo(<)f Fr(1)p Fo(=)p Fr(2,)i(according)g(to)g(\(28\).)47 b(One)30 b(obtains)h(the)g(upp)r (er)g(inequalit)n(y)g(in)g(a)f(similar)g(w)n(a)n(y)-7 b(.)3930 2680 y Fe(2)-135 2880 y Fr(T)g(o)27 b(illustrate)g(this)h (result,)f(w)n(e)h(write)f(do)n(wn)g(the)h(\(in)n(teger\))f(limits)1157 3083 y(~)-45 b Fo(\015)1202 3049 y Fh(\003)1197 3104 y Fj(j)1263 3083 y Fr(=)23 b(\(2)18 b Fn(\000)g Fr(\012\))p Fo(\015)1666 3049 y Fh(\003)1661 3104 y Fj(j)1727 3083 y Fr(=)1815 3013 y Fi(\014)1815 3063 y(\014)1842 3083 y Fo(j)1881 3049 y Fk(2)1937 3083 y Fn(\000)g Fo(j)5 b(m)p Fr(\()p Fo(j)g Fr(\))19 b Fn(\000)f Fo(m)p Fr(\()p Fo(j)5 b Fr(\))2513 3049 y Fk(2)2550 3013 y Fi(\014)2550 3063 y(\014)3845 3083 y Fr(\(31\))-260 3307 y(for)22 b(the)h(sequences)f(generated)g(b)n(y)g(the)h(\014rst)g(few)g(primitiv) n(es)1691 3286 y(^)1689 3307 y Fo(k)s Fr(\()p Fo(j)5 b Fr(\))23 b(with)h Fo(j)k(>)22 b Fr(0,)i(as)e(w)n(ell)g(as)g(a)g(lo)n (w)n(er)g(b)r(ound)h(for)f(the)h(remaining)f(ones.)-260 3407 y(Note)27 b(that)g(the)g(lo)n(w)n(er)e(and)h(upp)r(er)h(b)r(ounds) g(giv)n(en)e(in)i(Prop)r(osition)e(3)h(imply)h(a)g(wide)f(separation)f (among)h(the)h(limit)g(n)n(umerators)e(in)-260 3507 y(the)j(sequences,) f(except)h(for)f(some)g(of)g(them)h(whose)f(limits)h(coincide.)1170 3778 y(^)1168 3800 y Fo(k)s Fr(\()p Fo(j)5 b Fr(\))p 1436 3833 4 113 v 174 w(~)-46 b Fo(\015)1535 3770 y Fh(\003)1530 3821 y Fj(j)p 1047 3836 576 4 v 1118 3906 a Fr(\()p Fn(\000)p Fr(1)p Fo(;)14 b Fr(1\))p 1436 3936 4 100 v 142 w(1)1118 4005 y(\()p Fn(\000)p Fr(2)p Fo(;)g Fr(4\))p 1436 4035 V 142 w(4)1118 4105 y(\()p Fn(\000)p Fr(4)p Fo(;)g Fr(7\))p 1436 4135 V 142 w(5)1118 4205 y(\()p Fn(\000)p Fr(6)p Fo(;)g Fr(9\))p 1436 4234 V 142 w(9)1097 4304 y(\()p Fn(\000)p Fr(7)p Fo(;)g Fr(12\))p 1436 4334 V 100 w(11)1097 4404 y(\()p Fn(\000)p Fr(9)p Fo(;)g Fr(14\))p 1436 4434 V 100 w(11)2126 3776 y(^)2124 3798 y Fo(k)s Fr(\()p Fo(j)5 b Fr(\))p 2413 3831 4 113 v 237 w(~)-45 b Fo(\015)2555 3768 y Fh(\003)2550 3820 y Fj(j)p 1983 3834 702 4 v 2033 3904 a Fr(\()p Fn(\000)p Fr(11)p Fo(;)14 b Fr(17\))p 2413 3934 4 100 v 141 w(19)2033 4004 y(\()p Fn(\000)p Fr(12)p Fo(;)g Fr(20\))p 2413 4034 V 141 w(16)2033 4103 y(\()p Fn(\000)p Fr(14)p Fo(;)g Fr(22\))p 2413 4133 V 141 w(20)2033 4203 y(\()p Fn(\000)p Fr(15)p Fo(;)g Fr(25\))p 2413 4233 V 141 w(25)2033 4303 y(\()p Fn(\000)p Fr(17)p Fo(;)g Fr(27\))p 2413 4332 V 141 w(19)p 1983 4336 702 4 v 2082 4405 a Fo(j)28 b Fn(\025)23 b Fr(28)p 2413 4435 4 100 v 148 w Fo(>)g Fr(14)-260 4711 y(In)31 b(fact,)h(it)f(is)g(easy)f (to)g(see)h(that)g(there)g(exist)f(in\014nite)i(primitiv)n(es.)46 b(Indeed,)32 b(taking)e(t)n(w)n(o)g(primitiv)n(es)3062 4689 y(^)3059 4711 y Fo(k)s Fr(\()p Fo(j)3171 4723 y Fk(1)3209 4711 y Fr(\))h(and)3439 4689 y(^)3437 4711 y Fo(k)s Fr(\()p Fo(j)3549 4723 y Fk(2)3586 4711 y Fr(\))g(suc)n(h)g (that)-260 4819 y Fo(d)p Fr(\()p Fo(j)-151 4831 y Fk(1)-113 4819 y Fr(\))e(and)f Fo(d)p Fr(\()p Fo(j)219 4831 y Fk(2)257 4819 y Fr(\))h(ha)n(v)n(e)e(the)i(same)e(sign,)h(and)h(denoting)f Fo(k)f Fr(=)1723 4797 y(^)1721 4819 y Fo(k)s Fr(\()p Fo(j)1833 4831 y Fk(1)1870 4819 y Fr(\))20 b(+)2007 4797 y(^)2005 4819 y Fo(k)s Fr(\()p Fo(j)2117 4831 y Fk(2)2155 4819 y Fr(\),)29 b(one)f(sees)f(that)i(\012)24 b Fo(<)g Fn(jh)p Fo(k)s(;)14 b(!)s Fn(ij)25 b Fo(<)e Fr(1,)29 b(and)f(one)g(of)g Fo(U)9 b(k)31 b Fr(or)-260 4919 y Fo(U)-194 4889 y Fk(2)-157 4919 y Fo(k)g Fr(is)c(a)g(new)h(primitiv)n (e)f(v)n(ector,)g(whose)g(second)g(comp)r(onen)n(t)g(is)g(greater)f (than)i Fo(j)2398 4931 y Fk(1)2454 4919 y Fr(+)18 b Fo(j)2571 4931 y Fk(2)2608 4919 y Fr(.)-260 5118 y Fs(Remark.)47 b Fr(It)25 b(is)f(an)g(ob)n(vious)f(consequence)g(of)h(Prop)r(osition)e (3)i(that)h(the)f(exp)r(onen)n(t)g(in)h(the)f(Diophan)n(tine)g (condition)g(\(11\))g(is)g Fo(\034)33 b Fr(=)23 b(1.)-260 5218 y(Besides,)37 b(it)f(is)g(not)f(hard)g(to)h(\014nd)g(the)g (constan)n(t)e Fo(\015)41 b Fr(as)35 b(the)h(minim)n(um)g(of)f(the)h(n) n(umerators)e Fo(\015)2850 5230 y Fj(k)2891 5218 y Fr(,)k Fo(k)h Fn(6)p Fr(=)d(0.)60 b(A)36 b(simple)g(insp)r(ection)-260 5317 y(\(including)28 b(v)n(ectors)d Fo(k)35 b(=)-51 b Fn(2)23 b(A)p Fr(\))28 b(giv)n(es)e(the)h(rigorous)e(v)-5 b(alue)26 b Fo(\015)i Fr(=)23 b Fo(\015)1765 5332 y Fk(\(0)p Fj(;)p Fk(1\))1930 5317 y Fr(=)f(\012.)37 b(Nev)n(ertheless,)26 b(it)i(is)e(more)h(signi\014can)n(t)f(to)h(giv)n(e)f(for)g Fo(\015)32 b Fr(the)-260 5417 y(follo)n(wing)27 b(asymptotic)g(v)-5 b(alue)27 b(\(see)h(also)e(remark)g(3)i(after)f(Theorem)g(8\):)638 5660 y Fo(\015)h Fn(')22 b Fr(lim)14 b(inf)807 5717 y Fh(j)p Fj(k)q Fh(j!1)1040 5660 y Fo(\015)1083 5672 y Fj(k)1147 5660 y Fr(=)23 b(min)1387 5593 y Fi(\010)1436 5660 y Fo(\015)1484 5626 y Fh(\003)1479 5681 y Fj(j)1544 5660 y Fr(:)h Fo(j)32 b Fr(is)c(primitiv)n(e)2068 5593 y Fi(\011)2140 5660 y Fr(=)22 b Fo(\015)2275 5626 y Fh(\003)2270 5681 y Fk(1)2336 5660 y Fr(=)2515 5604 y(1)p 2434 5641 203 4 v 2434 5717 a(2)c Fn(\000)g Fr(\012)2670 5660 y(=)k(0)p Fo(:)p Fr(723607)p Fo(:)748 b Fr(\(32\))1845 5989 y(9)p eop %%Page: 10 10 10 9 bop -260 -82 a Fq(3)135 b(The)44 b(Melnik)l(o)l(v)i(p)t(oten)l (tial)h(of)e(a)g(generalized)i(example)-260 199 y Fr(The)28 b(Hamiltonian)g(w)n(e)g(consider)g(in)g(this)h(and)f(the)g(subsequen)n (t)g(sections,)g(with)h(3)e(degrees)g(of)i(freedom,)f(is)g(giv)n(en)f (as)h(in)g(\(1{6\))g(but)-260 299 y(w)n(e)f(consider,)g(instead)g(of)h (\(7\),)g(a)f(more)g(general)f(p)r(erturbation:)740 499 y Fo(f)9 b Fr(\()p Fo(')p Fr(\))23 b(=)1026 420 y Fi(X)1019 598 y Fj(k)q Fh(2Z)1167 499 y Fo(f)1208 511 y Fj(k)1262 499 y Fr(cos)o(\()p Fn(h)q Fo(k)s(;)14 b(')p Fn(i)k(\000)g Fo(\033)1755 511 y Fj(k)1797 499 y Fr(\))p Fo(;)97 b Fr(with)28 b Fo(f)2179 511 y Fj(k)2243 499 y Fr(=)22 b Fo(e)2369 468 y Fh(\000)p Fj(\032)p Fh(j)p Fj(k)q Fh(j)2563 499 y Fr(and)28 b Fo(\033)2772 511 y Fj(k)2836 499 y Fn(2)23 b Fl(T)p Fr(,)852 b(\(33\))-260 770 y(where)29 b Fn(Z)34 b(\032)26 b Fl(Z)228 740 y Fk(2)289 770 y Fr(is)k(de\014ned)g (as)f(in)h(\(8\).)43 b(Recall)29 b(the)h(di\013erence)g(b)r(et)n(w)n (een)g(the)g(t)n(w)n(o)f(v)-5 b(alues)29 b(of)h Fo(\027)35 b Fr(in)30 b(\(6\):)41 b(for)29 b Fo(\027)j Fr(=)26 b(1)j(the)h(whisk)n (ered)-260 870 y(torus)d(remains)g(\014xed)g(under)h(the)g(p)r (erturbation,)f(and)g(for)g Fo(\027)i Fr(=)22 b(0)28 b(it)g(do)r(es)f(not.)-135 1069 y(In)h(fact,)h(a)f(quite)h(general)e (condition)h(on)h(the)f(phases)g Fo(\033)1662 1081 y Fj(k)1732 1069 y Fr(will)h(ha)n(v)n(e)e(to)h(b)r(e)h(ful\014lled)g(in)g (order)e(to)i(guaran)n(tee)d(the)j(con)n(tin)n(uation)-260 1169 y(of)35 b(the)g(homo)r(clinic)g(orbits)f(\(see)g(this)h(condition) g(in)g(Section)f(6\).)59 b(F)-7 b(or)34 b(instance,)i(taking)e Fo(\033)2752 1181 y Fj(k)2828 1169 y Fr(=)h Fo(\031)s(=)p Fr(2)f(for)g(ev)n(ery)g Fo(k)s Fr(,)i(one)f(has)f(the)-260 1269 y(example)27 b(\(7\),)h(and)g(the)g(condition)f(on)g(the)h(phases) f(will)h(b)r(e)g(ful\014lled)g(for)f(this)h(concrete)f(case.)-135 1468 y(On)j(the)g(other)g(hand,)h(w)n(e)e(ha)n(v)n(e)g Fo(\013)f Fr(=)f(2)j(in)g(h)n(yp)r(othesis)g(\(H4\))g(of)h([DGS02)o(,)g (Sect.)g(1.1],)f(since)g Fo(\013)g Fr(is)h(the)f(exp)r(onen)n(t)g (obtained)g(in)-260 1567 y(the)e(follo)n(wing)f(estimate:)1050 1702 y Fn(k)o Fo(f)9 b Fn(k)1183 1727 y Fj(\032)p Fh(\000)p Fj(\016)1328 1702 y Fr(:=)1439 1623 y Fi(X)1481 1802 y Fj(k)1573 1702 y Fn(j)p Fo(f)1637 1714 y Fj(k)1677 1702 y Fn(j)14 b Fo(e)1753 1668 y Fk(\()p Fj(\032)p Fh(\000)p Fj(\016)r Fk(\))p Fh(j)p Fj(k)q Fh(j)2027 1702 y Fr(=)2114 1623 y Fi(X)2156 1802 y Fj(k)2248 1702 y Fo(e)2287 1668 y Fh(\000)p Fj(\016)r Fh(j)p Fj(k)q Fh(j)2474 1702 y Fn(\030)2590 1646 y Fr(1)p 2572 1683 78 4 v 2572 1759 a Fo(\016)2612 1735 y Fk(2)2659 1702 y Fo(;)-260 1931 y Fr(where)29 b(w)n(e)g(ha)n(v)n(e)f(considered)h(a)g(norm)g(whic)n(h)g (tak)n(es)g(in)n(to)g(accoun)n(t)f(the)i(F)-7 b(ourier)29 b(expansion)f(in)i(the)g(angular)d(v)-5 b(ariables)29 b(\(see)g(more)-260 2031 y(details)f(in)f([DGS02,)h(Sect.)g(1.5]\).)36 b(This)28 b(v)-5 b(alue)27 b(of)h Fo(\013)g Fr(has)f(b)r(een)h(used)g (to)f(giv)n(e)g(the)h(exp)r(onen)n(ts)f(in)h(Theorem)f(0.)-135 2230 y(W)-7 b(e)28 b(can)f(obtain)g(the)h(F)-7 b(ourier)27 b(expansion)f(of)i(the)g(Melnik)n(o)n(v)e(p)r(oten)n(tial)i(using)f (\(17\),)g(together)g(with)h(\(16\).)37 b(W)-7 b(e)28 b(ha)n(v)n(e:)673 2460 y Fo(L)p Fr(\()p Fo(s;)14 b(\022)r Fr(\))23 b(=)g Fn(\000)1175 2381 y Fi(X)1101 2563 y Fj(k)q Fh(2Z)5 b(nf)o Fk(0)p Fh(g)1383 2460 y Fo(f)1424 2472 y Fj(k)1479 2347 y Fi(Z)1562 2368 y Fh(1)1525 2536 y(\0001)1647 2460 y Fr(\(cos)13 b Fo(x)1851 2472 y Fk(0)1889 2460 y Fr(\()p Fo(s)19 b Fr(+)f Fo(bt)p Fr(\))g Fn(\000)g Fr(1\))c(cos)f(\()p Fn(h)p Fo(k)s(;)h(\022)21 b Fr(+)j(~)-48 b Fo(!)2816 2472 y Fj(")2851 2460 y Fo(t)p Fn(i)19 b(\000)f Fo(\033)3062 2472 y Fj(k)3103 2460 y Fr(\))c(d)p Fo(t)934 2706 y Fr(=)1097 2627 y Fi(X)1022 2809 y Fj(k)q Fh(2Z)5 b(nf)p Fk(0)p Fh(g)1305 2706 y Fo(L)1362 2718 y Fj(k)1416 2706 y Fr(cos)13 b(\()p Fn(h)q Fo(k)s(;)h(\022)20 b Fn(\000)25 b Fr(^)-49 b Fo(!)1883 2718 y Fj(")1918 2706 y Fo(s)p Fn(i)19 b(\000)f Fo(\033)2138 2718 y Fj(k)2179 2706 y Fr(\))c Fo(;)673 2995 y(L)730 3007 y Fj(k)794 2995 y Fr(=)22 b(2)p Fo(f)964 3007 y Fj(k)1018 2882 y Fi(Z)1101 2903 y Fh(1)1064 3071 y(\0001)1210 2939 y Fr(cos)13 b Fn(h)p Fo(k)s(;)21 b Fr(~)-49 b Fo(!)1502 2951 y Fj(")1537 2939 y Fo(t)p Fn(i)p 1210 2976 390 4 v 1268 3061 a Fr(cosh)1425 3024 y Fk(2)1476 3061 y Fo(bt)1610 2995 y Fr(d)p Fo(t)23 b Fr(=)1887 2939 y(2)p Fo(\031)17 b Fn(h)p Fo(k)s(;)j Fr(^)-48 b Fo(!)2160 2951 y Fj(")2195 2939 y Fn(i)14 b Fo(f)2282 2951 y Fj(k)p 1806 2976 597 4 v 1806 3057 a Fo(b)g Fr(sinh)2018 2989 y Fi(\000)2066 3024 y Fj(\031)p 2066 3038 41 4 v 2070 3085 a Fk(2)2130 3057 y Fn(h)q Fo(k)s(;)20 b Fr(^)-48 b Fo(!)2298 3069 y Fj(")2333 3057 y Fn(i)2365 2989 y Fi(\001)2436 2995 y Fr(=)2534 2939 y(2)p Fo(\031)16 b Fn(jh)q Fo(k)s(;)k Fr(^)-48 b Fo(!)2830 2951 y Fj(")2865 2939 y Fn(ij)14 b Fo(e)2973 2909 y Fh(\000)p Fj(\032)p Fh(j)o Fj(k)q Fh(j)p 2534 2976 606 4 v 2548 3057 a Fo(b)g Fr(sinh)2760 2986 y Fi(\014)2760 3036 y(\014)2797 3024 y Fj(\031)p 2797 3038 41 4 v 2801 3085 a Fk(2)2862 3057 y Fn(h)p Fo(k)s(;)21 b Fr(^)-49 b Fo(!)3029 3069 y Fj(")3064 3057 y Fn(i)3097 2986 y Fi(\014)3097 3036 y(\014)3845 2995 y Fr(\(34\))-260 3240 y(\(w)n(e)23 b(tak)n(e)f Fo(L)122 3252 y Fk(0)182 3240 y Fr(=)g(0)h(to)g(ha)n(v)n(e)e(zero)h(a)n (v)n(erage\).)33 b(The)23 b(in)n(tegral)e(has)i(b)r(een)g(computed)g(b) n(y)f(residues,)i(and)e(w)n(e)h(ha)n(v)n(e)e(also)h(used)h(the)g(form)n (ula)-260 3339 y(cos)13 b Fo(x)-88 3351 y Fk(0)-50 3339 y Fr(\()p Fo(bt)p Fr(\))d Fn(\000)g Fr(1)23 b(=)f Fn(\000)p Fr(2)p Fo(=)14 b Fr(cosh)636 3302 y Fk(2)687 3339 y Fo(bt)p Fr(.)35 b(Notice)24 b(that)g(the)f(v)-5 b(alue)24 b(of)f Fo(\027)29 b Fr(do)r(es)23 b(not)h(in\015uence)f(the)h(Melnik)n(o)n(v)f (p)r(oten)n(tial,)h(and)f(that)h(the)g(phases)-260 3439 y Fo(\033)-213 3451 y Fj(k)-144 3439 y Fr(in)k(the)g(F)-7 b(ourier)27 b(expansion)f(of)i Fo(L)p Fr(\(0)p Fo(;)14 b(\022)r Fr(\))27 b(are)g(the)h(same)f(as)g(in)h(the)f(function)i Fo(f)9 b Fr(\()p Fo(')p Fr(\))28 b(giv)n(en)f(in)g(\(33\).)-260 3814 y Fq(4)135 b(Dominan)l(t)46 b(harmonics)f(of)g(the)g(Melnik)l(o)l (v)h(p)t(oten)l(tial)-260 4096 y Fr(T)-7 b(o)20 b(sho)n(w)e(that)j(the) f(Melnik)n(o)n(v)e(p)r(oten)n(tial)i(has)f(nondegenerate)g(critical)g (p)r(oin)n(ts,)i(w)n(e)e(ha)n(v)n(e)g(to)h(consider)e(at)i(least)f(the) i(2)e(most)h(dominan)n(t)-260 4195 y(harmonics)27 b(in)h(its)f(F)-7 b(ourier)27 b(expansion.)36 b(As)28 b(w)n(e)g(see)f(b)r(elo)n(w,)g (whic)n(h)h(the)g(dominan)n(t)f(harmonics)g(are)g(dep)r(ends)h(on)f Fo(")p Fr(.)37 b(Nev)n(ertheless,)-260 4295 y(for)29 b(some)g(v)-5 b(alues)29 b(of)h Fo(")f Fr(w)n(e)g(will)h(ha)n(v)n(e)f (to)g(consider)g(the)h(3)f(most)g(dominan)n(t)g(harmonics)g(b)r(ecause) g(the)h(second)f(and)g(the)h(third)g(ones)-260 4394 y(can)d(b)r(e)h(of) g(the)g(same)f(magnitude.)-135 4594 y(Studying)j(the)h(size)f(of)g(the) h(co)r(e\016cien)n(ts)e Fo(L)1248 4606 y Fj(k)1319 4594 y Fr(requires)g(to)h(use)g(the)h(arithmetic)f(prop)r(erties)f(of)h(our) g(frequency)g(v)n(ector.)43 b(T)-7 b(aking)-260 4693 y(in)n(to)28 b(accoun)n(t)e(the)i(de\014nition)g(of)g Fo(\015)865 4705 y Fj(k)933 4693 y Fr(in)g(\(26\),)f(w)n(e)h(ha)n(v)n (e)1249 4926 y Fn(jh)p Fo(k)s(;)21 b Fr(^)-49 b Fo(!)1439 4938 y Fj(")1474 4926 y Fn(i)q(j)23 b Fr(=)1640 4806 y Fi(\014)1640 4856 y(\014)1640 4905 y(\014)1640 4955 y(\014)1668 4809 y(\034)1730 4926 y Fo(k)s(;)1838 4870 y(b)1874 4840 y Fh(0)1897 4870 y Fo(!)p 1823 4907 144 4 v 1823 4983 a(b)1859 4924 y Fn(p)p 1928 4924 39 4 v 59 x Fo(")1976 4809 y Fi(\035)2039 4806 y(\014)2039 4856 y(\014)2039 4905 y(\014)2039 4955 y(\014)2089 4926 y Fr(=)2247 4870 y Fo(b)2283 4840 y Fh(0)2306 4870 y Fo(\015)2349 4882 y Fj(k)p 2187 4907 264 4 v 2187 4983 a Fo(b)14 b Fn(j)p Fo(k)s Fn(j)2342 4924 y(p)p 2412 4924 39 4 v 2412 4983 a Fo(")2460 4926 y(:)-260 5164 y Fr(Recall)22 b(from)g(Theorem)g (0)g(that)g Fo(b)g Fr(and)g Fo(b)1010 5134 y Fh(0)1056 5164 y Fr(are)f(1)8 b(+)g Fn(O)15 b Fr(\()q Fo(\026\016)1516 5134 y Fh(\000)p Fj(q)1598 5142 y Fg(2)1634 5164 y Fr(\))q(.)35 b(Using)22 b(the)h(equalit)n(y)f(sinh)13 b Fo(x)24 b Fr(=)f Fo(e)2761 5134 y Fj(x)2802 5164 y Fr(\(1)8 b Fn(\000)g Fo(e)2996 5134 y Fh(\000)p Fk(2)p Fj(x)3122 5164 y Fr(\))p Fo(=)p Fr(2,)23 b(w)n(e)e(obtain)i(from)f(\(34\))-260 5264 y(the)28 b(follo)n(wing)f(expression)f(for)h(the)h(co)r(e\016cien) n(ts:)1619 5364 y Fo(L)1676 5376 y Fj(k)1740 5364 y Fr(=)22 b Fo(\013)1880 5376 y Fj(k)1921 5364 y Fo(e)1960 5330 y Fh(\000)p Fj(\014)2050 5339 y Fd(k)2090 5364 y Fo(;)1732 b Fr(\(35\))-260 5514 y(where)723 5641 y Fo(\013)776 5653 y Fj(k)840 5641 y Fr(=)1352 5585 y(4)p Fo(\031)s(b)1480 5555 y Fh(0)1502 5585 y Fo(\015)1545 5597 y Fj(k)p 937 5622 1063 4 v 937 5728 a Fo(b)973 5704 y Fk(2)1024 5728 y Fn(j)p Fo(k)s Fn(j)1130 5668 y(p)p 1199 5668 39 4 v 60 x Fo(")1252 5636 y Fi(h)1291 5728 y Fr(1)18 b Fn(\000)g Fr(exp)1575 5636 y Fi(n)1630 5728 y Fn(\000)1718 5691 y Fj(\031)r(b)1788 5674 y Fc(0)1811 5691 y Fj(\015)1846 5700 y Fd(k)p 1704 5709 192 4 v 1704 5760 a Fj(b)p Fh(j)p Fj(k)q Fh(j)1810 5718 y(p)p 1864 5718 32 3 v 1864 5760 a Fj(")1906 5636 y Fi(oi)2010 5641 y Fo(;)180 b(\014)2260 5653 y Fj(k)2324 5641 y Fr(=)23 b Fo(\032)14 b Fn(j)o Fo(k)s Fn(j)19 b Fr(+)2728 5585 y Fo(\031)s(b)2814 5555 y Fh(0)2837 5585 y Fo(\015)2880 5597 y Fj(k)p 2672 5622 305 4 v 2672 5699 a Fr(2)p Fo(b)14 b Fn(j)o Fo(k)s Fn(j)2869 5639 y(p)p 2938 5639 39 4 v 60 x Fo(")2986 5641 y(:)836 b Fr(\(36\))1825 5989 y(10)p eop %%Page: 11 11 11 10 bop -260 -82 a Fr(The)28 b(largest)e(co)r(e\016cien)n(ts)h Fo(L)656 -70 y Fj(k)724 -82 y Fr(will)h(b)r(e)g(giv)n(en)f(b)n(y)g(the) h(smallest)f(exp)r(onen)n(ts)g Fo(\014)2225 -70 y Fj(k)2266 -82 y Fr(.)-135 117 y(A)h(more)e(suitable)i(expression)e(for)h(those)g (exp)r(onen)n(ts)h(is:)1093 346 y Fo(\014)1140 358 y Fj(k)1204 346 y Fr(=)1301 290 y Fo(C)1360 302 y Fj(\026)1405 229 y Fn(p)p 1474 229 84 4 v 1477 290 a Fr(~)-45 b Fo(\015)1517 302 y Fj(k)p 1301 327 257 4 v 1337 405 a Fr(2)p Fo(")1418 381 y Fk(1)p Fj(=)p Fk(4)1582 229 y Fi(\022)1662 290 y Fn(j)p Fo(k)s Fn(j)14 b Fo(")1807 260 y Fk(1)p Fj(=)p Fk(4)p 1653 327 267 4 v 1653 405 a Fo(D)1722 417 y Fj(\026)1766 344 y Fn(p)p 1836 344 84 4 v 1839 405 a Fr(~)-45 b Fo(\015)1879 417 y Fj(k)1948 346 y Fr(+)2041 290 y Fo(D)2110 302 y Fj(\026)2154 229 y Fn(p)p 2223 229 V 2226 290 a Fr(~)g Fo(\015)2266 302 y Fj(k)p 2041 327 267 4 v 2050 405 a Fn(j)p Fo(k)s Fn(j)13 b Fo(")2194 381 y Fk(1)p Fj(=)p Fk(4)2317 229 y Fi(\023)2392 346 y Fo(;)1430 b Fr(\(37\))1093 616 y Fo(C)1152 628 y Fj(\026)1220 616 y Fr(=)1307 469 y Fi(s)p 1390 469 324 4 v 1455 560 a Fr(2)p Fo(\031)s(b)1583 536 y Fh(0)1606 560 y Fo(\032)p 1400 597 304 4 v 1400 673 a(b)p Fr(\(2)18 b Fn(\000)g Fr(\012\))1714 616 y Fo(;)96 b(D)1902 628 y Fj(\026)1970 616 y Fr(=)2057 469 y Fi(s)p 2140 469 408 4 v 2290 560 a Fo(\031)s(b)2376 536 y Fh(0)p 2150 597 388 4 v 2150 673 a Fr(2)p Fo(b)p Fr(\(2)18 b Fn(\000)g Fr(\012\))p Fo(\032)2571 616 y Fr(=)2669 560 y Fo(C)2728 572 y Fj(\026)p 2669 597 104 4 v 2678 673 a Fr(2)p Fo(\032)2782 616 y(;)-260 864 y Fr(where)28 b(w)n(e)g(denote)j(~)-45 b Fo(\015)415 876 y Fj(k)480 864 y Fr(=)23 b(\(2)c Fn(\000)f Fr(\012\))p Fo(\015)879 876 y Fj(k)920 864 y Fr(,)29 b(analogously)d(to)i(\(31\).) 38 b(Note)28 b(that)h Fo(C)2175 876 y Fj(\026)2244 864 y Fr(=)23 b Fo(C)2391 876 y Fk(0)2448 864 y Fr(+)18 b Fn(O)e Fr(\()q Fo(\026\016)2736 834 y Fh(\000)p Fj(q)2818 842 y Fg(2)2855 864 y Fr(\),)28 b Fo(D)3007 876 y Fj(\026)3076 864 y Fr(=)23 b Fo(D)3233 876 y Fk(0)3289 864 y Fr(+)c Fn(O)d Fr(\()p Fo(\026\016)3577 834 y Fh(\000)p Fj(q)3659 842 y Fg(2)3696 864 y Fr(\),)29 b(where)-260 964 y Fo(C)-201 976 y Fk(0)-136 964 y Fr(is)f(giv)n(en)f(in)h(\(24\))f(and)g Fo(D)667 976 y Fk(0)727 964 y Fr(=)c Fo(C)874 976 y Fk(0)911 964 y Fo(=)p Fr(2)p Fo(\032)p Fr(.)36 b(W)-7 b(e)28 b(deduce)g(from)f (\(37\))h(the)g(lo)n(w)n(er)e(b)r(ound)1617 1192 y Fo(\014)1664 1204 y Fj(k)1728 1192 y Fn(\025)1825 1136 y Fo(C)1884 1148 y Fj(\026)1929 1074 y Fn(p)p 1998 1074 84 4 v 2001 1136 a Fr(~)-45 b Fo(\015)2041 1148 y Fj(k)p 1825 1173 257 4 v 1882 1251 a Fo(")1921 1227 y Fk(1)p Fj(=)p Fk(4)2092 1192 y Fo(;)1730 b Fr(\(38\))-260 1396 y(whic)n(h)28 b(suggests)f(that)i(the)g(size)f(of)g(the)g(exp)r(onen)n(t)h Fo(\014)1430 1408 y Fj(k)1499 1396 y Fr(is)f(strongly)f(related)h(to)g (the)g(sequence)g Fo(s)p Fr(\()p Fo(j;)14 b Fn(\001)p Fr(\),)30 b(de\014ned)e(in)h(\(29\),)f(to)g(whic)n(h)g Fo(k)-260 1496 y Fr(b)r(elongs,)g(due)h(to)g(the)g(fact)g(that)g(the)g (n)n(umerators)d(tend)k(to)e(a)g(constan)n(t)g(for)g(eac)n(h)g (sequence.)40 b(Indeed,)29 b(recall)e(from)i(Prop)r(osition)e(3)-260 1595 y(that,)d(for)e Fo(k)k Fr(b)r(elonging)c(to)g(the)h(\(classical\)) f(Fib)r(onacci)h(sequence)f Fo(s)p Fr(\(1)p Fo(;)14 b Fn(\001)p Fr(\),)24 b(the)f(limit)g(of)g(the)j(~)-45 b Fo(\015)2682 1607 y Fj(k)2745 1595 y Fr(is)26 b(~)-45 b Fo(\015)2872 1565 y Fh(\003)2867 1616 y Fk(1)2933 1595 y Fr(=)22 b(1,)i(whereas)d(the)i(limits)g(for)f(all)-260 1695 y(other)f(sequences)f Fo(s)p Fr(\()p Fo(j;)14 b Fn(\001)p Fr(\),)24 b Fo(j)k(>)22 b Fr(1,)h(are)g(~)-45 b Fo(\015)982 1665 y Fh(\003)977 1717 y Fj(j)1043 1695 y Fn(\025)25 b Fr(~)-45 b Fo(\015)1178 1665 y Fh(\003)1173 1716 y Fk(4)1239 1695 y Fr(=)23 b(4)e(\(recall)f(that)i(the)j(~)-46 b Fo(\015)1995 1665 y Fh(\003)1990 1717 y Fj(j)2055 1695 y Fr(ha)n(v)n(e)20 b(b)r(een)h(de\014ned)h(in)g(\(31\)\).)34 b(This)22 b(sa)n(ys)d(that)j(the)g(smallest)-260 1795 y(exp)r(onen)n(ts)28 b(can)g(b)r(e)h(found)g(among)f(the)g(Fib)r (onacci)h(sequence.)39 b(W)-7 b(e)29 b(sho)n(w)e(b)r(elo)n(w)h(in)h (Lemma)f(4)g(that)h(the)g(3)f(dominan)n(t)g(harmonics)-260 1894 y(are)35 b(giv)n(en)g(b)n(y)g(3)g(consecutiv)n(e)g(Fib)r(onacci)h (harmonics,)g(and)g(w)n(e)f(pro)n(vide)f(an)i(estimate)f(for)h(their)f (size)h(through)e(the)j(asso)r(ciated)-260 1994 y(exp)r(onen)n(ts)30 b Fo(\014)180 2006 y Fj(k)221 1994 y Fr(.)44 b(W)-7 b(e)31 b(also)e(pro)n(vide)f(an)i(estimate)g(for)g(the)g(di\013erence)g(b)r (et)n(w)n(een)g(the)h(Melnik)n(o)n(v)e(p)r(oten)n(tial)h(and)g(its)g (appro)n(ximation)-260 2094 y(b)n(y)d(the)h(2)g(or)e(3)i(dominan)n(t)f (harmonics,)f(in)i(terms)f(of)h(the)g(follo)n(wing)e(dominan)n(t)i (harmonic.)-135 2293 y(W)-7 b(e)28 b(recall)e(the)i(decreasing)e (sequence)h Fo(")1157 2305 y Fj(n)1230 2293 y Fr(previously)f (de\014ned)i(in)g(\(21\),)f(and)h(de\014ne)g(a)f(new)h(sequence)f Fo(")3233 2263 y Fh(0)3233 2313 y Fj(n)3305 2293 y Fr(as)g(follo)n(ws:) 698 2467 y Fo(")737 2479 y Fj(n)805 2467 y Fr(=)22 b(\(\(2)d Fn(\000)f Fr(\012\)\012)1252 2433 y Fj(n)p Fk(+1)1382 2467 y Fo(D)1451 2479 y Fk(0)1487 2467 y Fr(\))1519 2433 y Fk(4)1580 2467 y Fr(=)23 b(\012)1728 2433 y Fk(4)p Fj(n)1806 2467 y Fo(")1845 2479 y Fk(0)1882 2467 y Fo(;)180 b(")2124 2433 y Fh(0)2124 2488 y Fj(n)2192 2467 y Fr(=)2279 2413 y Fn(p)p 2349 2413 253 4 v 2349 2467 a Fo(")2388 2479 y Fj(n)2433 2467 y Fo(")2472 2479 y Fj(n)p Fh(\000)p Fk(1)2625 2467 y Fr(=)22 b(\012)2772 2433 y Fk(4)p Fj(n)p Fh(\000)p Fk(2)2935 2467 y Fo(")2974 2479 y Fk(0)3011 2467 y Fo(:)-260 2642 y Fr(W)-7 b(e)28 b(no)n(w)f(in)n(tro)r(duce)g (the)h(functions)1009 2871 y Fo(g)1049 2883 y Fj(n)1094 2871 y Fr(\()p Fo(")p Fr(\))23 b(=)1318 2814 y(1)p 1318 2851 42 4 v 1318 2927 a(2)1383 2729 y Fi(")1432 2753 y(\022)1525 2814 y Fo(")p 1503 2851 84 4 v 1503 2927 a(")1542 2939 y Fj(n)1596 2753 y Fi(\023)1658 2771 y Fk(1)p Fj(=)p Fk(4)1780 2871 y Fr(+)1863 2778 y Fi(\020)1923 2814 y Fo(")1962 2826 y Fj(n)p 1923 2851 V 1946 2927 a Fo(")2017 2778 y Fi(\021)2066 2796 y Fk(1)p Fj(=)p Fk(4)2171 2729 y Fi(#)2242 2871 y Fr(=)g Fo(g)2370 2883 y Fk(0)2407 2871 y Fr(\(\012)2499 2836 y Fh(\000)p Fk(4)p Fj(n)2629 2871 y Fo(")p Fr(\))p Fo(:)-260 3115 y Fr(It)31 b(is)g(clear)f(that)h(eac)n(h)f Fo(g)535 3127 y Fj(n)610 3115 y Fr(has)g(its)h(minim)n(um)h(at)e Fo(")e Fr(=)g Fo(")1558 3127 y Fj(n)1603 3115 y Fr(.)47 b(Notice)30 b(that,)i(as)e(a)h(function)g(of)g(ln)14 b Fo(")p Fr(,)31 b(the)g(graph)f(of)h Fo(g)3446 3127 y Fj(n)3521 3115 y Fr(is)g(simply)g(the)-260 3215 y(graph)d(of)h Fo(g)113 3227 y Fk(0)180 3215 y Fr(translated)f(a)h(distance)g(4)p Fo(n)14 b Fr(ln)f(\012.)42 b(This)29 b(is)g(illustrated)g(in)g(Figure)g (1,)g(using)g(logarithmic)f(scale)g(for)h Fo(")g Fr(for)g(the)g(sak)n (e)f(of)-260 3314 y(clarit)n(y)-7 b(.)-135 3514 y(W)g(e)28 b(also)e(de\014ne)i(the)g(follo)n(wing)f(functions:)525 3684 y Fo(h)573 3696 y Fk(1)610 3684 y Fr(\()p Fo(")p Fr(\))c(=)g Fo(g)864 3696 y Fj(n)909 3684 y Fr(\()p Fo(")p Fr(\))p Fo(;)1524 b Fr(for)27 b Fo(")c Fn(2)2827 3616 y Fi(\002)2861 3684 y Fo(")2900 3654 y Fh(0)2900 3704 y Fj(n)p Fk(+1)3029 3684 y Fo(;)14 b(")3105 3654 y Fh(0)3105 3704 y Fj(n)3150 3616 y Fi(\003)3185 3684 y Fr(,)525 3817 y Fo(h)573 3829 y Fk(2)610 3817 y Fr(\()p Fo(")p Fr(\))23 b(=)g Fo(g)864 3829 y Fj(n)p Fk(+1)993 3817 y Fr(\()p Fo(")p Fr(\))p Fo(;)84 b(h)1251 3829 y Fk(3)1288 3817 y Fr(\()p Fo(")p Fr(\))23 b(=)g Fo(g)1542 3829 y Fj(n)p Fh(\000)p Fk(1)1672 3817 y Fr(\()p Fo(")p Fr(\))p Fo(;)83 b(h)1929 3829 y Fk(4)1966 3817 y Fr(\()p Fo(")p Fr(\))24 b(=)e Fo(g)2220 3829 y Fj(n)p Fk(+2)2349 3817 y Fr(\()p Fo(")p Fr(\))p Fo(;)84 b Fr(for)27 b Fo(")c Fn(2)2827 3750 y Fi(\002)2861 3817 y Fo(")2900 3787 y Fh(0)2900 3838 y Fj(n)p Fk(+1)3029 3817 y Fo(;)14 b(")3105 3829 y Fj(n)3150 3750 y Fi(\003)3185 3817 y Fr(,)525 3950 y Fo(h)573 3962 y Fk(2)610 3950 y Fr(\()p Fo(")p Fr(\))23 b(=)g Fo(g)864 3962 y Fj(n)p Fh(\000)p Fk(1)994 3950 y Fr(\()p Fo(")p Fr(\))p Fo(;)83 b(h)1251 3962 y Fk(3)1288 3950 y Fr(\()p Fo(")p Fr(\))23 b(=)g Fo(g)1542 3962 y Fj(n)p Fk(+1)1671 3950 y Fr(\()p Fo(")p Fr(\))p Fo(;)84 b(h)1929 3962 y Fk(4)1966 3950 y Fr(\()p Fo(")p Fr(\))24 b(=)e Fo(g)2220 3962 y Fj(n)p Fh(\000)p Fk(2)2350 3950 y Fr(\()p Fo(")p Fr(\))p Fo(;)83 b Fr(for)27 b Fo(")c Fn(2)h Fr([)p Fo(")2889 3962 y Fj(n)2934 3950 y Fo(;)14 b(")3010 3920 y Fh(0)3010 3971 y Fj(n)3054 3950 y Fr(].)3845 3817 y(\(39\))-260 4134 y(By)34 b(connecting)g(the)g(successiv)n(e)f (in)n(terv)-5 b(als)1178 4067 y Fi(\002)1213 4134 y Fo(")1252 4104 y Fh(0)1252 4155 y Fj(n)p Fk(+1)1381 4134 y Fo(;)14 b(")1457 4104 y Fh(0)1457 4154 y Fj(n)1502 4067 y Fi(\003)1536 4134 y Fr(,)36 b(w)n(e)e(get)g(that)h(these)f(functions)g(are)f(con)n (tin)n(uous)h(on)f(the)i(whole)f(in)n(terv)-5 b(al)-260 4242 y(\(0)p Fo(;)14 b Fn(1)p Fr(\),)33 b(and)e(satisfy)h(the)g (equalit)n(y)f Fo(h)966 4254 y Fj(i)1007 4175 y Fi(\000)1045 4242 y Fr(\012)1105 4212 y Fk(4)1142 4242 y Fo(")1181 4175 y Fi(\001)1249 4242 y Fr(=)e Fo(h)1391 4254 y Fj(i)1418 4242 y Fr(\()p Fo(")p Fr(\))j(for)f(an)n(y)g Fo(")f(>)f Fr(0.)49 b(In)31 b(other)g(w)n(ords,)h(the)g(functions)g Fo(h)3274 4254 y Fj(i)3333 4242 y Fr(are)e(4)14 b(ln)g(\012-p)r(erio)r (dic)-260 4342 y(in)27 b(ln)14 b Fo(")p Fr(.)37 b(The)27 b(functions)g Fo(h)593 4354 y Fj(i)621 4342 y Fr(\()p Fo(")p Fr(\))g(are)f(also)g(illustrated)h(in)h(Figure)e(1.)36 b(It)28 b(is)f(not)g(hard)f(to)h(c)n(hec)n(k)f(that)i(the)f(functions)g Fo(h)3475 4354 y Fk(1)3512 4342 y Fr(\()p Fo(")p Fr(\))h(and)f Fo(h)3852 4354 y Fk(2)3889 4342 y Fr(\()p Fo(")p Fr(\))-260 4441 y(are)g(the)h(ones)f(de\014ned)h(in)g(\(22{23\).)-135 4640 y(W)-7 b(e)28 b(can)f(easily)g(c)n(hec)n(k)f(that)491 4815 y(1)c Fn(\024)h Fo(h)691 4827 y Fk(1)728 4815 y Fr(\()p Fo(")p Fr(\))g Fn(\024)g Fr(1)p Fo(:)p Fr(029086)d Fn(\024)i Fo(h)1414 4827 y Fk(2)1452 4815 y Fr(\()p Fo(")p Fr(\))h Fn(\024)g Fr(1)p Fo(:)p Fr(118034)c Fn(\024)k Fo(h)2138 4827 y Fk(3)2175 4815 y Fr(\()p Fo(")p Fr(\))g Fn(\024)g Fr(1)p Fo(:)p Fr(272020)d Fn(\024)i Fo(h)2861 4827 y Fk(4)2898 4815 y Fr(\()p Fo(")p Fr(\))i Fn(\024)e Fr(1)p Fo(:)p Fr(5)p Fo(;)603 b Fr(\(40\))-260 4990 y(and)24 b(equalities)f(can)g(tak)n(e)g(place)h(only)f(for)g Fo(")g Fr(=)g Fo(")1284 5002 y Fj(n)1329 4990 y Fo(;)14 b(")1405 4959 y Fh(0)1405 5010 y Fj(n)1449 4990 y Fr(.)36 b(More)23 b(precisely)-7 b(,)24 b(for)f Fo(")g Fr(=)f Fo(")2383 5002 y Fj(n)2452 4990 y Fr(w)n(e)h(ha)n(v)n(e)g Fo(h)2806 5002 y Fk(1)2866 4990 y Fo(<)g(h)3002 5002 y Fk(2)3062 4990 y Fr(=)g Fo(h)3198 5002 y Fk(3)3258 4990 y Fo(<)f(h)3393 5002 y Fk(4)3430 4990 y Fr(,)j(and)f(for)f Fo(")g Fr(=)f Fo(")3947 4959 y Fh(0)3947 5010 y Fj(n)-260 5089 y Fr(w)n(e)27 b(ha)n(v)n(e)g Fo(h)102 5101 y Fk(1)162 5089 y Fr(=)c Fo(h)298 5101 y Fk(2)358 5089 y Fo(<)f(h)493 5101 y Fk(3)553 5089 y Fr(=)h Fo(h)689 5101 y Fk(4)726 5089 y Fr(.)-135 5288 y(F)-7 b(or)30 b(an)n(y)g(giv)n(en)g Fo(")e(<)g(")596 5300 y Fk(1)633 5288 y Fr(,)k(w)n(e)e(de\014ne)h Fo(N)1123 5300 y Fj(i)1179 5288 y Fr(=)d Fo(N)1339 5300 y Fj(i)1366 5288 y Fr(\()p Fo(")p Fr(\),)k Fo(i)c Fr(=)g(1)p Fo(;)14 b Fr(2)p Fo(;)g Fr(3)p Fo(;)g Fr(4,)29 b(as)i(the)g(4)f(in)n(tegers)g 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b(the)f(\014rst)g(mimim)n(um)h(is)e(giv)n(en)g(b)n(y)h Fo(N)2657 5644 y Fk(1)2717 5632 y Fr(=)g Fo(n)p Fr(.)35 b(The)22 b(second)f(and)h(third)g(mimima)-260 5740 y(are)33 b Fo(N)-48 5752 y Fk(2)24 5740 y Fr(=)g Fo(n)23 b Fn(\006)g Fr(1,)35 b Fo(N)450 5752 y Fk(3)521 5740 y Fr(=)f Fo(n)23 b Fn(\007)f Fr(1)34 b(dep)r(ending)h(on)f(the)g(subin)n(terv)-5 b(al)34 b(to)g(whic)n(h)g Fo(")g Fr(b)r(elongs:)2734 5673 y Fi(\000)2772 5740 y Fo(")2811 5710 y Fh(0)2811 5761 y Fj(n)p Fk(+1)2940 5740 y Fo(;)14 b(")3016 5752 y Fj(n)3061 5673 y Fi(\001)3133 5740 y Fr(or)34 b(\()p Fo(")3313 5752 y Fj(n)3358 5740 y Fo(;)14 b(")3434 5710 y Fh(0)3434 5761 y Fj(n)3479 5740 y Fr(\).)57 b(The)34 b(fourth)1825 5989 y(11)p eop %%Page: 12 12 12 11 bop -260 -82 a Fr(minim)n(um)26 b(is)g Fo(N)258 -70 y Fk(4)318 -82 y Fr(=)c Fo(n)14 b Fn(\006)g Fr(2.)35 b(The)26 b(main)f(fact)h(that)f(w)n(e)g(shall)g(use)h(is)f(that)h(the)f (v)-5 b(alues)25 b(of)h(the)f(4)g(minima)h(are)e(giv)n(en)h(b)n(y)g (the)h(functions)-260 17 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0 V 61 0 V 60 0 V 61 0 V 61 0 V 61 0 V 61 0 V 61 0 V 61 0 V 60 0 V 61 0 V 61 0 V 61 0 V 61 0 V 61 0 V 61 0 V 60 0 V 61 0 V 61 0 V 61 0 V 61 0 V 61 0 V 60 0 V 61 0 V 61 0 V 61 0 V 61 0 V 61 0 V 61 0 V 60 0 V 61 0 V 61 0 V 61 0 V 61 0 V 61 0 V 60 0 V 61 0 V 61 0 V 61 0 V 61 0 V 61 0 V 61 0 V 60 0 V 61 0 V 61 0 V 61 0 V 61 0 V 61 0 V 61 0 V 60 0 V 61 0 V 61 0 V 61 0 V 61 0 V 61 0 V 60 0 V 61 0 V 61 0 V 61 0 V 1.000 UP 1.000 UL LT6 756 1300 Pnt 817 1300 Pnt 878 1300 Pnt 939 1300 Pnt 999 1300 Pnt 1060 1300 Pnt 1121 1300 Pnt 1182 1300 Pnt 1243 1300 Pnt 1304 1300 Pnt 1364 1300 Pnt 1425 1300 Pnt 1486 1300 Pnt 1547 1300 Pnt 1608 1300 Pnt 1669 1300 Pnt 1730 1300 Pnt 1790 1300 Pnt 1851 1300 Pnt 1912 1300 Pnt 1973 1300 Pnt 2034 1300 Pnt 2095 1300 Pnt 2156 1300 Pnt 2216 1300 Pnt 2277 1300 Pnt 2338 1300 Pnt 2399 1300 Pnt 2460 1300 Pnt 2521 1300 Pnt 2581 1300 Pnt 2642 1300 Pnt 2703 1300 Pnt 2764 1300 Pnt 2825 1300 Pnt 2886 1300 Pnt 2947 1300 Pnt 3007 1300 Pnt 3068 1300 Pnt 3129 1300 Pnt 3190 1300 Pnt 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1425 1449 Pnt 1486 1449 Pnt 1547 1449 Pnt 1608 1449 Pnt 1669 1449 Pnt 1730 1449 Pnt 1790 1449 Pnt 1851 1449 Pnt 1912 1449 Pnt 1973 1449 Pnt 2034 1449 Pnt 2095 1449 Pnt 2156 1449 Pnt 2216 1449 Pnt 2277 1449 Pnt 2338 1449 Pnt 2399 1449 Pnt 2460 1449 Pnt 2521 1449 Pnt 2581 1449 Pnt 2642 1449 Pnt 2703 1449 Pnt 2764 1449 Pnt 2825 1449 Pnt 2886 1449 Pnt 2947 1449 Pnt 3007 1449 Pnt 3068 1449 Pnt 3129 1449 Pnt 3190 1449 Pnt 3251 1449 Pnt 3312 1449 Pnt 3372 1449 Pnt 3433 1449 Pnt 3494 1449 Pnt 3555 1449 Pnt 3616 1449 Pnt 3677 1449 Pnt 3738 1449 Pnt 3798 1449 Pnt 3859 1449 Pnt 3920 1449 Pnt 3981 1449 Pnt 4042 1449 Pnt 4103 1449 Pnt 4164 1449 Pnt 4224 1449 Pnt 4285 1449 Pnt 4346 1449 Pnt 4407 1449 Pnt 4468 1449 Pnt 4529 1449 Pnt 4589 1449 Pnt 4650 1449 Pnt 4711 1449 Pnt 4772 1449 Pnt 4833 1449 Pnt 4894 1449 Pnt 4955 1449 Pnt 5015 1449 Pnt 5076 1449 Pnt 5137 1449 Pnt 5198 1449 Pnt 5259 1449 Pnt 5320 1449 Pnt 5380 1449 Pnt 5441 1449 Pnt 5502 1449 Pnt 5563 1449 Pnt 5624 1449 Pnt 5685 1449 Pnt 5746 1449 Pnt 5806 1449 Pnt 5867 1449 Pnt 5928 1449 Pnt 5989 1449 Pnt 6050 1449 Pnt 6111 1449 Pnt 6172 1449 Pnt 6232 1449 Pnt 6293 1449 Pnt 6354 1449 Pnt 6415 1449 Pnt 6476 1449 Pnt 6537 1449 Pnt 6597 1449 Pnt 6658 1449 Pnt 6719 1449 Pnt 6780 1449 Pnt 1.000 UP 1.000 UL LT8 756 1903 Pnt 817 1903 Pnt 878 1903 Pnt 939 1903 Pnt 999 1903 Pnt 1060 1903 Pnt 1121 1903 Pnt 1182 1903 Pnt 1243 1903 Pnt 1304 1903 Pnt 1364 1903 Pnt 1425 1903 Pnt 1486 1903 Pnt 1547 1903 Pnt 1608 1903 Pnt 1669 1903 Pnt 1730 1903 Pnt 1790 1903 Pnt 1851 1903 Pnt 1912 1903 Pnt 1973 1903 Pnt 2034 1903 Pnt 2095 1903 Pnt 2156 1903 Pnt 2216 1903 Pnt 2277 1903 Pnt 2338 1903 Pnt 2399 1903 Pnt 2460 1903 Pnt 2521 1903 Pnt 2581 1903 Pnt 2642 1903 Pnt 2703 1903 Pnt 2764 1903 Pnt 2825 1903 Pnt 2886 1903 Pnt 2947 1903 Pnt 3007 1903 Pnt 3068 1903 Pnt 3129 1903 Pnt 3190 1903 Pnt 3251 1903 Pnt 3312 1903 Pnt 3372 1903 Pnt 3433 1903 Pnt 3494 1903 Pnt 3555 1903 Pnt 3616 1903 Pnt 3677 1903 Pnt 3738 1903 Pnt 3798 1903 Pnt 3859 1903 Pnt 3920 1903 Pnt 3981 1903 Pnt 4042 1903 Pnt 4103 1903 Pnt 4164 1903 Pnt 4224 1903 Pnt 4285 1903 Pnt 4346 1903 Pnt 4407 1903 Pnt 4468 1903 Pnt 4529 1903 Pnt 4589 1903 Pnt 4650 1903 Pnt 4711 1903 Pnt 4772 1903 Pnt 4833 1903 Pnt 4894 1903 Pnt 4955 1903 Pnt 5015 1903 Pnt 5076 1903 Pnt 5137 1903 Pnt 5198 1903 Pnt 5259 1903 Pnt 5320 1903 Pnt 5380 1903 Pnt 5441 1903 Pnt 5502 1903 Pnt 5563 1903 Pnt 5624 1903 Pnt 5685 1903 Pnt 5746 1903 Pnt 5806 1903 Pnt 5867 1903 Pnt 5928 1903 Pnt 5989 1903 Pnt 6050 1903 Pnt 6111 1903 Pnt 6172 1903 Pnt 6232 1903 Pnt 6293 1903 Pnt 6354 1903 Pnt 6415 1903 Pnt 6476 1903 Pnt 6537 1903 Pnt 6597 1903 Pnt 6658 1903 Pnt 6719 1903 Pnt 6780 1903 Pnt 1.000 UP 1.000 UL LT0 756 2688 Pnt 817 2688 Pnt 878 2688 Pnt 939 2688 Pnt 999 2688 Pnt 1060 2688 Pnt 1121 2688 Pnt 1182 2688 Pnt 1243 2688 Pnt 1304 2688 Pnt 1364 2688 Pnt 1425 2688 Pnt 1486 2688 Pnt 1547 2688 Pnt 1608 2688 Pnt 1669 2688 Pnt 1730 2688 Pnt 1790 2688 Pnt 1851 2688 Pnt 1912 2688 Pnt 1973 2688 Pnt 2034 2688 Pnt 2095 2688 Pnt 2156 2688 Pnt 2216 2688 Pnt 2277 2688 Pnt 2338 2688 Pnt 2399 2688 Pnt 2460 2688 Pnt 2521 2688 Pnt 2581 2688 Pnt 2642 2688 Pnt 2703 2688 Pnt 2764 2688 Pnt 2825 2688 Pnt 2886 2688 Pnt 2947 2688 Pnt 3007 2688 Pnt 3068 2688 Pnt 3129 2688 Pnt 3190 2688 Pnt 3251 2688 Pnt 3312 2688 Pnt 3372 2688 Pnt 3433 2688 Pnt 3494 2688 Pnt 3555 2688 Pnt 3616 2688 Pnt 3677 2688 Pnt 3738 2688 Pnt 3798 2688 Pnt 3859 2688 Pnt 3920 2688 Pnt 3981 2688 Pnt 4042 2688 Pnt 4103 2688 Pnt 4164 2688 Pnt 4224 2688 Pnt 4285 2688 Pnt 4346 2688 Pnt 4407 2688 Pnt 4468 2688 Pnt 4529 2688 Pnt 4589 2688 Pnt 4650 2688 Pnt 4711 2688 Pnt 4772 2688 Pnt 4833 2688 Pnt 4894 2688 Pnt 4955 2688 Pnt 5015 2688 Pnt 5076 2688 Pnt 5137 2688 Pnt 5198 2688 Pnt 5259 2688 Pnt 5320 2688 Pnt 5380 2688 Pnt 5441 2688 Pnt 5502 2688 Pnt 5563 2688 Pnt 5624 2688 Pnt 5685 2688 Pnt 5746 2688 Pnt 5806 2688 Pnt 5867 2688 Pnt 5928 2688 Pnt 5989 2688 Pnt 6050 2688 Pnt 6111 2688 Pnt 6172 2688 Pnt 6232 2688 Pnt 6293 2688 Pnt 6354 2688 Pnt 6415 2688 Pnt 6476 2688 Pnt 6537 2688 Pnt 6597 2688 Pnt 6658 2688 Pnt 6719 2688 Pnt 6780 2688 Pnt 1.000 UP 1.000 UL LT1 756 3852 Pnt 817 3852 Pnt 878 3852 Pnt 939 3852 Pnt 999 3852 Pnt 1060 3852 Pnt 1121 3852 Pnt 1182 3852 Pnt 1243 3852 Pnt 1304 3852 Pnt 1364 3852 Pnt 1425 3852 Pnt 1486 3852 Pnt 1547 3852 Pnt 1608 3852 Pnt 1669 3852 Pnt 1730 3852 Pnt 1790 3852 Pnt 1851 3852 Pnt 1912 3852 Pnt 1973 3852 Pnt 2034 3852 Pnt 2095 3852 Pnt 2156 3852 Pnt 2216 3852 Pnt 2277 3852 Pnt 2338 3852 Pnt 2399 3852 Pnt 2460 3852 Pnt 2521 3852 Pnt 2581 3852 Pnt 2642 3852 Pnt 2703 3852 Pnt 2764 3852 Pnt 2825 3852 Pnt 2886 3852 Pnt 2947 3852 Pnt 3007 3852 Pnt 3068 3852 Pnt 3129 3852 Pnt 3190 3852 Pnt 3251 3852 Pnt 3312 3852 Pnt 3372 3852 Pnt 3433 3852 Pnt 3494 3852 Pnt 3555 3852 Pnt 3616 3852 Pnt 3677 3852 Pnt 3738 3852 Pnt 3798 3852 Pnt 3859 3852 Pnt 3920 3852 Pnt 3981 3852 Pnt 4042 3852 Pnt 4103 3852 Pnt 4164 3852 Pnt 4224 3852 Pnt 4285 3852 Pnt 4346 3852 Pnt 4407 3852 Pnt 4468 3852 Pnt 4529 3852 Pnt 4589 3852 Pnt 4650 3852 Pnt 4711 3852 Pnt 4772 3852 Pnt 4833 3852 Pnt 4894 3852 Pnt 4955 3852 Pnt 5015 3852 Pnt 5076 3852 Pnt 5137 3852 Pnt 5198 3852 Pnt 5259 3852 Pnt 5320 3852 Pnt 5380 3852 Pnt 5441 3852 Pnt 5502 3852 Pnt 5563 3852 Pnt 5624 3852 Pnt 5685 3852 Pnt 5746 3852 Pnt 5806 3852 Pnt 5867 3852 Pnt 5928 3852 Pnt 5989 3852 Pnt 6050 3852 Pnt 6111 3852 Pnt 6172 3852 Pnt 6232 3852 Pnt 6293 3852 Pnt 6354 3852 Pnt 6415 3852 Pnt 6476 3852 Pnt 6537 3852 Pnt 6597 3852 Pnt 6658 3852 Pnt 6719 3852 Pnt 6780 3852 Pnt stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument @endspecial 515 3773 a /End PSfrag 515 3773 a 515 1092 a /Hide PSfrag 515 1092 a -225 1150 a Ff(PSfrag)26 b(replacemen)n(ts)p -225 1180 741 4 v 515 1183 a /Unhide PSfrag 515 1183 a 431 1255 a { 431 1255 a 347 1263 a Fe(")386 1275 y Fd(n)p Fc(+2)431 1255 y } 0/Place PSfrag 431 1255 a 431 1337 a { 431 1337 a 347 1355 a Fe(")386 1324 y Fb(0)386 1375 y Fd(n)p Fc(+2)431 1337 y } 1/Place PSfrag 431 1337 a 431 1454 a { 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Fj(S)950 -35 y Fd(l)p Fg(+1)1048 -56 y Fo(;)99 b Fr(0)22 b Fn(\024)h Fo(l)h Fn(\024)f Fr(3)p Fo(;)98 b(m)23 b Fn(\025)g Fr(0)p Fo(:)-260 311 y Fp(Pr)l(o)l(of.)69 b Fr(W)-7 b(e)34 b(need)f(to)h(sho)n(w)e(that)i (the)g(largest)e(co)r(e\016cien)n(ts)h Fo(L)1785 323 y Fj(k)1859 311 y Fr(are)g(the)h(ones)e(giv)n(en)h(b)n(y)g Fo(k)j Fr(=)d Fo(S)2915 323 y Fj(i)2942 311 y Fr(.)55 b(According)32 b(to)i(\(35{36\),)f(this)-260 411 y(corresp)r(onds)22 b(essen)n(tially)g(to)h(\014nd)h(the)g(smallest)f(exp)r(onen)n(ts)f Fo(\014)1730 423 y Fj(k)1771 411 y Fr(.)36 b(T)-7 b(o)23 b(b)r(egin,)h(w)n(e)f(study)h(the)f(co)r(e\016cien)n(ts)g(indexed)h(b)n (y)f(the)g(Fib)r(onacci)-260 511 y(sequence:)37 b Fo(k)25 b Fr(=)e Fo(s)p Fr(\(1)p Fo(;)14 b(n)p Fr(\).)37 b(W)-7 b(e)28 b(deduce)f(from)h(Prop)r(osition)e(3,)h(using)g Fo(m)p Fr(\(1\))c(=)g(1)k(and)g Fo(d)p Fr(\(1\))d(=)e(\012)d Fn(\000)f Fr(1,)27 b(that)777 747 y Fn(j)p Fo(s)p Fr(\(1)p Fo(;)14 b(n)p Fr(\))p Fn(j)23 b Fr(=)1175 691 y(\012)1235 660 y Fh(\000)p Fk(\()p Fj(n)p Fk(+1\))p 1175 728 293 4 v 1220 804 a Fr(2)18 b Fn(\000)g Fr(\012)1497 747 y(+)g Fn(O)1662 680 y Fi(\000)1700 747 y Fr(\012)1760 713 y Fj(n)p Fk(+1)1889 680 y Fi(\001)1941 747 y Fo(;)183 b Fr(~)-45 b Fo(\015)2187 762 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(\))2391 747 y Fr(=)23 b(1)18 b(+)g Fn(O)2704 680 y Fi(\000)2742 747 y Fr(\012)2802 713 y Fk(2)p Fj(n)2881 680 y Fi(\001)2932 747 y Fo(:)890 b Fr(\(44\))-260 958 y(Then,)28 b(from)f(\(37\))h(and)f(the)h(de\014nition)g(of)f Fo(")1158 970 y Fj(n)1231 958 y Fr(w)n(e)g(obtain)h(the)g(appro)n (ximation)801 1195 y Fo(\014)848 1210 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(\))1052 1195 y Fn(')1193 1139 y Fo(C)1252 1151 y Fk(0)p 1149 1176 185 4 v 1149 1253 a Fr(2)p Fo(")1230 1229 y Fk(1)p Fj(=)p Fk(4)1358 1078 y Fi(\022)1429 1139 y Fr(\012)1489 1108 y Fh(\000)p Fk(\()p Fj(n)p Fk(+1\))1722 1139 y Fo(")1761 1108 y Fk(1)p Fj(=)p Fk(4)p 1429 1176 437 4 v 1460 1252 a Fr(\(2)18 b Fn(\000)g Fr(\012\))p Fo(D)1796 1264 y Fk(0)1893 1195 y Fr(+)2018 1139 y(\(2)g Fn(\000)g Fr(\012\))p Fo(D)2354 1151 y Fk(0)p 1986 1176 V 1986 1253 a Fr(\012)2046 1229 y Fh(\000)p Fk(\()p Fj(n)p Fk(+1\))2279 1253 y Fo(")2318 1229 y Fk(1)p Fj(=)p Fk(4)2432 1078 y Fi(\023)2516 1195 y Fr(=)2614 1139 y Fo(C)2673 1151 y Fk(0)2711 1139 y Fo(g)2751 1151 y Fj(n)2795 1139 y Fr(\()p Fo(")p Fr(\))p 2614 1176 285 4 v 2685 1253 a Fo(")2724 1229 y Fk(1)p Fj(=)p Fk(4)2909 1195 y Fo(;)913 b Fr(\(45\))-260 1417 y(and)28 b(hence)f(the)h(functions)g Fo(g)673 1429 y Fj(n)746 1417 y Fr(con)n(tain)e(the)i(main)g (information)f(on)g(the)h(size)f(of)h(the)g(exp)r(onen)n(ts)f Fo(\014)2929 1432 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(\))3110 1417 y Fr(.)-135 1616 y(Since)j(the)h(4)g(smallest)f(v)-5 b(alues)30 b(of)h Fo(g)1013 1628 y Fj(n)1057 1616 y Fr(\()p Fo(")p Fr(\),)h(for)e Fo(")h Fr(small)f(enough,)h(are)e(the)i(ones)f (obtained)h(for)f Fo(n)e Fr(=)f Fo(N)3135 1628 y Fj(i)3163 1616 y Fr(\()p Fo(")p Fr(\),)32 b(and)e(these)h(smallest)-260 1716 y(v)-5 b(alues)27 b(coincide)h(with)g(the)g(functions)g Fo(h)1044 1728 y Fj(i)1071 1716 y Fr(\()p Fo(")p Fr(\),)g(the)g (smallest)f(exp)r(onen)n(ts)g(among)g(the)h Fo(\014)2535 1731 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(\))2744 1716 y Fr(are)1283 1949 y Fo(\014)1330 1961 y Fj(S)1371 1969 y Fd(i)1424 1949 y Fn(')1522 1893 y Fo(C)1581 1905 y Fk(0)1618 1893 y Fo(h)1666 1905 y Fj(i)1694 1893 y Fr(\()p Fo(")p Fr(\))p 1522 1930 276 4 v 1588 2008 a Fo(")1627 1984 y Fk(1)p Fj(=)p Fk(4)1807 1949 y Fo(;)180 b(i)23 b Fr(=)f(1)p Fo(;)14 b Fr(2)p Fo(;)g Fr(3)p Fo(;)g Fr(4)p Fo(:)-260 2155 y Fr(In)29 b(fact,)g(when)f(w)n(e)h(lo)r(ok)e(at)i(the)g(true)f (exp)r(onen)n(ts)g Fo(\014)1415 2167 y Fj(S)1456 2175 y Fd(i)1486 2155 y Fr(,)h(w)n(e)f(actually)g(ha)n(v)n(e)f(a)i(p)r (erturbation)e(of)i(this)g(situation.)39 b(Substituting)30 b(\(44\))-260 2255 y(in)n(to)e(\(37\))f(and)g(using)g(that)h(\012)702 2225 y Fk(2)p Fj(N)788 2233 y Fd(i)841 2255 y Fn(\030)929 2195 y(p)p 998 2195 39 4 v 60 x Fo(")p Fr(,)g(w)n(e)f(obtain:)1292 2484 y Fo(\014)1339 2496 y Fj(S)1380 2504 y Fd(i)1433 2484 y Fr(=)1530 2428 y Fo(C)1589 2440 y Fk(0)1627 2428 y Fo(h)1675 2440 y Fj(i)1702 2428 y Fr(\()p Fo(")p Fr(\))19 b(+)f Fn(O)e Fr(\()2022 2368 y Fn(p)p 2091 2368 V 60 x Fo(";)e(\026\016)2257 2398 y Fh(\000)p Fj(q)2339 2406 y Fg(2)2375 2428 y Fr(\))p 1530 2465 878 4 v 1898 2542 a Fo(")1937 2518 y Fk(1)p Fj(=)p Fk(4)2418 2484 y Fo(;)-260 2702 y Fr(and)29 b(the)g(p)r(erturbativ)n(e)e(part)i(can)f(b)r(e)h (neglected)f(under)h(the)g(smallness)e(conditions)i(for)f Fo(")g Fr(and)h Fo(\026)p Fr(,)g(since)f(w)n(e)g(ha)n(v)n(e)3465 2642 y Fn(p)p 3534 2642 39 4 v 60 x Fo(")c Fn(\026)h Fo(")3726 2672 y Fk(1)p Fj(=)p Fk(4)3858 2702 y Fr(and)-260 2801 y Fo(\026\016)-170 2771 y Fh(\000)p Fj(q)-88 2779 y Fg(2)-28 2801 y Fn(\026)e Fo(")99 2771 y Fk(1)p Fj(=)p Fk(4)203 2801 y Fr(.)-135 3001 y(W)-7 b(e)29 b(ha)n(v)n(e)f(sho)n(wn)h (that)g(the)g(exp)r(onen)n(ts)g Fo(\014)1220 3013 y Fj(S)1261 3021 y Fd(i)1321 3001 y Fr(are)f(the)h(smallest)g(among)f(the)i(exp)r (onen)n(ts)e Fo(\014)2778 3013 y Fj(k)2848 3001 y Fr(for)h Fo(k)j Fr(in)d(the)h(Fib)r(onacci)f(sequence)-260 3100 y Fo(s)p Fr(\(1)p Fo(;)14 b Fn(\001)p Fr(\),)25 b(but)g(it)g(is)f(also) g(imp)r(ortan)n(t)g(to)g(stress)f(\(and)i(w)n(e)f(use)g(this)h(fact)f (b)r(elo)n(w)g(in)h(this)g(pro)r(of)6 b(\))24 b(that)h(the)g(exp)r (onen)n(ts)f Fo(\014)3434 3112 y Fj(S)3475 3120 y Fd(i)3505 3100 y Fr(,)h Fo(i)e Fr(=)f(1)p Fo(;)14 b Fr(2)p Fo(;)g Fr(3)p Fo(;)g Fr(4,)-260 3200 y(are)27 b(also)f(the)i(smaller)f(than)h (an)n(y)f(non-Fib)r(onacci)f(exp)r(onen)n(t)i Fo(\014)1771 3212 y Fj(k)1812 3200 y Fr(.)37 b(Indeed,)28 b(if)g Fo(k)i Fr(b)r(elongs)d(to)h(another)e(sequence)i Fo(s)p Fr(\()p Fo(j;)14 b Fn(\001)p Fr(\),)28 b(w)n(e)f(use)h(the)-260 3299 y(lo)n(w)n(er)e(b)r(ound)i(\(38\))f(and)h(the)g(fact)g(that)f(the) h(limit)h(of)e(the)1615 3238 y Fn(p)p 1684 3238 84 4 v 1687 3299 a Fr(~)-45 b Fo(\015)1727 3311 y Fj(k)1796 3299 y Fr(is)27 b(giv)n(en)g(b)n(y)1315 3412 y Fi(q)p 1398 3412 86 4 v 1401 3502 a Fr(~)-45 b Fo(\015)1446 3474 y Fh(\003)1441 3525 y Fj(j)1507 3502 y Fn(\025)23 b Fr(2)f Fn(\035)h Fo(h)1813 3514 y Fk(4)1850 3502 y Fr(\()p Fo(")p Fr(\))p Fo(;)181 b Fn(8)p Fo(j)27 b(>)22 b Fr(1)p Fo(:)1428 b Fr(\(46\))-260 3706 y(On)23 b(the)f(other)g(hand,) i(for)e(a)g(non-admissible)g Fo(k)s Fr(,)i(i.e.)e(not)h(b)r(elonging)f (to)g(an)n(y)g(sequence)g Fo(s)p Fr(\()p Fo(j;)14 b Fn(\001)p Fr(\))24 b(\(see)e(Section)h(2\),)g(w)n(e)f(ha)n(v)n(e)g Fo(\015)3652 3718 y Fj(k)3716 3706 y Fo(>)g Fn(j)p Fo(k)s Fn(j)14 b Fo(=)p Fr(2)-260 3806 y(and)28 b(w)n(e)f(alw)n(a)n(ys)e (\014nd)k(in)e(\(36\))h(that)g Fo(\014)955 3818 y Fj(k)1018 3806 y Fn(\027)23 b Fr(1)p Fo(=)1190 3746 y Fn(p)p 1258 3746 39 4 v 1258 3806 a Fo(")28 b Fr(for)f(this)h(case.)-135 4005 y(Once)22 b(w)n(e)g(ha)n(v)n(e)g(found)h(the)g(smallest)f(exp)r (onen)n(ts)g Fo(\014)1488 4017 y Fj(k)1529 4005 y Fr(,)h(w)n(e)g(sho)n (w)e(that)i(the)g(dominance)f(in)h(the)g(co)r(e\016cien)n(ts)g Fo(L)3319 4017 y Fj(k)3382 4005 y Fr(is)f(not)h(a\013ected)f(b)n(y)-260 4105 y(the)29 b(m)n(ultiplicativ)n(e)f(term)g Fo(\013)660 4117 y Fj(k)729 4105 y Fr(in)g(\(35{36\).)36 b(Indeed,)29 b(w)n(e)e(deduce)i(from)e(\(36\))h(that)g Fo(\013)2458 4117 y Fj(k)2523 4105 y Fn(\026)c Fo(\014)2659 4117 y Fj(k)2727 4105 y Fr(in)29 b(general,)e(and)g(hence)i Fn(j)p Fr(ln)14 b Fo(\013)3687 4117 y Fj(k)3728 4105 y Fn(j)23 b(\034)h Fo(\014)3928 4117 y Fj(k)3969 4105 y Fr(.)-260 4205 y(In)30 b(fact,)f(the)h(only)f(p)r(ossible)g (exception)g(could)g(o)r(ccur)f(if)i(the)g(denominator)e([1)18 b Fn(\000)g Fr(exp)13 b Fn(f\001)h(\001)g(\001g)o Fr(])30 b(in)f(\(36\))g(is)g(to)r(o)g(small.)42 b(But)29 b(this)h(only)-260 4304 y(happ)r(ens)21 b(if)h Fn(j)p Fo(k)s Fn(j)h(\027)g Fo(\015)375 4316 y Fj(k)415 4304 y Fo(=)457 4244 y Fn(p)p 526 4244 V 60 x Fo(")p Fr(,)f(and)f(one)g(then)h(obtains)e Fo(\013)1431 4316 y Fj(k)1495 4304 y Fn(\026)j Fr(1.)34 b(F)-7 b(or)21 b(the)g(dominan)n(t)g(co)r(e\016cien)n(ts)g Fo(L)2797 4316 y Fj(S)2838 4324 y Fd(i)2868 4304 y Fr(,)h(w)n(e)f(see)g (that)g([1)d Fn(\000)g Fr(exp)c Fn(f)o(\001)g(\001)g(\001g)p Fr(])23 b Fn(\030)g Fr(1)-260 4404 y(and)28 b(easily)e(\014nd)i(the)g (estimate)g Fo(\013)825 4416 y Fj(S)866 4424 y Fd(i)919 4404 y Fn(\030)23 b Fo(")1046 4374 y Fh(\000)p Fk(1)p Fj(=)p Fk(4)1202 4404 y Fr(.)37 b(This)27 b(estimate,)h(together)f (with)h(the)g(one)f(for)g Fo(\014)2793 4416 y Fj(S)2834 4424 y Fd(i)2864 4404 y Fr(,)h(implies)g(part)f(\(a\).)-135 4603 y(Let)f(us)h(pro)n(v)n(e)e(the)i(b)r(ound)g(of)f(part)h(\(b\))g (for)f Fo(l)e Fr(=)f(1.)36 b(T)-7 b(o)26 b(b)r(ound)h(the)g(sum)g(of)g (the)g(co)r(e\016cien)n(ts)f Fo(L)2917 4615 y Fj(k)2984 4603 y Fr(for)g Fo(k)g Fn(6)p Fr(=)d Fo(S)3318 4615 y Fk(1)3355 4603 y Fr(,)k(w)n(e)f(\014rst)g(consider)-260 4703 y(the)j(ones)e(corresp)r(onding)g(to)h(Fib)r(onacci)g(harmonics:) 37 b Fo(k)27 b Fr(=)c Fo(s)p Fr(\(1)p Fo(;)14 b(n)p Fr(\),)29 b(with)f Fo(n)c Fn(6)p Fr(=)g Fo(N)2378 4715 y Fk(1)2415 4703 y Fr(,)29 b(and)f(divide)g(them)h(in)n(to)f(the)h(ones)e(previous) g(to)-260 4802 y Fo(S)-209 4814 y Fk(1)-140 4802 y Fr(and)32 b(the)g(ones)g(after)f Fo(S)614 4814 y Fk(1)652 4802 y Fr(.)50 b(W)-7 b(e)32 b(call)g Fo(S)1085 4772 y Fh(0)1080 4823 y Fk(2)1149 4802 y Fr(the)g(last)g(Fib)r(onacci)f(previous)g(to)h Fo(S)2333 4814 y Fk(1)2370 4802 y Fr(,)i(and)d Fo(S)2648 4772 y Fh(00)2643 4823 y Fk(2)2722 4802 y Fr(the)i(\014rst)f(Fib)r (onacci)f(after)h Fo(S)3681 4814 y Fk(1)3718 4802 y Fr(;)i(notice)-260 4902 y(that)28 b Fo(S)-29 4914 y Fk(2)36 4902 y Fr(is)f(one)h(of)f Fo(S)422 4872 y Fh(0)417 4923 y Fk(2)454 4902 y Fr(,)h Fo(S)561 4872 y Fh(00)556 4923 y Fk(2)603 4902 y Fr(.)37 b(F)-7 b(or)27 b(an)n(y)g Fo(s)p Fr(\(1)p Fo(;)14 b(n)p Fr(\))27 b(previous)g(to)g Fo(S)1711 4914 y Fk(1)1776 4902 y Fr(\(i.e.)h Fo(n)23 b(<)g(N)2170 4914 y Fk(1)2207 4902 y Fr(\),)28 b(w)n(e)f(use)g(\(45\))h(and)f(\(41\):)1621 5121 y Fo(\014)1668 5136 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(\))1873 5121 y Fn(\030)1971 5065 y Fr(\012)2031 5035 y Fj(n)p 1970 5102 108 4 v 1970 5119 a Fn(p)p 2039 5119 39 4 v 59 x Fo(")2088 5121 y(;)-260 5342 y Fr(and)e(w)n(e)f(get)h(that)g(the)g (corresp)r(onding)e(co)r(e\016cien)n(ts)h Fo(L)1476 5357 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(\))1682 5342 y Fr(gro)n(w)f(faster)h(than) h(geometrically)-7 b(,)24 b(and)g(the)h(same)g(can)f(b)r(e)h(said)g (for)f(the)-260 5441 y(co)r(e\016cien)n(ts)j Fn(j)p Fo(s)p Fr(\(1)p Fo(;)14 b(n)p Fr(\))p Fn(j)438 5399 y Fj(m)515 5441 y Fo(L)572 5456 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(\))780 5441 y Fr(of)28 b(the)g(deriv)-5 b(ativ)n(e.)36 b(Then,)28 b(their)f(sum)h(can)f(b)r(e)h(\(essen)n(tially\))g(estimated)f(b)n(y)h (the)g(last)f(co)r(e\016cien)n(t:)1047 5587 y Fi(X)1018 5765 y Fj(n)g(N)1069 -70 y Fk(1)1106 -82 y Fr(\),)j(w)n(e)d(pro)r(ceed)g(analogously)-7 b(,)33 b(but)i(using)e(\(42\),)i(and)f(deduce)g(that)g(the)g(corresp)r (onding)-260 17 y(co)r(e\016cien)n(ts)27 b Fo(L)217 32 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(\))426 17 y Fr(decrease)f(faster)h(than)h (in)f(a)h(geometric)e(series.)36 b(As)28 b(b)r(efore,)f(w)n(e)g(obtain) 1030 177 y Fi(X)1001 355 y Fj(n>N)1147 363 y Fg(1)1193 256 y Fn(j)p Fo(s)p Fr(\(1)p Fo(;)14 b(n)p Fr(\))p Fn(j)1471 214 y Fj(m)1548 256 y Fo(L)1605 271 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(\))1809 256 y Fn(\030)22 b(j)p Fo(S)1975 222 y Fh(00)1970 277 y Fk(2)2018 256 y Fn(j)2041 210 y Fj(m)2118 256 y Fo(L)2175 270 y Fj(S)2219 250 y Fc(00)2216 291 y Fg(2)2286 256 y Fn(\030)2447 200 y Fr(1)p 2384 237 169 4 v 2384 314 a Fo(")2423 290 y Fj(m=)p Fk(4)2562 256 y Fo(L)2619 270 y Fj(S)2663 250 y Fc(00)2660 291 y Fg(2)2708 256 y Fo(:)1114 b Fr(\(48\))-260 529 y(T)-7 b(o)33 b(b)r(ound)g(the)g(sum)g(of)f(the)i(non-Fib)r(onacci)e(co)r (e\016cien)n(ts)g(w)n(e)g(ma)n(y)g(pro)r(ceed)g(in)h(a)g(similar)f(w)n (a)n(y)-7 b(,)33 b(dividing)f(them)i(in)n(to)e(the)h(cases)-260 629 y Fn(j)p Fo(k)s Fn(j)28 b(\026)f Fo(")-9 598 y Fh(\000)p Fk(1)p Fj(=)p Fk(4)177 629 y Fr(and)j Fn(j)p Fo(k)s Fn(j)d(\027)g Fo(")591 598 y Fh(\000)p Fk(1)p Fj(=)p Fk(4)747 629 y Fr(.)45 b(In)30 b(the)h(\014rst)f(case,)g(w)n(e)g(can)g(use)g(that)g (the)h(n)n(um)n(b)r(er)f(of)g(suc)n(h)g(terms)g(is)g Fn(O)16 b Fr(\(1)p Fo(=)3307 569 y Fn(p)p 3375 569 39 4 v 3375 629 a Fo(")p Fr(\),)31 b(together)f(with)-260 728 y(the)c(b)r(ound)g(\(38\))f(and)g(the)h(inequalit)n(y)f(\(46\),)h (obtaining)f(an)g(upp)r(er)h(b)r(ound)g(smaller)e(than)i(\(47{48\).)34 b(In)26 b(the)g(second)f(case,)g(the)h(same)-260 828 y(is)i(true)f(b)r(ecause)g(those)h(terms)f(can)g(b)r(e)h(b)r(ounded)g (b)n(y)f(a)h(geometric)e(series.)-135 1027 y(Finally)-7 b(,)31 b(w)n(e)g(p)r(oin)n(t)g(out)g(that)g(the)g(pro)r(of)f(of)h (\(b\))h(in)f(the)g(other)f(cases)g Fo(l)g Fn(6)p Fr(=)e(1)j(is)f (analogous.)45 b(The)31 b(only)f(di\013erence)h(is)f(that)i(one)-260 1127 y(has)f(to)h(exclude)f(from)g(the)h(sum)g(some)f(dominan)n(t)h (Fib)r(onacci)f(harmonics,)g(whic)n(h)h(are)f(consecutiv)n(e,)g (instead)h(of)f(excluding)h(only)-260 1226 y(the)c(most)g(dominan)n(t)f (harmonic.)3111 b Fe(2)-260 1618 y Fq(5)135 b(Dominan)l(t)46 b(harmonics)f(of)g(the)g(splitting)i(p)t(oten)l(tial)-260 1903 y Fr(The)29 b(estimates)f(giv)n(en)f(in)i(Lemma)f(4)g(for)g(the)h (dominan)n(t)f(harmonics)g(of)g(the)h(Melnik)n(o)n(v)e(p)r(oten)n(tial) i Fo(L)p Fr(\()p Fo(s;)14 b(\022)r Fr(\))28 b(can)g(b)r(e)h(used)g(as)e (a)h(\014rst)-260 2002 y(appro)n(ximation)h(to)h(study)h(the)g(simple)g (zeros)e(of)i(the)g(Melnik)n(o)n(v)e(function)j Fo(M)k Fr(=)28 b Fo(@)2439 2014 y Fj(\022)2477 2002 y Fo(L)p Fr(,)j(on)f(a)g(concrete)g(section)g Fo(s)e Fr(=)g(const.)46 b(No)n(w,)-260 2102 y(w)n(e)30 b(w)n(an)n(t)g(to)h(sho)n(w)e(that,)j (assuming)e Fo(\026)e Fr(=)f Fo(")1164 2072 y Fj(p)1233 2102 y Fr(for)j(a)g(suitable)h Fo(p)c(>)h Fr(0,)j(the)g(dominan)n(t)f (harmonics)g(remain)f(essen)n(tially)h(unc)n(hanged)-260 2202 y(when)e(one)f(considers)f(the)i(whole)g(splitting)g(function)g Fn(M)p Fr(\()p Fo(s;)14 b(\022)r Fr(\),)28 b(including)g(the)g(error)d (term)j(\(18\).)-135 2401 y(Recalling)f(that)g Fn(M)c Fr(=)g Fo(@)660 2413 y Fj(\022)698 2401 y Fn(L)p Fr(,)28 b(and)f(taking)g(in)n(to)g(accoun)n(t)g(that)h(the)g(splitting)g(p)r (oten)n(tial)f Fn(L)h Fr(is)34 b(^)-48 b Fo(!)2920 2413 y Fj(")2955 2401 y Fr(-quasip)r(erio)r(dic,)26 b(w)n(e)i(can)f(write) 780 2604 y Fn(L)p Fr(\()p Fo(s;)14 b(\022)r Fr(\))24 b(=)1146 2525 y Fi(X)1129 2707 y Fj(k)q Fh(2)p Ff(Z)1255 2690 y Fg(2)1296 2604 y Fn(L)1353 2569 y Fh(\003)1353 2624 y Fj(k)1394 2604 y Fo(e)1433 2569 y Fj(i)p Fh(h)p Fj(k)q(;\022)r Fh(\000)5 b Fk(^)-38 b Fj(!)1667 2577 y Fd(")1699 2569 y Fj(s)p Fh(i)1784 2604 y Fr(=)1879 2525 y Fi(X)1872 2704 y Fj(k)q Fh(2Z)2019 2604 y Fn(L)2076 2616 y Fj(k)2131 2604 y Fr(cos)13 b(\()q Fn(h)p Fo(k)s(;)h(\022)20 b Fn(\000)25 b Fr(^)-49 b Fo(!)2598 2616 y Fj(")2634 2604 y Fo(s)p Fn(i)18 b(\000)g Fo(\034)2842 2616 y Fj(k)2883 2604 y Fr(\))d Fo(;)892 b Fr(\(49\))-260 2881 y(where)31 b Fn(L)41 2893 y Fj(k)82 2881 y Fr(,)i Fo(\034)174 2893 y Fj(k)246 2881 y Fr(are)e(real,)g Fn(L)635 2893 y Fj(k)706 2881 y Fn(\025)e Fr(0)i(\(recall)g(that)g Fn(Z)39 b Fr(is)31 b(de\014ned)h(in)f(\(8\)\).)49 b(F)-7 b(or)31 b(ev)n(ery)f Fo(k)j Fn(2)c(Z)7 b Fr(,)33 b(the)e(co)r(e\016cien)n(ts)h(of)f(the)h (exp)r(onen)n(tial)-260 2981 y(form)c(and)h(the)g(trigonometric)e(form) h(are)f(related)h(b)n(y)h Fn(L)1550 2950 y Fh(\003)1550 3004 y Fj(k)1615 2981 y Fr(=)1715 2948 y Fk(1)p 1715 2962 34 4 v 1715 3009 a(2)1758 2981 y Fn(L)1815 2993 y Fj(k)1856 2981 y Fo(e)1895 2950 y Fh(\000)p Fj(i\034)2001 2959 y Fd(k)2041 2981 y Fr(,)g Fn(L)2150 2950 y Fh(\003)2150 3004 y(\000)p Fj(k)2267 2981 y Fr(=)p 2357 2914 99 4 v 25 w Fn(L)2414 2952 y Fh(\003)2414 3006 y Fj(k)2479 2981 y Fr(=)2579 2948 y Fk(1)p 2579 2962 34 4 v 2579 3009 a(2)2622 2981 y Fn(L)2679 2993 y Fj(k)2720 2981 y Fo(e)2759 2950 y Fj(i\034)2813 2959 y Fd(k)2853 2981 y Fr(.)40 b(The)28 b(analogous)f(co)r(e\016cien)n(ts)h(for)-260 3080 y(the)e(splitting)g(function)g Fn(M)p Fr(\()p Fo(s;)14 b(\022)r Fr(\))25 b(are)g Fn(M)1073 3050 y Fh(\003)1073 3104 y Fj(k)1136 3080 y Fr(=)e Fo(ik)s Fn(L)1356 3050 y Fh(\003)1356 3104 y Fj(k)1422 3080 y Fr(\(in)j(the)g(exp)r(onen)n (tial)e(form\).)36 b(In)26 b(the)g(next)f(lemma,)h(w)n(e)f(use)g(the)h (b)r(ound)g(on)f(the)-260 3180 y(error)c(term)h Fn(R)p Fr(\()p Fo(s;)14 b(\022)r Fr(\),)25 b(giv)n(en)c(in)i(Theorem)f(0,)h (to)f(compare)f(the)i(co)r(e\016cien)n(ts)f(of)g(the)h(splitting)g(p)r (oten)n(tial)f(with)h(those)f(of)h(the)f(Melnik)n(o)n(v)-260 3279 y(p)r(oten)n(tial.)37 b(W)-7 b(e)26 b(also)g(compare,)f(for)h(the) h(dominan)n(t)f(co)r(e\016cien)n(ts,)h(the)g(phases)e Fo(\034)2305 3291 y Fj(k)2346 3279 y Fr(,)i Fo(\033)2443 3291 y Fj(k)2511 3279 y Fr(of)g(b)r(oth)f(p)r(oten)n(tials)h(\(this)g (is)f(a\013ected)h(b)n(y)f(the)-260 3379 y(translation)h Fo(s)200 3391 y Fk(0)264 3379 y Fr(that)h(app)r(ears)f(in)h(\(18\)\).) -260 3674 y Fs(Lemma)i(5)41 b Fp(Assuming)1114 3820 y Fo(")23 b Fn(\026)g Fr(1)p Fo(;)183 b(\026)23 b Fr(=)f Fo(")1711 3786 y Fj(p)1834 3820 y Fp(with)31 b Fo(p)23 b(>)f(p)2209 3786 y Fh(\003)2270 3820 y Fr(:=)2391 3764 y Fo(q)2428 3776 y Fk(3)2484 3764 y Fn(\000)c Fr(2)p 2391 3801 218 4 v 2479 3877 a(4)3845 3820 y(\(50\))-260 4009 y Fp(\(we)30 b(de\014ne)l(d)g Fo(q)214 4021 y Fk(3)281 4009 y Fp(in)g(The)l(or)l(em)h(0\),)f(one)g(has:)-200 4333 y Fr(\(a\))42 b Fn(L)5 4345 y Fj(S)46 4353 y Fd(i)99 4333 y Fn(\030)243 4277 y Fo(\026)p 197 4314 144 4 v 197 4391 a(")236 4367 y Fk(1)p Fj(=)p Fk(4)364 4333 y Fr(exp)504 4216 y Fi(\032)567 4333 y Fn(\000)642 4277 y Fo(C)701 4289 y Fk(0)738 4277 y Fo(h)786 4289 y Fj(i)813 4277 y Fr(\()p Fo(")p Fr(\))p 642 4314 276 4 v 707 4391 a Fo(")746 4367 y Fk(1)p Fj(=)p Fk(4)927 4216 y Fi(\033)1003 4333 y Fo(;)-52 4512 y Fn(j)p Fo(\034)7 4524 y Fj(S)48 4532 y Fd(i)97 4512 y Fn(\000)18 b Fo(\033)227 4524 y Fj(S)268 4532 y Fd(i)317 4512 y Fn(\000)g Fo(s)439 4524 y Fk(0)490 4512 y Fn(h)p Fo(S)573 4524 y Fj(i)601 4512 y Fo(;)i Fr(^)-48 b Fo(!)690 4524 y Fj(")725 4512 y Fn(ij)23 b(\026)932 4456 y Fo(\026)p 901 4493 112 4 v 901 4569 a(")940 4545 y Fj(p)974 4528 y Fc(\003)1023 4512 y Fo(;)98 b(i)23 b Fr(=)g(1)p Fo(;)14 b Fr(2)p Fo(;)g Fr(3)p Fo(;)g Fr(4)p Fo(:)-204 4769 y Fr(\(b\))50 4691 y Fi(X)-52 4869 y Fj(k)q Fh(6)p Fk(=)p Fj(S)76 4877 y Fg(1)109 4869 y Fj(;:::)n(;S)248 4878 y Fd(l)286 4769 y Fn(j)p Fo(k)s Fn(j)378 4728 y Fj(m)455 4769 y Fn(L)512 4781 y Fj(k)576 4769 y Fn(\030)737 4713 y Fr(1)p 673 4750 169 4 v 673 4828 a Fo(")712 4804 y Fj(m=)p Fk(4)852 4769 y Fn(L)909 4781 y Fj(S)950 4790 y Fd(l)p Fg(+1)1049 4769 y Fo(;)99 b Fr(0)23 b Fn(\024)f Fo(l)j Fn(\024)e Fr(3)p Fo(;)98 b(m)23 b Fn(\025)f Fr(0)p Fo(:)-260 5154 y Fp(Pr)l(o)l(of.)53 b Fr(Assuming)26 b(the)g(smallness)f(conditions)g(\(19\))g(on)h Fo(\026)p Fr(,)g(with)g Fo(\016)j Fr(to)d(b)r(e)g(c)n(hosen,)f(w)n(e)g (kno)n(w)g(from)h(Theorem)e(0)i(that)g(the)g(splitting)-260 5254 y(function)g Fn(M)p Fr(\()p Fo(s;)14 b(\022)r Fr(\))26 b(can)g(b)r(e)g(de\014ned)g(on)f 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x Fo(")1449 5586 y Fi(\023)1524 5703 y Fo(e)1563 5669 y Fh(\000)1624 5653 y Fk(~)1615 5669 y Fj(\014)1653 5678 y Fd(k)1693 5703 y Fo(;)1907 5681 y Fr(~)1896 5703 y Fo(\014)1943 5715 y Fj(k)2007 5703 y Fr(=)f(\()p Fo(\032)d Fn(\000)f Fo(\016)s Fr(\))c Fn(j)p Fo(k)s Fn(j)k Fr(+)2560 5647 y(\()p Fo(\031)s(=)p Fr(2)g Fn(\000)g Fo(\016)s Fr(\))p Fo(b)2935 5617 y Fh(0)2959 5647 y Fo(\015)3002 5659 y Fj(k)p 2560 5684 483 4 v 2670 5760 a Fo(b)c Fn(j)o Fo(k)s Fn(j)2825 5700 y(p)p 2894 5700 39 4 v 60 x Fo(")3053 5703 y(;)1825 5989 y Fr(14)p eop %%Page: 15 15 15 14 bop -260 -82 a Fr(for)27 b(an)n(y)g Fo(k)f Fn(6)p Fr(=)d(0.)36 b(Since)28 b Fn(R)569 -112 y Fh(\003)569 -59 y Fj(k)633 -82 y Fr(=)23 b Fo(ik)809 -149 y Fi(\000)847 -82 y Fn(L)904 -112 y Fh(\003)904 -59 y Fj(k)964 -82 y Fn(\000)18 b Fo(\026L)1154 -112 y Fh(\003)1154 -59 y Fj(k)1194 -82 y Fo(e)1233 -112 y Fh(\000)p Fj(is)1339 -104 y Fg(0)1372 -112 y Fh(h)p Fj(k)q(;)5 b Fk(^)-38 b Fj(!)1497 -104 y Fd(")1529 -112 y Fh(i)1560 -149 y Fi(\001)1598 -82 y Fr(,)28 b(taking)f(mo)r(dulus)h(and)f(argumen)n(t)g (w)n(e)g(get)851 149 y Fn(jL)931 161 y Fj(k)991 149 y Fn(\000)18 b Fo(\026L)1181 161 y Fj(k)1221 149 y Fn(j)23 b(\026)1365 92 y(jR)1458 62 y Fh(\003)1458 116 y Fj(k)1499 92 y Fn(j)p 1365 129 158 4 v 1398 205 a(j)p Fo(k)s Fn(j)1532 149 y Fo(;)180 b Fn(j)p Fo(\034)1794 161 y Fj(k)1854 149 y Fn(\000)18 b Fo(\033)1984 161 y Fj(k)2044 149 y Fn(\000)g Fo(s)2166 161 y Fk(0)2217 149 y Fn(h)p Fo(k)s(;)i Fr(^)-48 b Fo(!)2384 161 y Fj(")2419 149 y Fn(ij)23 b(\026)2643 92 y(jR)2736 62 y Fh(\003)2736 116 y Fj(k)2777 92 y Fn(j)p 2595 129 254 4 v 2595 205 a(j)p Fo(k)s Fn(j)14 b Fo(\026L)2808 217 y Fj(k)2858 149 y Fo(:)-135 471 y Fr(Pro)r(ceeding)26 b(as)h(in)h(\(37\),)f(but)h(no)n(w)f(with)h Fo(\031)s(=)p Fr(2)18 b Fn(\000)g Fo(\016)31 b Fr(and)c Fo(\032)18 b Fn(\000)g Fo(\016)31 b Fr(instead)d(of)f Fo(\031)s(=)p Fr(2)g(and)h Fo(\032)p Fr(,)f(w)n(e)g(ha)n(v)n(e:)1139 683 y(~)1127 705 y Fo(\014)1174 717 y Fj(k)1238 705 y Fr(=)1336 649 y Fo(C)1395 661 y Fj(\026;\016)1491 588 y Fn(p)p 1561 588 84 4 v 1564 649 a Fr(~)-45 b Fo(\015)1604 661 y Fj(k)p 1336 686 309 4 v 1398 764 a Fr(2)p Fo(")1479 740 y Fk(1)p Fj(=)p Fk(4)1668 588 y Fi(\022)1774 649 y Fn(j)p Fo(k)s Fn(j)14 b Fo(")1919 619 y Fk(1)p Fj(=)p Fk(4)p 1739 686 319 4 v 1739 764 a Fo(D)1808 776 y Fj(\026;\016)1905 703 y Fn(p)p 1974 703 84 4 v 1977 764 a Fr(~)-45 b Fo(\015)2017 776 y Fj(k)2086 705 y Fr(+)2179 649 y Fo(D)2248 661 y Fj(\026;\016)2344 588 y Fn(p)p 2413 588 V 2416 649 a Fr(~)g Fo(\015)2456 661 y Fj(k)p 2179 686 319 4 v 2214 764 a Fn(j)p Fo(k)s Fn(j)13 b Fo(")2358 740 y Fk(1)p Fj(=)p Fk(4)2507 588 y Fi(\023)2582 705 y Fo(;)-260 938 y Fr(no)n(w)27 b(with)h Fo(C)161 950 y Fj(\026;\016)281 938 y Fr(=)23 b Fo(C)428 950 y Fk(0)484 938 y Fr(+)18 b Fn(O)e Fr(\()p Fo(\026\016)771 907 y Fh(\000)p Fj(q)853 915 y Fg(2)890 938 y Fo(;)e(\016)s Fr(\))q(,)27 b Fo(D)1119 950 y Fj(\026;\016)1239 938 y Fr(=)22 b Fo(D)1395 950 y Fk(0)1451 938 y Fr(+)c Fn(O)e Fr(\()p Fo(\026\016)1738 907 y Fh(\000)p Fj(q)1820 915 y Fg(2)1857 938 y Fo(;)e(\016)s Fr(\).)37 b(Pro)r(ceeding)26 b(as)h(in)h(the)g(pro)r(of)f(of)h(Lemma)f (4,)g(w)n(e)g(obtain)1265 1145 y(~)1253 1167 y Fo(\014)1300 1179 y Fj(S)1341 1187 y Fd(i)1394 1167 y Fr(=)1492 1111 y Fo(C)1551 1123 y Fk(0)1588 1111 y Fo(h)1636 1123 y Fj(i)1664 1111 y Fr(\()p Fo(")p Fr(\))19 b(+)f Fn(O)e Fr(\()1983 1051 y Fn(p)p 2052 1051 39 4 v 60 x Fo(";)e(\026\016)2218 1081 y Fh(\000)p Fj(q)2300 1089 y Fg(2)2337 1111 y Fo(;)g(\016)s Fr(\))p 1492 1148 955 4 v 1898 1226 a Fo(")1937 1202 y Fk(1)p Fj(=)p Fk(4)2456 1167 y Fo(;)-260 1383 y Fr(and)28 b(the)g(p)r(erturbativ)n(e)e(part)i(can)f(b)r(e)h(neglected)f(if)h Fo(\026)23 b Fn(\026)g Fo(\016)1609 1352 y Fj(q)1639 1360 y Fg(2)1676 1383 y Fo(")1715 1352 y Fk(1)p Fj(=)p Fk(4)1847 1383 y Fr(and)k Fo(\016)f Fn(\026)d Fo(")2198 1352 y Fk(1)p Fj(=)p Fk(4)2302 1383 y Fr(.)37 b(So)27 b(w)n(e)g(c)n(ho)r(ose)1708 1557 y Fo(\016)f Fr(=)c Fo(")1897 1523 y Fk(1)p Fj(=)p Fk(4)2002 1557 y Fo(;)-260 1732 y Fr(and)i(the)g(smallness)e(conditions)i(\(19\))f(can)g(b)r(e)h (rewritten)f(as)g Fo(\026)h Fn(\026)e Fo(")1874 1702 y Fj(q)1904 1710 y Fg(1)1937 1702 y Fj(=)p Fk(4)2008 1732 y Fr(.)36 b(Note)23 b(that)h(the)g(condition)g(on)f Fo(\026)h Fr(con)n(taining)e(the)i(exp)r(onen)n(t)-260 1831 y Fo(q)-223 1843 y Fk(2)-158 1831 y Fr(in)k(\(19\))f(can)g(b)r(e)h (ignored,)f(since)g Fo(q)938 1843 y Fk(1)999 1831 y Fn(\025)22 b Fo(q)1123 1843 y Fk(2)1179 1831 y Fr(+)c(3.)37 b(Then,)27 b(using)h(also)e(\(43\),)i(w)n(e)f(get)1081 2059 y Fn(jL)1161 2071 y Fj(S)1202 2079 y Fd(i)1251 2059 y Fn(\000)18 b Fo(\026L)1441 2071 y Fj(S)1482 2079 y Fd(i)1512 2059 y Fn(j)23 b(\026)1767 2003 y Fo(\026)1817 1973 y Fk(2)p 1656 2040 310 4 v 1656 2118 a Fo(")1695 2094 y Fk(\()p Fj(q)1751 2102 y Fg(3)1783 2094 y Fh(\000)p Fk(1\))p Fj(=)p Fk(4)1989 2059 y Fr(exp)2130 1942 y Fi(\032)2192 2059 y Fn(\000)2267 2003 y Fo(C)2326 2015 y Fk(0)2363 2003 y Fo(h)2411 2015 y Fj(i)2439 2003 y Fr(\()p Fo(")p Fr(\))p 2267 2040 276 4 v 2333 2118 a Fo(")2372 2094 y Fk(1)p Fj(=)p Fk(4)2552 1942 y Fi(\033)2628 2059 y Fo(:)1194 b Fr(\(51\))-260 2284 y(As)26 b(b)r(efore,)f(the)h(term)f (con)n(taining)g(the)h(exp)r(onen)n(t)f Fo(q)1403 2296 y Fk(4)1466 2284 y Fr(can)g(b)r(e)g(ignored,)g(since)g Fo(q)2281 2296 y Fk(3)2342 2284 y Fn(\025)d Fo(q)2466 2296 y Fk(4)2518 2284 y Fr(+)13 b(2.)36 b(The)26 b(upp)r(er)f(b)r(ound) h(\(51\))f(is)g(dominated)-260 2383 y(b)n(y)i(the)g(term)h Fn(j)p Fo(\026L)326 2395 y Fj(S)367 2403 y Fd(i)396 2383 y Fn(j)p Fr(,)g(estimated)f(in)g(Lemma)g(4,)g(if)h(one)e(tak)n(es)h Fo(\026)c Fr(=)f Fo(")1971 2353 y Fj(p)2037 2383 y Fr(with)27 b Fo(p)c(>)g Fr(\()p Fo(q)2447 2395 y Fk(3)2502 2383 y Fn(\000)17 b Fr(2\))p Fo(=)p Fr(4.)35 b(Since)28 b Fo(q)3054 2395 y Fk(3)3108 2383 y Fn(\000)17 b Fr(2)23 b Fn(\025)g Fo(q)3380 2395 y Fk(1)3417 2383 y Fr(,)k(it)h(is)f(enough)f (to)-260 2483 y(consider)i Fo(p)107 2453 y Fh(\003)174 2483 y Fr(as)g(de\014ned)h(in)g(\(50\),)g(and)f(this)h(implies)g(the)g (\014rst)g(statemen)n(t)f(of)h(\(a\).)41 b(The)28 b(second)g(statemen)n (t)h(of)g(\(a\))g(is)f(pro)n(v)n(ed)g(in)h(a)-260 2582 y(similar)e(w)n(a)n(y)-7 b(.)-135 2782 y(The)30 b(remaining)f(co)r (e\016cien)n(ts)g Fn(jL)930 2794 y Fj(k)990 2782 y Fn(\000)18 b Fo(\026L)1180 2794 y Fj(k)1220 2782 y Fn(j)p Fr(,)31 b(for)e Fo(k)h Fn(6)p Fr(=)d Fo(S)1642 2794 y Fj(i)1669 2782 y Fr(,)k(can)f(b)r(e)g(b)r(ounded)g(from)g(ab)r(o)n(v)n(e)f (analogously)e(to)j(the)g(co)r(e\016cien)n(ts)g Fo(L)3952 2794 y Fj(k)-260 2881 y Fr(in)e(the)g(pro)r(of)f(of)h(Lemma)f(4,)g(and) h(this)f(giv)n(es)g(statemen)n(t)g(\(b\).)2205 b Fe(2)-260 3081 y Fs(Remarks.)-158 3340 y Fr(1.)41 b(Notice)31 b(that)g(the)h(exp) r(onen)n(t)f Fo(p)945 3310 y Fh(\003)1014 3340 y Fr(is)g(the)h(one)e (announced)h(in)g(Section)h(1.3)e(\(see)h(Theorem)f(1\):)44 b(for)31 b Fo(\027)j Fr(=)29 b(0)h(w)n(e)h(ha)n(v)n(e)f Fo(q)3749 3352 y Fk(3)3816 3340 y Fr(=)e(14)-52 3439 y(and)f(hence)h Fo(p)382 3409 y Fh(\003)443 3439 y Fr(=)22 b(3,)28 b(and)f(for)g Fo(\027)h Fr(=)23 b(1)k(w)n(e)h(ha)n(v)n(e)e Fo(q)1488 3451 y Fk(3)1548 3439 y Fr(=)d(10)k(and)g Fo(p)1950 3409 y Fh(\003)2011 3439 y Fr(=)c(2.)-158 3602 y(2.)41 b(In)30 b(fact,)h(w)n(e)f(should)g(recall)f(that)h(the)h(results)f(of)g (this)g(pap)r(er)g(rely)f(on)h(the)h(application)e(of)h(the)h(h)n(yp)r (erb)r(olic)f(KAM)g(theorem)-52 3702 y([DGS02)o(,)j(Th.)e(1])g(with)h Fo(n)d Fr(=)g(2)i(as)f(the)i(n)n(um)n(b)r(er)f(of)g(frequencies,)h(and) f(with)h Fo(\034)39 b Fr(=)29 b(1)i(as)f(the)i(exp)r(onen)n(t)f(in)h (the)f(Diophan)n(tine)-52 3801 y(condition)i(\(11\).)55 b(T)-7 b(o)33 b(b)r(e)h(strictly)g(rigorous,)f(the)h(quoted)f(theorem)h (cannot)f(b)r(e)h(directly)f(applied)h(b)r(ecause)f(the)h(condition)-52 3901 y Fo(\034)h(>)26 b(n)19 b Fn(\000)g Fr(1)29 b(should)g(b)r(e)h (required.)41 b(Nev)n(ertheless,)28 b(it)i(is)f(clear)f(that)i Fo(!)i Fr(also)c(satis\014es)h(\(11\))g(with)g(an)n(y)g Fo(\034)35 b(>)26 b Fr(1,)j(and)g(one)g(easily)-52 4001 y(sees)e(that)h(this)f(do)r(es)h(not)f(a\013ect)h(the)g(\014nal)f (restriction)g Fo(p)c(>)g(p)1935 3971 y Fh(\003)2000 4001 y Fr(in)28 b(\(50\).)-260 4374 y Fq(6)135 b(Critical)41 b(p)t(oin)l(ts)g(of)f(the)g(splitting)i(p)t(oten)l(tial:)59 b(transv)l(erse)41 b(homo)t(clinic)g(orbits)-58 4523 y(and)k(their)g(con)l(tin)l(uation)-260 4804 y Fr(W)-7 b(e)31 b(are)e(going)h(to)g(use)g(in)h(this)f(section)g(the)h (estimates)f(giv)n(en)g(in)h(Lemma)f(5,)g(to)h(sho)n(w)e(that)i(the)g (splitting)f(p)r(oten)n(tial)h Fn(L)p Fr(\(0)p Fo(;)14 b(\022)r Fr(\))31 b(has)-260 4904 y(nondegenerate)e(critical)i(p)r(oin) n(ts)f(\(\014xing)h Fo(s)d Fr(=)g(0\).)47 b(First,)31 b(w)n(e)g(will)g(study)f(the)i(critical)e(p)r(oin)n(ts)g(for)h(the)g (appro)n(ximations)e(giv)n(en)g(b)n(y)-260 5004 y(the)f(2)f(or)g(3)g (most)h(dominan)n(t)f(harmonics:)414 5178 y Fn(L)471 5144 y Fk(\(2\))560 5178 y Fr(\()p Fo(\022)r Fr(\))d(=)797 5099 y Fi(X)777 5276 y Fj(i)p Fk(=1)p Fj(;)p Fk(2)951 5178 y Fn(L)1008 5190 y Fj(S)1049 5198 y Fd(i)1093 5178 y Fr(cos)13 b(\()p Fn(h)q Fo(S)1334 5190 y Fj(i)1361 5178 y Fo(;)h(\022)r Fn(i)19 b(\000)f Fo(\034)1609 5190 y Fj(S)1650 5198 y Fd(i)1681 5178 y Fr(\))c Fo(;)180 b Fn(L)1987 5144 y Fk(\(3\))2076 5178 y Fr(\()p Fo(\022)r Fr(\))24 b(=)2339 5099 y Fi(X)2293 5276 y Fj(i)p Fk(=1)p Fj(;)p Fk(2)p Fj(;)p Fk(3)2520 5178 y Fn(L)2577 5190 y Fj(S)2618 5198 y Fd(i)2662 5178 y Fr(cos)13 b(\()p Fn(h)q Fo(S)2903 5190 y Fj(i)2930 5178 y Fo(;)h(\022)r Fn(i)19 b(\000)f Fo(\034)3178 5190 y Fj(S)3219 5198 y Fd(i)3250 5178 y Fr(\))c Fo(:)526 b Fr(\(52\))-260 5441 y(Afterw)n(ards,)31 b(w)n(e)g(discuss)f(the)i(v)-5 b(alidit)n(y)30 b(of)h(these)g(critical)g(p)r(oin)n(ts)g(in)g(the)g(whole)g(function)h Fn(L)p Fr(\(0)p Fo(;)14 b(\022)r Fr(\),)32 b(b)n(y)f(means)f(of)h(a)g (quan)n(titativ)n(e)-260 5541 y(v)n(ersion)d(of)i(the)g(implicit)h (function)f(theorem,)g(giv)n(en)f(in)h(the)g(App)r(endix.)44 b(As)30 b(the)g(functions)g Fo(h)2809 5553 y Fj(i)2836 5541 y Fr(\()p Fo(")p Fr(\))g(de\014ned)h(in)f(\(39\))f(suggest,)g(it)h (is)-260 5641 y(enough)h(to)h(consider)e(the)i(2)f(dominan)n(t)h (harmonics)e(for)h(most)g(v)-5 b(alues)32 b(of)f Fo(")p Fr(,)i(but)f(for)f Fo(")g Fr(close)g(to)h(a)f(critical)g(v)-5 b(alue)31 b Fo(")3520 5653 y Fj(n)3597 5641 y Fr(w)n(e)g(ha)n(v)n(e)f (to)-260 5740 y(consider)d(the)h(3)f(dominan)n(t)g(harmonics.)1825 5989 y(15)p eop %%Page: 16 16 16 15 bop -135 -82 a Fr(T)-7 b(o)26 b(start,)g(w)n(e)g(study)h(the)g (function)g Fn(L)1079 -112 y Fk(\()q(2\))1195 -82 y Fr(for)f Fo(")d Fn(6)p Fr(=)g Fo(")1510 -70 y Fj(n)1555 -82 y Fr(,)k(and)f(the)h(function)g Fn(L)2288 -112 y Fk(\()q(3\))2404 -82 y Fr(for)f Fo(")d Fn(6)p Fr(=)g Fo(")2719 -112 y Fh(0)2719 -62 y Fj(n)2764 -82 y Fr(.)36 b(T)-7 b(o)27 b(\014x)f(ideas,)h(w)n(e)f(lo)r(ok)g(at)g(concrete)-260 17 y(in)n(terv)-5 b(als:)41 b(w)n(e)29 b(assume)h Fo(")c Fn(2)i Fr(\()p Fo(")743 29 y Fj(n)788 17 y Fo(;)14 b(")864 29 y Fj(n)p Fh(\000)p Fk(1)994 17 y Fr(\))30 b(in)g(the)h(\014rst)e (case,)h(and)g Fo(")c Fn(2)1988 -50 y Fi(\000)2026 17 y Fo(")2065 -13 y Fh(0)2065 38 y Fj(n)p Fk(+1)2194 17 y Fo(;)14 b(")2270 -13 y Fh(0)2270 38 y Fj(n)2315 -50 y Fi(\001)2383 17 y Fr(in)30 b(the)g(second)f(case.)44 b(Recalling)29 b(Figure)g(1,)h(note)-260 117 y(that)519 196 y Fo(S)570 208 y Fk(1)630 196 y Fr(=)23 b Fo(s)p Fr(\(1)p Fo(;)14 b(n)p Fr(\))p Fo(;)226 b(S)1250 208 y Fk(2)1310 196 y Fr(=)23 b Fo(s)p Fr(\(1)p Fo(;)14 b(n)j Fr(+)h(1\))p Fo(;)83 b(S)1929 208 y Fk(3)1990 196 y Fr(=)22 b Fo(s)p Fr(\(1)p Fo(;)14 b(n)k Fn(\000)g Fr(1\))p Fo(;)83 b Fr(for)27 b Fo(")c Fn(2)2825 129 y Fi(\000)2863 196 y Fo(")2902 166 y Fh(0)2902 217 y Fj(n)p Fk(+1)3031 196 y Fo(;)14 b(")3107 208 y Fj(n)3152 129 y Fi(\001)3190 196 y Fr(,)519 329 y Fo(S)570 341 y Fk(1)630 329 y Fr(=)23 b Fo(s)p Fr(\(1)p Fo(;)14 b(n)p Fr(\))p Fo(;)226 b(S)1250 341 y Fk(2)1310 329 y Fr(=)23 b Fo(s)p Fr(\(1)p Fo(;)14 b(n)j Fn(\000)h Fr(1\))p Fo(;)83 b(S)1929 341 y Fk(3)1990 329 y Fr(=)22 b Fo(s)p Fr(\(1)p Fo(;)14 b(n)k Fr(+)g(1\))p Fo(;)83 b Fr(for)27 b Fo(")c Fn(2)g Fr(\()q Fo(")2897 341 y Fj(n)2941 329 y Fo(;)14 b(")3017 299 y Fh(0)3017 350 y Fj(n)3062 329 y Fr(\))q(,)519 462 y Fo(S)570 474 y Fk(1)630 462 y Fr(=)23 b Fo(s)p Fr(\(1)p Fo(;)14 b(n)k Fn(\000)g Fr(1\))p Fo(;)83 b(S)1250 474 y Fk(2)1310 462 y Fr(=)23 b Fo(s)p Fr(\(1)p Fo(;)14 b(n)p Fr(\))p Fo(;)905 b Fr(for)27 b Fo(")c Fn(2)g Fr(\()q Fo(")2897 432 y Fh(0)2897 482 y Fj(n)2941 462 y Fo(;)14 b(")3017 474 y Fj(n)p Fh(\000)p Fk(1)3147 462 y Fr(\))q(.)-260 624 y(In)26 b(order)e(to)i(ha)n(v)n(e)e (a)i(simpler)f(expression)f(for)h(the)h(functions)g Fn(L)1780 594 y Fk(\()q Fj(i)p Fk(\))1860 624 y Fr(\()p Fo(\022)r Fr(\),)h(w)n(e)e(carry)f(out)i(in)g(b)r(oth)g(cases)f(the)h(linear)f(c) n(hange)f(\()p Fo(\022)3703 636 y Fk(1)3741 624 y Fo(;)14 b(\022)3817 636 y Fk(2)3854 624 y Fr(\))23 b Fn(7!)-260 724 y Fr(\()p Fo( )-174 736 y Fk(1)-136 724 y Fo(;)14 b( )-45 736 y Fk(2)-8 724 y Fr(\))28 b(de\014ned)g(b)n(y)744 824 y Fo( )798 836 y Fk(1)858 824 y Fr(=)23 b Fn(h)p Fo(s)p Fr(\(1)p Fo(;)14 b(n)k Fn(\000)g Fr(1\))p Fo(;)c(\022)r Fn(i)19 b(\000)f Fo(\034)1601 839 y Fj(s)p Fk(\(1)p Fj(;n)p Fh(\000)p Fk(1\))1867 824 y Fo(;)180 b( )2124 836 y Fk(2)2184 824 y Fr(=)23 b Fn(h)p Fo(s)p Fr(\(1)p Fo(;)14 b(n)p Fr(\))p Fo(;)g(\022)r Fn(i)19 b(\000)f Fo(\034)2784 839 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(\))2966 824 y Fo(;)856 b Fr(\(53\))-260 974 y(whic)n(h)28 b(can)f(b)r(e)h(written)g(as)543 1186 y Fo( )e Fr(=)d Fn(A)777 1198 y Fj(n)822 1186 y Fo(\022)e Fn(\000)d Fo(b)1001 1198 y Fj(n)1046 1186 y Fo(;)180 b Fr(where)55 b Fn(A)1583 1198 y Fj(n)1651 1186 y Fr(=)1739 1069 y Fi(\022)1841 1136 y Fo(s)p Fr(\(1)p Fo(;)14 b(n)k Fn(\000)g Fr(1\))2216 1105 y Fh(>)1913 1236 y Fo(s)p Fr(\(1)p Fo(;)c(n)p Fr(\))2145 1206 y Fh(>)2314 1069 y Fi(\023)2389 1186 y Fo(;)41 b(b)2489 1198 y Fj(n)2557 1186 y Fr(=)2645 1069 y Fi(\022)2747 1136 y Fo(\034)2783 1151 y Fj(s)p Fk(\(1)p Fj(;n)p Fh(\000)p Fk(1\))2790 1235 y Fo(\034)2826 1250 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(\))3091 1069 y Fi(\023)3166 1186 y Fo(:)-260 1425 y Fr(This)32 b(c)n(hange)e(is)h(one-to-one)f(on)h Fl(T)884 1395 y Fk(2)921 1425 y Fr(,)i(b)r(ecause)e(det)14 b Fn(A)1483 1437 y Fj(n)1558 1425 y Fr(=)29 b(\()p Fn(\000)p Fr(1\))1823 1395 y Fj(n)p Fh(\000)p Fk(1)1985 1425 y Fr(as)h(one)i(ma)n(y)e(c)n (hec)n(k)h(b)n(y)g(induction.)49 b(With)33 b(this)e(c)n(hange,)h(the) -260 1525 y(functions)c Fn(L)155 1495 y Fk(\(2\))244 1525 y Fr(\()p Fo(\022)r Fr(\),)h Fn(L)458 1495 y Fk(\(3\))547 1525 y Fr(\()p Fo(\022)r Fr(\))g(mo)n(v)n(e)d(resp)r(ectiv)n(ely)h(to)h (the)g(follo)n(wing)e(ones:)751 1708 y Fn(K)815 1673 y Fk(\()q(2\))905 1708 y Fr(\()p Fo( )s Fr(\))e(=)e Fo(A)14 b Fr(cos)f Fo( )1392 1720 y Fk(1)1448 1708 y Fr(+)18 b Fo(B)g Fr(cos)13 b Fo( )1791 1720 y Fk(2)1829 1708 y Fo(;)1993 b Fr(\(54\))751 1832 y Fn(K)815 1798 y Fk(\()q(3\))905 1832 y Fr(\()p Fo( )s Fr(\))24 b(=)e Fo(B)t(\021)s Fr(\(1)d Fn(\000)f Fo(Q)p Fr(\))c(cos)f Fo( )1715 1844 y Fk(1)1770 1832 y Fr(+)18 b Fo(B)g Fr(cos)13 b Fo( )2113 1844 y Fk(2)2169 1832 y Fr(+)18 b Fo(B)t(\021)s(Q)c Fr(cos)o(\()p Fo( )2640 1844 y Fk(1)2696 1832 y Fr(+)k Fo( )2833 1844 y Fk(2)2889 1832 y Fn(\000)g(4)p Fo(\034)9 b Fr(\))p Fo(;)-260 2015 y Fr(where)27 b(w)n(e)g(denote)896 2198 y Fo(A)d Fr(=)e Fn(L)1126 2213 y Fj(s)p Fk(\(1)p Fj(;n)p Fh(\000)p Fk(1\))1393 2198 y Fo(;)97 b(B)27 b Fr(=)22 b Fn(L)1747 2213 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(\))1929 2198 y Fo(;)896 2379 y(\021)k Fr(=)1061 2317 y Fn(L)1118 2332 y Fj(s)p Fk(\(1)p Fj(;n)p Fh(\000)p Fk(1\))1403 2317 y Fr(+)18 b Fn(L)1543 2332 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(+1\))p 1061 2360 748 4 v 1316 2436 a Fn(L)1373 2451 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(\))1819 2379 y Fo(;)179 b(Q)23 b Fr(=)2420 2317 y Fn(L)2477 2332 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(+1\))p 2208 2360 V 2208 2436 a Fn(L)2265 2451 y Fj(s)p Fk(\(1)p Fj(;n)p Fh(\000)p Fk(1\))2550 2436 y Fr(+)18 b Fn(L)2690 2451 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(+1\))2965 2379 y Fo(;)896 2561 y Fn(4)p Fo(\034)32 b Fr(=)22 b Fo(\034)1162 2576 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(+1\))1446 2561 y Fn(\000)c Fo(\034)1565 2576 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(\))1765 2561 y Fn(\000)g Fo(\034)1884 2576 y Fj(s)p Fk(\(1)p Fj(;n)p Fh(\000)p Fk(1\))2174 2561 y Fn(2)23 b Fl(T)g Fr(=)f Fl(R)p Fo(=)p Fr(2)p Fo(\031)s Fl(Z)p Fo(:)1154 b Fr(\(55\))-260 2756 y(Note)29 b(that)g Fo(A)p Fr(,)g Fo(B)t Fr(,)g Fo(\021)j Fr(and)d Fo(Q)f Fr(are)g(p)r(ositiv)n(e,)g(b)r(ecause)g(so)g(are)g(the)h(co)r (e\016cien)n(ts)f Fn(L)2330 2768 y Fj(k)2400 2756 y Fr(in)h(\(49\).)40 b(Lo)r(oking)27 b(at)i Fn(K)3196 2726 y Fk(\(2\))3285 2756 y Fr(,)g(w)n(e)g(ha)n(v)n(e)e Fo(B)i Fr(=)24 b Fn(L)3891 2768 y Fj(S)3932 2776 y Fg(1)3969 2756 y Fr(,)-260 2856 y Fo(A)f Fr(=)g Fn(L)-30 2868 y Fj(S)11 2876 y Fg(2)74 2856 y Fr(for)i Fo(")e Fn(2)g Fr(\()p Fo(")410 2868 y Fj(n)455 2856 y Fo(;)14 b(")531 2826 y Fh(0)531 2876 y Fj(n)576 2856 y Fr(\),)26 b(and)g Fo(A)d Fr(=)g Fn(L)1047 2868 y Fj(S)1088 2876 y Fg(1)1125 2856 y Fr(,)j Fo(B)h Fr(=)c Fn(L)1409 2868 y Fj(S)1450 2876 y Fg(2)1512 2856 y Fr(for)i Fo(")e Fn(2)g Fr(\()q Fo(")1849 2826 y Fh(0)1849 2876 y Fj(n)1894 2856 y Fo(;)14 b(")1970 2868 y Fj(n)p Fh(\000)p Fk(1)2099 2856 y Fr(\))q(,)26 b(i.e.)g(the)g(\014rst)f(and)h (second)f(dominan)n(t)h(harmonics)e(sw)n(ap)-260 2955 y(when)k Fo(")f Fr(go)r(es)g(across)f(the)i(v)-5 b(alue)27 b Fo(")848 2925 y Fh(0)848 2976 y Fj(n)893 2955 y Fr(.)-135 3155 y(Instead,)h(when)h(lo)r(oking)e(at)h Fn(K)857 3124 y Fk(\()q(3\))975 3155 y Fr(w)n(e)g(ha)n(v)n(e)g Fo(B)g Fr(=)c Fn(L)1528 3167 y Fj(S)1569 3175 y Fg(1)1634 3155 y Fr(for)k(an)n(y)f Fo(")e Fn(2)2062 3087 y Fi(\000)2100 3155 y Fo(")2139 3124 y Fh(0)2139 3175 y Fj(n)p Fk(+1)2268 3155 y Fo(;)14 b(")2344 3124 y Fh(0)2344 3175 y Fj(n)2389 3087 y Fi(\001)2427 3155 y Fr(.)39 b(So)29 b(the)f(\014rst)g(dominan)n (t)h(harmonic)e(of)h Fn(K)3818 3124 y Fk(\()q(3\))3936 3155 y Fr(is)-260 3254 y(alw)n(a)n(ys)g(cos)13 b Fo( )187 3266 y Fk(2)225 3254 y Fr(,)31 b(whereas)e(the)h(second)g(and)g(third)g (ones)g(sw)n(ap)f(when)h Fo(")g Fr(go)r(es)f(across)g Fo(")2543 3266 y Fj(n)2588 3254 y Fr(.)44 b(Note)30 b(that)h Fo(\021)i Fr(measures)c(the)i(size)f(of)g(the)-260 3354 y(second)h(and)g(third)g(harmonics)f(with)h(resp)r(ect)g(to)g(the)h (\014rst)f(one,)g(and)g Fo(Q)g Fr(is)g(an)g(indicator)f(of)h(the)h (relativ)n(e)e(w)n(eigh)n(t)g(of)h(the)h(second)-260 3453 y(and)g(third)h(harmonics)e(\(0)g Fo(<)g(Q)g(<)g Fr(1\).)51 b(W)-7 b(e)33 b(study)g Fn(K)1509 3423 y Fk(\(3\))1631 3453 y Fr(in)g(terms)f(of)g Fo(\021)k Fr(and)c Fo(Q)p Fr(,)i(considering)d Fo(\021)36 b Fr(as)c(a)g(p)r(erturbation)g (parameter)-260 3553 y(\(note)d(that)g Fo(\021)f Fn(\030)d(L)355 3565 y Fj(S)396 3573 y Fg(2)432 3553 y Fo(=)p Fn(L)531 3565 y Fj(S)572 3573 y Fg(1)637 3553 y Fr(is)k(small)f(except)h(for)f Fo(")g Fr(close)g(to)h(the)g(endp)r(oin)n(ts)f Fo(")2262 3523 y Fh(0)2262 3574 y Fj(n)2307 3553 y Fr(,)h Fo(")2398 3523 y Fh(0)2398 3574 y Fj(n)p Fk(+1)2527 3553 y Fr(\).)41 b(Ho)n(w)n(ev)n(er,)27 b(w)n(e)h(ha)n(v)n(e)g(to)g(p)r(oin)n(t)h(out)g (that)g Fo(\021)-260 3653 y Fr(and)f Fo(Q)f Fr(are)f(not)i(indep)r (enden)n(t)h(parameters,)c(b)r(ecause)j(they)f(are)g(b)r(oth)h(link)n (ed)g(to)f Fo(")p Fr(.)-135 3852 y(In)d(the)g(next)g(lemma,)h(w)n(e)e (sho)n(w)g(the)i(existence)e(of)h(4)f(critical)h(p)r(oin)n(ts)g(for)f Fo(\021)k Fr(small)d(enough)f(and)h(an)n(y)f Fo(Q)p Fr(,)h(pro)n(vided) f(the)h(di\013erence)-260 3952 y(of)k(phases)f Fn(4)p Fo(\034)k Fn(2)23 b Fl(T)28 b Fr(is)f(not)h(v)n(ery)e(close)h(to)g(0)g (or)g Fo(\031)k Fr(\(mo)r(d)42 b(2)p Fo(\031)s Fr(\).)37 b(T)-7 b(o)27 b(measure)g(this)h(closeness,)e(w)n(e)h(denote)1389 4134 y Fo(\034)1434 4100 y Fh(\003)1496 4134 y Fr(=)c(min\()p Fn(j4)p Fo(\034)9 b Fn(j)14 b Fo(;)g Fn(j4)p Fo(\034)27 b Fn(\000)18 b Fo(\031)s Fn(j)p Fr(\))p Fo(:)-260 4417 y Fs(Lemma)30 b(6)-200 4683 y Fr(\(a\))42 b Fp(The)36 b(function)g Fn(K)520 4653 y Fk(\()q(2\))645 4683 y Fp(has)h(exactly)f (4)g(critic)l(al)h(p)l(oints,)h(al)t(l)f(nonde)l(gener)l(ate:)51 b Fo( )2492 4640 y Fk(\(2\))2489 4711 y(\(1\))2615 4683 y Fr(=)33 b(\(0)p Fo(;)14 b Fr(0\))p Fp(,)37 b Fo( )3017 4640 y Fk(\(2\))3014 4711 y(\(2\))3140 4683 y Fr(=)d(\(0)p Fo(;)14 b(\031)s Fr(\))p Fp(,)38 b Fo( )3552 4640 y Fk(\(2\))3549 4711 y(\(3\))3675 4683 y Fr(=)33 b(\()p Fo(\031)s(;)14 b Fr(0\))p Fp(,)-52 4830 y Fo( )5 4786 y Fk(\(2\))2 4858 y(\(4\))117 4830 y Fr(=)23 b(\()p Fo(\031)s(;)14 b(\031)s Fr(\))p Fp(.)39 b(A)n(t)29 b(the)h(critic)l(al)h(p)l(oints,)1274 4734 y Fi(\014)1274 4784 y(\014)1274 4834 y(\014)1301 4830 y Fr(det)14 b(D)1493 4799 y Fk(2)1531 4830 y Fn(K)1595 4799 y Fk(\(2\))1698 4737 y Fi(\020)1748 4830 y Fo( )1805 4786 y Fk(\(2\))1802 4858 y(\()p Fj(j)s Fk(\))1894 4737 y Fi(\021)1944 4734 y(\014)1944 4784 y(\014)1944 4834 y(\014)1995 4830 y Fr(=)22 b Fo(AB)t Fp(.)-204 5033 y Fr(\(b\))42 b Fp(Assume)29 b Fo(\034)303 5002 y Fh(\003)365 5033 y Fo(>)22 b Fr(0)29 b Fp(and)i(de\014ne)f Fo(E)993 5002 y Fk(\()p Fh(\006)p Fk(\))1101 5033 y Fp(,)g Fo(\013)1209 5002 y Fk(\()p Fh(\006)p Fk(\))1347 5033 y Fp(by)1029 5227 y Fo(E)1095 5192 y Fk(\()p Fh(\006)p Fk(\))1226 5227 y Fr(=)1314 5152 y Fi(p)p 1397 5152 976 4 v 75 x Fr(1)18 b Fn(\000)g Fr(2)p Fo(Q)p Fr(\(1)f Fn(\000)h Fo(Q)p Fr(\)\(1)h Fn(\007)f Fr(cos)13 b Fn(4)p Fo(\034)c Fr(\))o Fo(;)1029 5399 y Fr(cos)k Fo(\013)1207 5365 y Fk(\()q Fh(\006)p Fk(\))1339 5399 y Fr(=)1436 5343 y(\(1)18 b Fn(\000)g Fo(Q)p Fr(\))h Fn(\006)f Fo(Q)c Fr(cos)e Fn(4)p Fo(\034)p 1436 5380 699 4 v 1699 5458 a(E)1765 5434 y Fk(\()p Fh(\006)p Fk(\))2145 5399 y Fo(;)99 b Fr(sin)13 b Fo(\013)2435 5365 y Fk(\()q Fh(\006)p Fk(\))2567 5399 y Fr(=)2664 5343 y Fn(\006)p Fo(Q)h Fr(sin)f Fn(4)p Fo(\034)p 2664 5380 380 4 v 2767 5458 a(E)2833 5434 y Fk(\()p Fh(\006)p Fk(\))3054 5399 y Fo(:)-52 5623 y Fp(Then,)41 b(for)e(any)f Fo(Q)g Fn(2)g Fr([0)p Fo(;)14 b Fr(1])37 b Fp(and)h Fr(0)g Fo(<)f(\021)k Fn(\026)d Fo(\034)1490 5593 y Fh(\003)1566 5623 y Fp(the)h(function)e Fn(K)2111 5593 y Fk(\()q(3\))2239 5623 y Fp(has)i(exactly)f(4)g(critic)l(al)i(p)l (oints,)g(al)t(l)f(nonde)l(gener)l(ate:)-52 5739 y Fo( )5 5696 y Fk(\(3\))2 5767 y(\()p Fj(j)s Fk(\))121 5739 y Fr(=)27 b Fo( )270 5696 y Fk(\(3\))267 5767 y(\()p Fj(j)s Fk(\))q Fj(;)p Fk(0)427 5739 y Fr(+)20 b Fn(O)r Fr(\()p Fo(\021)s Fr(\))p Fp(,)34 b Fo(j)f Fr(=)27 b(1)p Fo(;)14 b Fr(2)p Fo(;)g Fr(3)p Fo(;)g Fr(4)p Fp(,)30 b(wher)l(e)j Fo( )1534 5696 y Fk(\(3\))1531 5767 y(\(1\))p Fj(;)p Fk(0)1701 5739 y Fr(=)1793 5672 y Fi(\000)1831 5739 y Fo(\013)1884 5709 y Fk(\(+\))1991 5739 y Fo(;)14 b Fr(0)2070 5672 y Fi(\001)2108 5739 y Fp(,)33 b Fo( )2223 5696 y Fk(\(3\))2220 5767 y(\(2\))p Fj(;)p Fk(0)2389 5739 y Fr(=)2481 5672 y Fi(\000)2519 5739 y Fo(\013)2572 5709 y Fk(\()q Fh(\000)p Fk(\))2681 5739 y Fo(;)14 b(\031)2768 5672 y Fi(\001)2806 5739 y Fp(,)33 b Fo( )2921 5696 y Fk(\(3\))2918 5767 y(\(3\))p Fj(;)p Fk(0)3087 5739 y Fr(=)3179 5672 y Fi(\000)3217 5739 y Fo(\013)3270 5709 y Fk(\()q(+\))3396 5739 y Fr(+)18 b Fo(\031)s(;)c Fr(0)3608 5672 y Fi(\001)3646 5739 y Fp(,)33 b Fo( )3761 5696 y Fk(\(3\))3758 5767 y(\(4\))p Fj(;)p Fk(0)3928 5739 y Fr(=)1825 5989 y(16)p eop %%Page: 17 17 17 16 bop -52 -149 a Fi(\000)-14 -82 y Fo(\013)39 -112 y Fk(\()p Fh(\000)p Fk(\))166 -82 y Fr(+)18 b Fo(\031)s(;)c(\031)386 -149 y Fi(\001)424 -82 y Fp(.)39 b(A)n(t)29 b(the)h(critic)l(al)h(p)l (oints,)1210 29 y Fi(\014)1210 79 y(\014)1210 129 y(\014)1238 125 y Fr(det)14 b(D)1430 90 y Fk(2)1468 125 y Fn(K)1532 90 y Fk(\(3\))1635 32 y Fi(\020)1685 125 y Fo( )1742 82 y Fk(\(3\))1739 153 y(\(1)p Fj(;)p Fk(3\))1881 32 y Fi(\021)1930 29 y(\014)1930 79 y(\014)1930 129 y(\014)1981 125 y Fr(=)23 b Fo(B)2136 90 y Fk(2)2187 32 y Fi(\020)2237 125 y Fo(E)2303 90 y Fk(\(+\))2410 125 y Fo(\021)e Fr(+)d Fn(O)2638 57 y Fi(\000)2676 125 y Fo(\021)2720 90 y Fk(2)2757 57 y Fi(\001)2795 32 y(\021)2859 125 y Fo(;)1210 212 y Fi(\014)1210 262 y(\014)1210 311 y(\014)1238 307 y Fr(det)c(D)1430 273 y Fk(2)1468 307 y Fn(K)1532 273 y Fk(\(3\))1635 215 y Fi(\020)1685 307 y Fo( )1742 264 y Fk(\(3\))1739 336 y(\(2)p Fj(;)p Fk(4\))1881 215 y Fi(\021)1930 212 y(\014)1930 262 y(\014)1930 311 y(\014)1981 307 y Fr(=)23 b Fo(B)2136 273 y Fk(2)2187 215 y Fi(\020)2237 307 y Fo(E)2303 273 y Fk(\()p Fh(\000)p Fk(\))2411 307 y Fo(\021)e Fr(+)d Fn(O)2639 240 y Fi(\000)2677 307 y Fo(\021)2721 273 y Fk(2)2758 240 y Fi(\001)2796 215 y(\021)2860 307 y Fo(:)-260 605 y Fp(Pr)l(o)l(of.)61 b Fr(W)-7 b(e)30 b(do)f(not)g(pro)n(v)n(e)f(part)h(\(a\),)h(b)r(ecause)f(it)h(is)f(v)n (ery)f(simple.)43 b(Instead,)29 b(the)h(pro)r(of)f(of)h(\(b\))g (requires)e(some)h(more)f(w)n(ork.)41 b(The)-260 705 y(critical)27 b(p)r(oin)n(ts)h(of)f Fn(K)425 675 y Fk(\()q(3\))542 705 y Fr(are)g(the)h(solutions)f(of)g(the)h(follo)n(wing)f(system)g(of) h(equations:)576 879 y(sin)13 b Fo( )745 891 y Fk(2)806 879 y Fr(=)22 b Fo(\021)s Fr(\(1)d Fn(\000)f Fo(Q)p Fr(\))c(sin)f Fo( )1394 891 y Fk(1)1432 879 y Fo(;)180 b Fr(\(1)18 b Fn(\000)g Fo(Q)p Fr(\))c(sin)g Fo( )2092 891 y Fk(1)2147 879 y Fr(+)k Fo(Q)c Fr(sin\()p Fo( )2498 891 y Fk(1)2554 879 y Fr(+)k Fo( )2691 891 y Fk(2)2747 879 y Fn(\000)g(4)p Fo(\034)9 b Fr(\))22 b(=)h(0)p Fo(:)688 b Fr(\(56\))-260 1052 y(It)28 b(is)g(clear)e(that,)i(for)f Fo(\021)k Fr(small)c(enough,) g(the)h(solutions)f(of)h(the)g(\014rst)f(equation)g(of)h(\(56\))f(are)f (t)n(w)n(o)h(curv)n(es)g(in)g Fl(T)3250 1022 y Fk(2)3287 1052 y Fr(,)h(de\014ned)g(b)n(y)1094 1226 y Fo( )1148 1238 y Fk(2)1209 1226 y Fr(=)22 b Fo(f)1337 1238 y Fj(\021)1377 1226 y Fr(\()p Fo( )1463 1238 y Fk(1)1501 1226 y Fr(\))h(=)g Fn(O)r Fr(\()p Fo(\021)s Fr(\))p Fo(;)181 b( )2078 1238 y Fk(2)2139 1226 y Fr(=)22 b Fo(\031)g Fn(\000)c Fo(f)2419 1238 y Fj(\021)2459 1226 y Fr(\()p Fo( )2545 1238 y Fk(1)2583 1226 y Fr(\))p Fo(:)-260 1419 y Fr(Replacing)27 b Fo( )178 1431 y Fk(2)239 1419 y Fr(=)22 b Fo(f)367 1431 y Fj(\021)407 1419 y Fr(\()p Fo( )493 1431 y Fk(1)531 1419 y Fr(\))28 b(in)n(to)f(the)h(second)f(equation)g(of)h(\(56\),)f(one)g(obtains)g (the)h(equation)f Fo(F)2792 1376 y Fk(\(+\))2780 1429 y Fj(\021)2899 1419 y Fr(\()p Fo( )2985 1431 y Fk(1)3023 1419 y Fr(\))c(=)g(0,)k(with)871 1603 y Fo(F)936 1569 y Fk(\(+\))924 1623 y Fj(\021)1043 1603 y Fr(\()p Fo( )1129 1615 y Fk(1)1166 1603 y Fr(\))84 b(=)e(\(1)18 b Fn(\000)g Fo(Q)p Fr(\))c(sin)g Fo( )1886 1615 y Fk(1)1942 1603 y Fr(+)k Fo(Q)c Fr(sin)o(\()p Fo( )2292 1615 y Fk(1)2348 1603 y Fn(\000)k(4)p Fo(\034)9 b Fr(\))18 b(+)g Fn(O)r Fr(\()p Fo(\021)s Fr(\))1282 1763 y(=)82 b Fo(E)1495 1729 y Fk(\(+\))1616 1763 y Fr(sin)1732 1671 y Fi(\020)1781 1763 y Fo( )1835 1775 y Fk(1)1891 1763 y Fn(\000)18 b Fo(\013)2027 1729 y Fk(\()q(+\))2135 1671 y Fi(\021)2203 1763 y Fr(+)g Fn(O)r Fr(\()p Fo(\021)s Fr(\))p Fo(:)-260 1973 y Fr(F)-7 b(or)28 b Fo(\021)h Fr(=)24 b(0,)29 b(the)g(solutions)f (are)g(clearly)f Fo(\013)1094 1943 y Fk(\()q(+\))1231 1973 y Fr(and)h Fo(\013)1446 1943 y Fk(\()q(+\))1573 1973 y Fr(+)19 b Fo(\031)s Fr(,)29 b(except)g(for)f(the)i(case)d(that)i Fo(E)2717 1943 y Fk(\(+\))2850 1973 y Fr(=)24 b(0)29 b(\(whic)n(h)g(happ)r(ens)f(if)i Fn(4)p Fo(\034)j Fr(=)25 b Fo(\031)s Fr(,)-260 2073 y Fo(Q)e Fr(=)f(1)p Fo(=)p Fr(2\).)36 b(Note)27 b(that)g Fo(E)578 2043 y Fk(\(+\))708 2073 y Fn(\025)796 2002 y Fi(p)p 879 2002 536 4 v 71 x Fr(\(1)18 b(+)g(cos)13 b Fn(4)p Fo(\034)c Fr(\))p Fo(=)p Fr(2)22 b Fn(\027)h Fo(\034)1570 2043 y Fh(\003)1636 2073 y Fr(and,)k(consequen)n(tly)-7 b(,)26 b(these)h(solutions)f(p)r (ersist)h(for)f Fo(\021)h Fn(\026)22 b Fo(\034)3477 2043 y Fh(\003)3543 2073 y Fr(\(in)28 b(fact,)f(this)-260 2193 y(is)g(a)g(simple)g(consequence)f(of)h(Prop)r(osition)e(9\).)36 b(The)28 b(p)r(erturb)r(ed)f(solutions)f(obtained)h(giv)n(e)f(rise)g (to)h(the)g(critical)g(p)r(oin)n(ts)f Fo( )3683 2150 y Fk(\(3\))3680 2222 y(\(1\))3773 2193 y Fr(,)h Fo( )3880 2150 y Fk(\(3\))3877 2222 y(\(3\))3969 2193 y Fr(.)-135 2431 y(Analogously)-7 b(,)20 b(one)g(can)f(replace)h Fo( )964 2443 y Fk(2)1024 2431 y Fr(=)j Fo(\031)7 b Fn(\000)t Fo(f)1276 2443 y Fj(\021)1315 2431 y Fr(\()p Fo( )1401 2443 y Fk(1)1439 2431 y Fr(\))20 b(in)n(to)g(the)h(second)e(equation)h (of)g(\(56\),)h(obtaining)f(the)h(equation)e Fo(F)3553 2388 y Fk(\()p Fh(\000)p Fk(\))3541 2440 y Fj(\021)3661 2431 y Fr(\()p Fo( )3747 2443 y Fk(1)3785 2431 y Fr(\))k(=)g(0,)-260 2530 y(with)870 2704 y Fo(F)935 2670 y Fk(\()p Fh(\000)p Fk(\))923 2725 y Fj(\021)1043 2704 y Fr(\()p Fo( )1129 2716 y Fk(1)1167 2704 y Fr(\))83 b(=)g(\(1)18 b Fn(\000)g Fo(Q)p Fr(\))c(sin)f Fo( )1886 2716 y Fk(1)1942 2704 y Fn(\000)18 b Fo(Q)c Fr(sin\()p Fo( )2293 2716 y Fk(1)2349 2704 y Fn(\000)k(4)p Fo(\034)9 b Fr(\))18 b(+)g Fn(O)r Fr(\()p Fo(\021)s Fr(\))1282 2864 y(=)83 b Fo(E)1496 2830 y Fk(\()p Fh(\000)p Fk(\))1617 2864 y Fr(sin)1733 2772 y Fi(\020)1783 2864 y Fo( )1837 2876 y Fk(1)1893 2864 y Fn(\000)18 b Fo(\013)2029 2830 y Fk(\()p Fh(\000)p Fk(\))2137 2772 y Fi(\021)2205 2864 y Fr(+)g Fn(O)r Fr(\()p Fo(\021)s Fr(\))p Fo(;)-260 3075 y Fr(and)34 b(for)g Fo(\021)i Fr(=)e(0)f(the)i(solutions)e(are)g Fo(\013)994 3044 y Fk(\()q Fh(\000)p Fk(\))1137 3075 y Fr(and)g Fo(\013)1357 3044 y Fk(\()q Fh(\000)p Fk(\))1489 3075 y Fr(+)22 b Fo(\031)s Fr(,)36 b(except)e(for)f(the)i(case)e(that)h Fo(E)2669 3044 y Fk(\()p Fh(\000)p Fk(\))2811 3075 y Fr(=)f(0)h(\(whic)n(h)g(happ)r(ens)g(if)h Fn(4)p Fo(\034)42 b Fr(=)34 b(0,)-260 3190 y Fo(Q)24 b Fr(=)f(1)p Fo(=)p Fr(2\).)37 b(In)28 b(this)g(case,)g Fo(E)668 3160 y Fk(\()p Fh(\000)p Fk(\))799 3190 y Fn(\025)888 3119 y Fi(p)p 971 3119 V 71 x Fr(\(1)18 b Fn(\000)g Fr(cos)13 b Fn(4)p Fo(\034)c Fr(\))p Fo(=)p Fr(2)23 b Fn(\027)g Fo(\034)1663 3160 y Fh(\003)1730 3190 y Fr(and,)28 b(pro)r(ceeding)f(as)g(b)r (efore,)h(one)g(obtains)f(the)i(critical)e(p)r(oin)n(ts)h Fo( )3880 3147 y Fk(\(3\))3877 3219 y(\(2\))3969 3190 y Fr(,)-260 3328 y Fo( )-203 3285 y Fk(\(3\))-206 3356 y(\(4\))-114 3328 y Fr(.)-135 3536 y(The)f(determinan)n(t)h(is)f (easily)g(computed.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)408 3710 y(det)15 b(D)601 3676 y Fk(2)638 3710 y Fn(K)702 3676 y Fk(\()q(3\))792 3710 y Fr(\()p Fo( )s Fr(\))24 b(=)e Fo(B)1091 3676 y Fk(2)1142 3643 y Fi(\000)1180 3710 y Fo(\021)c Fr(cos)13 b Fo( )1418 3722 y Fk(2)1473 3710 y Fn(\001)19 b Fr(\(\(1)f Fn(\000)g Fo(Q)p Fr(\))c(cos)f Fo( )2013 3722 y Fk(1)2069 3710 y Fr(+)18 b Fo(Q)c Fr(cos)o(\()p Fo( )2429 3722 y Fk(1)2485 3710 y Fr(+)k Fo( )2622 3722 y Fk(2)2677 3710 y Fn(\000)g(4)p Fo(\034)9 b Fr(\)\))19 b(+)f Fn(O)3128 3643 y Fi(\000)3166 3710 y Fo(\021)3210 3676 y Fk(2)3248 3643 y Fi(\001\001)-260 3908 y Fr(for)27 b(an)n(y)g Fo( )f Fn(2)e Fl(T)239 3878 y Fk(2)275 3908 y Fr(.)37 b(A)n(t)28 b(the)g(p)r(oin)n(t)g Fo( )872 3865 y Fk(\(3\))869 3937 y(\(1\))984 3908 y Fr(=)1072 3841 y Fi(\000)1110 3908 y Fo(\013)1163 3878 y Fk(\(+\))1270 3908 y Fo(;)14 b Fr(0)1349 3841 y Fi(\001)1405 3908 y Fr(+)k Fn(O)r Fr(\()p Fo(\021)s Fr(\),)29 b(w)n(e)e(obtain)424 4154 y(det)14 b(D)616 4119 y Fk(2)654 4154 y Fn(K)718 4119 y Fk(\(3\))821 4062 y Fi(\020)871 4154 y Fo( )928 4111 y Fk(\(3\))925 4182 y(\(1\))1017 4062 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b(the)h(harmonic) -260 528 y(cos)o(\()p Fo( )-63 540 y Fk(1)-2 528 y Fr(+)23 b Fo( )140 540 y Fk(2)201 528 y Fn(\000)f(4)p Fo(\034)9 b Fr(\)\).)59 b(Assuming)35 b(0)f Fo(<)h Fn(4)p Fo(\034)43 b(<)35 b(\031)s Fr(,)i(the)e(p)r(oin)n(t)g Fo( )1914 485 y Fk(\(3\))1911 557 y(\(1\))p Fj(;)p Fk(0)2088 528 y Fr(drifts)g(on)f(a)h(line)g(from)f(\(0)p Fo(;)14 b Fr(0\))35 b(to)f(\()p Fn(4)p Fo(\034)5 b(;)14 b Fr(0\))34 b(with)h(the)g(\014rst)-260 666 y(comp)r(onen)n(t)c(increasing.)48 b(On)31 b(the)h(other)f(hand,)h(the)g(p)r(oin)n(t)g Fo( )1771 623 y Fk(\(3\))1768 694 y(\(2\))p Fj(;)p Fk(0)1941 666 y Fr(drifts)g(on)f(a)g(line)h(from)f(\(0)p Fo(;)14 b(\031)s Fr(\))32 b(to)g(\()p Fn(4)p Fo(\034)e Fr(+)20 b Fo(\031)s(;)14 b(\031)s Fr(\))33 b(with)f(the)f(\014rst)-260 804 y(comp)r(onen)n(t)25 b(decreasing.)34 b(Recall)25 b(that)g Fo( )1077 761 y Fk(\(3\))1074 832 y(\(3)p Fj(;)p Fk(4\))p Fj(;)p Fk(0)1292 804 y Fr(=)d Fo( )1436 761 y Fk(\(3\))1433 832 y(\(1)p Fj(;)p Fk(2\))p Fj(;)p Fk(0)1641 804 y Fr(+)13 b(\()p Fo(\031)s(;)h Fr(0\).)36 b(If)25 b(one)f(assumes)g Fo(\031)j(<)22 b Fn(4)p Fo(\034)32 b(<)23 b Fr(2)p Fo(\031)s Fr(,)i(the)g(situation)g (is)f(analogous,)-260 942 y(but)29 b(the)f(p)r(oin)n(ts)f(rev)n(erse)f (their)i(motion.)37 b(When)28 b(one)g(considers)e(the)j(p)r(erturb)r (ed)f(p)r(oin)n(ts)f Fo( )2665 899 y Fk(\(3\))2662 970 y(\()p Fj(j)s Fk(\))2778 942 y Fr(=)c Fo( )2923 899 y Fk(\(3\))2920 970 y(\()p Fj(j)s Fk(\))q Fj(;)p Fk(0)3078 942 y Fr(+)18 b Fn(O)r Fr(\()p Fo(\021)s Fr(\),)29 b(the)g(lines)e(b)r (ecome)-260 1055 y(close)21 b(curv)n(es.)34 b(This)22 b(picture)f(do)r(es)h(not)g(hold)f(for)h Fn(4)p Fo(\034)30 b Fr(near)21 b(to)h(0)f(or)g Fo(\031)k Fr(\(see)d(remark)e(2\),)j(b)r (ecause)e(in)h(this)g(case)f(there)h(are)f(bifurcations)-260 1155 y(of)28 b(some)f(of)g(the)h(critical)f(p)r(oin)n(ts.)-135 1354 y(Next,)i(w)n(e)f(translate)g(the)h(results)f(of)h(Lemma)f(6)g (from)h(the)g(functions)g Fn(K)2200 1324 y Fk(\()p Fj(i)p Fk(\))2280 1354 y Fr(\()p Fo( )s 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Fk(\))3221 1876 y Fr(;)h(w)n(e)e(denote)g Fo(m)3741 1833 y Fk(\()p Fj(i)p Fk(\))3741 1904 y(\()p Fj(j)s Fk(\))3858 1876 y Fr(this)-260 2014 y(eigen)n(v)-5 b(alue,)27 b(closely)g(related) g(with)h(the)g(transv)n(ersalit)n(y)d(of)i(the)h(homo)r(clinic)g(orbit) f(asso)r(ciated)f(to)h(the)h(critical)f(p)r(oin)n(t)h Fo(\022)3585 1971 y Fk(\()p Fj(i)p Fk(\))3583 2042 y(\()p Fj(j)s Fk(\))3670 2014 y Fr(.)-260 2296 y Fs(Lemma)i(7)-200 2555 y Fr(\(a\))42 b Fp(The)30 b(function)g Fn(L)501 2525 y Fk(\(2\))620 2555 y Fp(has)h(exactly)f(4)g(critic)l(al)h(p)l (oints)f Fo(\022)1681 2512 y Fk(\(2\))1679 2583 y(\()p Fj(j)s Fk(\))1770 2555 y Fp(,)h(given)f(by)g(\(57\),)h(al)t(l)g(nonde)l (gener)l(ate)f(and)g(satisfying)1384 2767 y Fo(m)1457 2724 y Fk(\(2\))1457 2795 y(\()p Fj(j)s Fk(\))1569 2767 y Fn(\030)1657 2703 y(p)p 1726 2703 39 4 v 64 x Fo(")p Fn(L)1822 2779 y Fj(S)1863 2787 y Fg(2)1899 2767 y Fo(;)184 b(j)28 b Fr(=)23 b(1)p Fo(;)14 b Fr(2)p Fo(;)g Fr(3)p Fo(;)g Fr(4)p Fo(:)-204 3011 y Fr(\(b\))42 b Fp(Assuming)d Fo(\034)385 2981 y Fh(\003)466 3011 y Fo(>)i Fr(0)f Fp(and)g Fn(L)882 3023 y Fj(S)923 3031 y Fg(2)1002 3011 y Fn(\026)h Fo(\034)1153 2981 y Fh(\003)1192 3011 y Fn(L)1249 3023 y Fj(S)1290 3031 y Fg(1)1327 3011 y Fp(,)i(the)d(function)g Fn(L)1937 2981 y Fk(\(3\))2067 3011 y Fp(has)h(exactly)f(4)h(critic)l (al)g(p)l(oints)f Fo(\022)3179 2968 y Fk(\(3\))3177 3039 y(\()p Fj(j)s Fk(\))3269 3011 y Fp(,)j(given)e(by)f(\(57\),)k(al)t(l) -52 3120 y(nonde)l(gener)l(ate)30 b(and)g(satisfying)1166 3231 y Fo(\034)1211 3196 y Fh(\003)1250 3167 y Fn(p)p 1319 3167 V 64 x Fo(")p Fn(L)1415 3243 y Fj(S)1456 3251 y Fg(2)1515 3231 y Fn(\026)23 b Fo(m)1676 3188 y Fk(\(3\))1676 3259 y(\()p Fj(j)s Fk(\))1788 3231 y Fn(\026)1876 3167 y(p)p 1945 3167 V 64 x Fo(")o Fn(L)2040 3243 y Fj(S)2081 3251 y Fg(2)2118 3231 y Fo(;)184 b(j)28 b Fr(=)22 b(1)p Fo(;)14 b Fr(2)p Fo(;)g Fr(3)p Fo(;)g Fr(4)p Fo(:)-260 3551 y Fp(Pr)l(o)l(of.)57 b Fr(F)-7 b(or)27 b(the)h(minim)n(um)g(eigen) n(v)-5 b(alue)27 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Grant, and David Carlisle. %% All rights reserved. %% %% This file is part of the PSfrag package. %% userdict begin /PSfragLib 90 dict def /PSfragDict 6 dict def /PSfrag { PSfragLib begin load exec end } bind def end PSfragLib begin /RO /readonly load def /CP /currentpoint load def /CM /currentmatrix load def /B { bind RO def } bind def /X { exch def } B /MD { { X } forall } B /OE { end exec PSfragLib begin } B /S false def /tstr 8 string def /islev2 { languagelevel } stopped { false } { 2 ge } ifelse def [ /sM /tM /srcM /dstM /dM /idM /srcFM /dstFM ] { matrix def } forall sM currentmatrix RO pop dM defaultmatrix RO idM invertmatrix RO pop srcFM identmatrix pop /Hide { gsave { CP } stopped not newpath clip { moveto } if } B /Unhide { { CP } stopped not grestore { moveto } if } B /setrepl islev2 {{ /glob currentglobal def true setglobal array astore globaldict exch /PSfrags exch put glob setglobal }} {{ array astore /PSfrags X }} ifelse B /getrepl islev2 {{ globaldict /PSfrags get aload length }} {{ PSfrags aload length }} ifelse B /convert { /src X src length string /c 0 def src length { dup c src c get dup 32 lt { pop 32 } if put /c c 1 add def } repeat } B /Begin { /saver save def srcFM exch 3 exch put 0 ne /debugMode X 0 setrepl dup /S exch dict def { S 3 1 roll exch convert exch put } repeat srcM CM dup invertmatrix pop mark { currentdict { end } stopped { pop exit } if } loop PSfragDict counttomark { begin } repeat pop } B /End { mark { currentdict end dup PSfragDict eq { pop exit } if } loop counttomark { begin } repeat pop getrepl saver restore 7 idiv dup /S exch dict def { 6 array astore /mtrx X tstr cvs /K X S K [ S K known { S K get aload pop } if mtrx ] put } repeat } B /Place { tstr cvs /K X S K known { bind /proc X tM CM pop CP /cY X /cX X 0 0 transform idtransform neg /aY X neg /aX X S K get dup length /maxiter X /iter 1 def { iter maxiter ne { /saver save def } if tM setmatrix aX aY translate [ exch aload pop idtransform ] concat cX neg cY neg translate cX cY moveto /proc load OE iter maxiter ne { saver restore /iter iter 1 add def } if } forall /noXY { CP /cY X /cX X } stopped def tM setmatrix noXY { newpath } { cX cY moveto } ifelse } { Hide OE Unhide } ifelse } B /normalize { 2 index dup mul 2 index dup mul add sqrt div dup 4 -1 roll exch mul 3 1 roll mul } B /replace { aload pop MD CP /bY X /lX X gsave sM setmatrix str stringwidth abs exch abs add dup 0 eq { pop } { 360 exch div dup scale } ifelse lX neg bY neg translate newpath lX bY moveto str { /ch X ( ) dup 0 ch put false charpath ch Kproc } forall flattenpath pathbbox [ /uY /uX /lY /lX ] MD CP grestore moveto currentfont /FontMatrix get dstFM copy dup 0 get 0 lt { uX lX /uX X /lX X } if 3 get 0 lt { uY lY /uY X /lY X } if /cX uX lX add 0.5 mul def /cY uY lY add 0.5 mul def debugMode { gsave 0 setgray 1 setlinewidth lX lY moveto lX uY lineto uX uY lineto uX lY lineto closepath lX bY moveto uX bY lineto lX cY moveto uX cY lineto cX lY moveto cX uY lineto stroke grestore } if dstFM dup invertmatrix dstM CM srcM 2 { dstM concatmatrix } repeat pop getrepl /temp X S str convert get { aload pop [ /rot /scl /loc /K ] MD /aX cX def /aY cY def loc { dup 66 eq { /aY bY def } { % B dup 98 eq { /aY lY def } { % b dup 108 eq { /aX lX def } { % l dup 114 eq { /aX uX def } { % r dup 116 eq { /aY uY def } % t if } ifelse } ifelse } ifelse } ifelse pop } forall K srcFM rot tM rotate dstM 2 { tM concatmatrix } repeat aload pop pop pop 2 { scl normalize 4 2 roll } repeat aX aY transform /temp temp 7 add def } forall temp setrepl } B /Rif { S 3 index convert known { pop replace } { exch pop OE } ifelse } B /XA { bind [ /Kproc /str } B /XC { ] 2 array astore def } B /xs { pop } XA XC /xks { /kern load OE } XA /kern XC /xas { pop ax ay rmoveto } XA /ay /ax XC /xws { c eq { cx cy rmoveto } if } XA /c /cy /cx XC /xaws { ax ay rmoveto c eq { cx cy rmoveto } if } XA /ay /ax /c /cy /cx XC /raws { xaws { awidthshow } Rif } B /rws { xws { widthshow } Rif } B /rks { xks { kshow } Rif } B /ras { xas { ashow } Rif } B /rs { xs { show } Rif } B /rrs { getrepl dup 2 add -1 roll //restore exec setrepl } B PSfragDict begin islev2 not { /restore { /rrs PSfrag } B } if /show { /rs PSfrag } B /kshow { /rks PSfrag } B /ashow { /ras PSfrag } B /widthshow { /rws PSfrag } B /awidthshow { /raws PSfrag } B end PSfragDict RO pop end %%EndProcSet %%BeginProcSet: special.pro %! 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copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 420 934 M (texbullet) Cshow 836 934 M (texcirc) Cshow 420 2243 M (texbullet) Cshow 2145 2243 M (texcirc) Cshow 1728 934 M (texbullet) Cshow 2145 934 M (texcirc) Cshow 1728 2243 M (texbullet) Cshow 836 2243 M (texcirc) Cshow 253 768 M (tex0) Cshow 1728 768 M (texpi) Cshow 253 2243 M (texpi) Cshow 836 768 M (texttau) Cshow 2228 768 M (texttaupi) Cshow 628 1222 M (texpsi1) Cshow 2591 2597 M (texpsi2) Cshow 1937 3263 M (texpsi3) Cshow 1282 2305 M (texpsi4) Cshow 1728 280 M (interval1) Cshow 4345 934 M (texbullet) Cshow 4762 934 M (texcirc) Cshow 4345 2243 M 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a -225 2392 a Fd(PSfrag)26 b(replacemen)n(ts)p -225 2422 741 4 v 515 2425 a /Unhide PSfrag 515 2425 a 494 2506 a { 494 2506 a 473 2524 a Fc(\017)494 2506 y } 0/Place PSfrag 494 2506 a 494 2606 a { 494 2606 a 473 2624 a Fc(\016)494 2606 y } 1/Place PSfrag 494 2606 a 494 2697 a { 494 2697 a 473 2724 a Fd(0)494 2697 y } 2/Place PSfrag 494 2697 a 490 2805 a { 490 2805 a 465 2823 a Fb(\031)490 2805 y } 3/Place PSfrag 490 2805 a 455 2886 a { 455 2886 a 395 2907 a Fc(4)p Fb(\034)455 2886 y } 4/Place PSfrag 455 2886 a 379 2986 a { 379 2986 a 243 3006 a Fc(4)p Fb(\034)h Fd(+)18 b Fb(\031)379 2986 y } 5/Place PSfrag 379 2986 a 442 3096 a { 442 3096 a 369 3118 a Fb( )426 3074 y Fa(\(3\))423 3146 y(\(1\))442 3096 y } 6/Place PSfrag 442 3096 a 442 3233 a { 442 3233 a 369 3255 a Fb( )426 3212 y Fa(\(3\))423 3284 y(\(2\))442 3233 y } 7/Place PSfrag 442 3233 a 442 3371 a { 442 3371 a 369 3393 a Fb( )426 3350 y Fa(\(3\))423 3422 y(\(3\))442 3371 y } 8/Place PSfrag 442 3371 a 442 3509 a { 442 3509 a 369 3531 a Fb( )426 3488 y Fa(\(3\))423 3559 y(\(4\))442 3509 y } 9/Place PSfrag 442 3509 a 298 3637 a { 298 3637 a 82 3657 a Fd(0)23 b Fb(<)f Fc(4)p Fb(\034)32 b(<)22 b(\031)298 3637 y } 10/Place PSfrag 298 3637 a 273 3736 a { 273 3736 a 32 3757 a Fb(\031)k(<)d Fc(4)p Fb(\034)31 b(<)23 b Fd(2)p Fb(\031)273 3736 y } 11/Place PSfrag 273 3736 a eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF %%EndDocument @endspecial 657 5724 a(Figure)27 b(2:)36 b(Con)n(tin)n(uation)27 b(of)g(the)h(critical)f(p)r(oin)n(ts)h(from)f Fo(Q)c Fr(=)f(0)28 b(to)f Fo(Q)c Fr(=)f(1)1825 5989 y(18)p eop %%Page: 19 19 19 18 bop -260 -70 a Fr(where)35 b(w)n(e)g(denote)g Fo(D)i Fr(=)e(det)14 b(D)791 -100 y Fk(2)829 -70 y Fn(L)886 -100 y Fk(\(2\))989 -162 y Fi(\020)1039 -70 y Fo(\022)1080 -113 y Fk(\(2\))1078 -42 y(\()p Fj(j)s Fk(\))1169 -162 y Fi(\021)1254 -70 y Fr(and)35 b Fo(T)47 b Fr(=)35 b(tr)14 b(D)1761 -100 y Fk(2)1798 -70 y Fn(L)1855 -100 y Fk(\(2\))1958 -162 y Fi(\020)2008 -70 y Fo(\022)2049 -113 y 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Fk(\(2\))1821 178 y Fi(\020)1870 271 y Fo( )1927 227 y Fk(\(2\))1924 299 y(\()p Fj(j)s Fk(\))2016 178 y Fi(\021)2089 271 y Fr(=)2177 154 y Fi(\022)2279 220 y Fo(k)2322 232 y Fk(11)2476 220 y Fo(k)2519 232 y Fk(12)2279 320 y Fo(k)2322 332 y Fk(12)2476 320 y Fo(k)2519 332 y Fk(22)2631 154 y Fi(\023)2720 271 y Fr(w)n(e)g(see)g(that)644 533 y(D)707 499 y Fk(2)744 533 y Fn(L)801 499 y Fk(\(2\))904 441 y Fi(\020)954 533 y Fo( )1011 490 y Fk(\(2\))1008 561 y(\()p Fj(j)s Fk(\))1100 441 y Fi(\021)1233 533 y Fr(=)82 b Fo(k)1423 545 y Fk(11)1494 533 y Fo(s)p Fr(\(1)p Fo(;)14 b(n)k Fn(\000)g Fr(1\))p Fo(s)p Fr(\(1)p Fo(;)c(n)k Fn(\000)g Fr(1\))2244 499 y Fh(>)1380 695 y Fr(+)p Fo(k)1488 707 y Fk(12)1572 628 y Fi(\000)1610 695 y Fo(s)p Fr(\(1)p Fo(;)c(n)k Fn(\000)g Fr(1\))p Fo(s)p Fr(\(1)p Fo(;)c(n)p Fr(\))2217 661 y Fh(>)2291 695 y Fr(+)k Fo(s)p Fr(\(1)p Fo(;)c(n)p Fr(\))p Fo(s)p Fr(\(1)p Fo(;)g(n)k Fn(\000)g Fr(1\))2981 661 y Fh(>)3037 628 y Fi(\001)1380 832 y Fr(+)p Fo(k)1488 844 y Fk(22)1559 832 y Fo(s)p Fr(\(1)p Fo(;)c(n)p Fr(\))p Fo(s)p Fr(\(1)p Fo(;)g(n)p Fr(\))2023 797 y Fh(>)-260 1011 y Fr(and,)28 b(using)f(\(30\),)g(w)n(e)g(obtain)414 1190 y Fo(T)94 b Fr(=)83 b Fo(k)748 1202 y Fk(11)832 1123 y Fi(\000)870 1190 y Fo(f)9 b Fr(\(1)p Fo(;)14 b(n)k Fn(\000)g Fr(2\))1256 1156 y Fk(2)1311 1190 y Fr(+)g Fo(f)9 b Fr(\(1)p Fo(;)14 b(n)k Fn(\000)g Fr(1\))1780 1156 y Fk(2)1817 1123 y Fi(\001)1873 1190 y Fr(+)g(2)p Fo(k)2041 1202 y Fk(12)2111 1190 y Fo(f)9 b Fr(\(1)p Fo(;)14 b(n)k Fn(\000)g Fr(1\))c(\()p Fo(f)9 b Fr(\(1)p Fo(;)14 b(n)j Fn(\000)i Fr(2\))f(+)g Fo(f)9 b Fr(\(1)p Fo(;)14 b(n)p Fr(\)\))705 1324 y(+)p Fo(k)813 1336 y Fk(22)897 1257 y Fi(\000)935 1324 y Fo(f)9 b Fr(\(1)p Fo(;)14 b(n)j Fn(\000)h Fr(1\))1320 1290 y Fk(2)1376 1324 y Fr(+)g Fo(f)9 b Fr(\(1)p Fo(;)14 b(n)p Fr(\))1702 1290 y Fk(2)1738 1257 y Fi(\001)1790 1324 y Fo(:)2032 b Fr(\(59\))-260 1504 y(Using)28 b(that)f Fn(j)p Fo(k)219 1516 y Fk(11)290 1504 y Fn(j)c Fr(=)g Fo(A)p Fr(,)28 b Fn(j)p Fo(k)603 1516 y Fk(22)673 1504 y Fn(j)c Fr(=)e Fo(B)32 b Fr(and)27 b Fo(k)1106 1516 y Fk(12)1200 1504 y Fr(=)c(0,)k(and)g(also)g(that)h Fn(L)1945 1516 y Fj(S)1986 1524 y Fg(1)2045 1504 y Fr(=)23 b(max\()p Fo(A;)14 b(B)t Fr(\))28 b(and)g(estimate)f(\(43\),)g(w)n(e)h(deduce)f(that)1615 1722 y Fn(j)p Fo(T)12 b Fn(j)22 b(\030)1875 1666 y Fr(1)p 1842 1703 108 4 v 1842 1720 a Fn(p)p 1911 1720 39 4 v 59 x Fo(")1960 1722 y Fn(L)2017 1734 y Fj(S)2058 1742 y Fg(1)2094 1722 y Fo(:)1728 b Fr(\(60\))-260 1955 y(Since)28 b Fn(j)p Fo(D)r Fn(j)23 b(\034)g Fo(T)264 1925 y Fk(2)301 1955 y Fr(,)k(w)n(e)g(see)h(from)f(\(58\))g(that)1473 2102 y Fo(m)1546 2059 y Fk(\(2\))1546 2131 y(\()p Fj(j)s Fk(\))1658 2102 y Fn(\030)1756 2046 y(j)p Fo(D)r Fn(j)p 1756 2083 118 4 v 1761 2159 a(j)p Fo(T)12 b Fn(j)1906 2102 y(\030)1994 2039 y(p)p 2063 2039 39 4 v 63 x Fo(")p Fn(L)2159 2114 y Fj(S)2200 2122 y Fg(2)2236 2102 y Fo(:)-135 2431 y Fr(T)-7 b(o)25 b(estimate)g(the)h(minim)n(um)g(eigen)n(v)-5 b(alue)24 b(of)h(D)1374 2401 y Fk(2)1412 2431 y Fn(L)1469 2401 y Fk(\(3\))1572 2339 y Fi(\020)1622 2431 y Fo(\022)1663 2388 y Fk(\(3\))1661 2460 y(\()p Fj(j)s Fk(\))1752 2339 y Fi(\021)1802 2431 y Fr(,)g(w)n(e)g(can)g(pro)r(ceed)g(analogously)-7 b(.)34 b(Applying)25 b(Lemma)g(6,)h(w)n(e)f(obtain)612 2669 y Fn(j)p Fo(D)r Fn(j)e Fr(=)g Fo(B)907 2635 y Fk(2)958 2577 y Fi(\020)1007 2669 y Fo(E)1073 2635 y Fk(\()p Fh(\006)p Fk(\))1181 2669 y Fo(\021)f Fr(+)c Fn(O)1409 2602 y Fi(\000)1447 2669 y Fo(\021)1491 2635 y Fk(2)1529 2602 y Fi(\001)1567 2577 y(\021)1639 2669 y Fn(\030)23 b(L)1795 2635 y Fk(2)1784 2689 y Fj(S)1825 2697 y Fg(1)1862 2669 y Fo(E)1928 2635 y Fk(\()p Fh(\006)p Fk(\))2036 2669 y Fo(\021)j Fn(\030)d Fo(E)2257 2635 y Fk(\()p Fh(\006)p Fk(\))2365 2669 y Fn(L)2422 2681 y Fj(S)2463 2689 y Fg(1)2499 2669 y Fn(L)2556 2681 y Fj(S)2597 2689 y Fg(2)2657 2669 y Fn(\027)f Fo(\034)2789 2635 y Fh(\003)2828 2669 y Fn(L)2885 2681 y Fj(S)2926 2689 y Fg(1)2963 2669 y Fn(L)3020 2681 y Fj(S)3061 2689 y Fg(2)3097 2669 y Fo(;)-260 2869 y Fr(under)28 b(the)g(additional)f (condition)1603 2990 y Fo(\021)f Fn(\030)1767 2934 y(L)1824 2946 y Fj(S)1865 2954 y Fg(2)p 1767 2971 135 4 v 1767 3047 a Fn(L)1824 3059 y Fj(S)1865 3067 y Fg(1)1935 2990 y Fn(\026)d Fo(\034)2068 2955 y Fh(\003)2107 2990 y Fo(:)-260 3197 y Fr(W)-7 b(e)23 b(ha)n(v)n(e)f(used)h(that)g Fo(B)k Fr(=)c Fn(L)660 3209 y Fj(S)701 3217 y Fg(1)760 3197 y Fr(and)g(the)g(fact)g(that)g Fo(\034)1434 3167 y Fh(\003)1496 3197 y Fn(\026)g Fo(E)1650 3167 y Fk(\()p Fh(\006)p Fk(\))1781 3197 y Fr(\(as)f(seen)h(in)g(the)g(pro)r(of)f(of)h(Lemma)f(6\).)36 b(On)22 b(the)h(other)g(hand,)g(to)g(giv)n(e)-260 3297 y(an)j(estimate)g(for)f Fo(T)37 b Fr(w)n(e)25 b(can)h(use)f(equalit)n (y)h(\(59\))f(with)i Fn(j)p Fo(k)1550 3309 y Fk(22)1620 3297 y Fn(j)c Fr(=)g Fo(B)18 b Fr(\(1)g(+)g Fn(O)r Fr(\()p Fo(\021)s Fr(\)\))25 b Fn(\030)d(L)2387 3309 y Fj(S)2428 3317 y Fg(1)2491 3297 y Fr(and)j Fn(j)q Fo(k)2717 3309 y Fk(11)2787 3297 y Fn(j)14 b Fo(;)g Fn(j)p Fo(k)2927 3309 y Fk(12)2998 3297 y Fn(j)23 b(\026)f Fo(B)t(\021)27 b Fn(\030)22 b(L)3410 3309 y Fj(S)3451 3317 y Fg(2)3488 3297 y Fr(,)k(obtaining)f(an)-260 3413 y(estimate)j(lik)n(e)f(\(60\))g (again)g(and,)g(consequen)n(tly)-7 b(,)27 b(the)h(exp)r(ected)g (estimate)f(for)g Fo(m)2335 3370 y Fk(\(3\))2335 3441 y(\()p Fj(j)s Fk(\))2424 3413 y Fr(.)1483 b Fe(2)-260 3626 y Fs(Remark.)56 b Fr(It)28 b(is)h(also)e(in)n(teresting)h(to)g (study)-7 b(,)29 b(applying)f(the)g(linear)g(c)n(hange)f(\(53\))h(to)g (Figure)g(2,)g(the)h(con)n(tin)n(uation)f(of)g(the)h(critical)-260 3741 y(p)r(oin)n(ts)f Fo(\022)31 3698 y Fk(\()q(3\))29 3770 y(\()p Fj(j)s Fk(\))149 3741 y Fr(of)g Fn(L)301 3711 y Fk(\()q(3\))419 3741 y Fr(as)g Fo(Q)g Fr(go)r(es)f(from)h(0)g (to)g(1.)38 b(Recall)28 b(that)h(the)g(critical)e(p)r(oin)n(ts)h Fo( )2431 3698 y Fk(\(3\))2428 3770 y(\()q Fj(j)s Fk(\))2549 3741 y Fr(of)g Fn(K)2708 3711 y Fk(\()q(3\))2826 3741 y Fr(drift)h(on)f(curv)n(es)f(close)g(to)i(the)f(lines)-260 3879 y Fo( )-206 3891 y Fk(2)-134 3879 y Fr(=)33 b(0,)j Fo( )119 3891 y Fk(2)190 3879 y Fr(=)e Fo(\031)s Fr(.)57 b(Then,)36 b(through)e(the)g(linear)g(c)n(hange)f(\(53\))h(w)n(e)g(see) f(that)i(the)f(p)r(oin)n(ts)h Fo(\022)2741 3836 y Fk(\(3\))2739 3908 y(\()p Fj(j)s Fk(\))2864 3879 y Fr(drift)g(on)f(curv)n(es)f(close) g(to)h(2)g(lines)-260 3993 y Fn(h)p Fo(s)p Fr(\(1)p Fo(;)14 b(n)p Fr(\))p Fo(;)g(\022)r Fn(i)24 b Fr(=)e(const.)37 b(Note)27 b(that)h(their)g(slop)r(e)f(is)h Fo(f)9 b Fr(\(1)p Fo(;)14 b(n)j Fn(\000)h Fr(1\))p Fo(=f)9 b Fr(\(1)p Fo(;)14 b(n)p Fr(\),)27 b(whic)n(h)g(tends)h(to)g(\012)f(as)g Fo(n)c Fn(!)g(1)28 b Fr(\(i.e.)g(as)f Fo(")c Fn(!)g Fr(0\).)-135 4289 y(After)28 b(ha)n(ving)f(studied)h(the)g(critical)f(p)r(oin)n(ts)h (of)g(the)g(appro)n(ximations)d Fn(L)2192 4259 y Fk(\()q(2\))2282 4289 y Fr(,)j Fn(L)2390 4259 y Fk(\(3\))2479 4289 y Fr(,)g(the)g(last)g (step)g(is)f(to)h(sho)n(w)f(their)h(p)r(ersistence)-260 4389 y(in)h(the)f(whole)g(splitting)h(p)r(oten)n(tial)f Fn(L)p Fr(.)39 b(T)-7 b(o)28 b(ensure)g(the)h(con)n(tin)n(uation)e(of)h (the)h(critical)f(p)r(oin)n(ts)g(as)g Fo(")c Fn(!)g Fr(0,)k(w)n(e)g (assume)f(in)i(\(33\))f(that)-260 4488 y(the)g(di\013erence)g(of)f (phases)1187 4588 y Fn(4)p Fo(\033)1309 4600 y Fj(n)1377 4588 y Fr(:=)22 b Fo(\033)1534 4603 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(+1\))1818 4588 y Fn(\000)c Fo(\033)1948 4603 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(\))2149 4588 y Fn(\000)g Fo(\033)2279 4603 y Fj(s)p Fk(\(1)p Fj(;n)p Fh(\000)p Fk(1\))3845 4588 y Fr(\(61\))-260 4735 y(k)n(eeps)30 b(far)g(a)n(w)n(a)n(y)f(from)i(0)f(or)g Fo(\031)k Fr(\(mo)r(d)42 b(2)p Fo(\031)s Fr(\))31 b(for)f(an)n(y)g Fo(n)p Fr(.)47 b(The)31 b(next)g(theorem)f(is)h(form)n(ulated)f(in)h(terms)f(of)h(the) g(splitting)h(function,)-260 4835 y Fn(M)p Fr(\(0)p Fo(;)14 b(\022)r Fr(\))37 b(=)g Fo(@)207 4847 y Fj(\022)244 4835 y Fn(L)p Fr(\(0)p Fo(;)14 b(\022)r Fr(\),)39 b(whic)n(h)d(giv)n(es)f(a) h(measure)f(of)h(the)g(distance)g(b)r(et)n(w)n(een)g(the)g(whisk)n (ers.)61 b(The)36 b(result)g(for)f Fn(M)p Fr(\(0)p Fo(;)14 b(\022)r Fr(\))36 b(can)g(b)r(e)-260 4934 y(extended)28 b(to)f Fn(M)p Fr(\()p Fo(s;)14 b(\022)r Fr(\))29 b(in)e(a)h(natural)e (w)n(a)n(y)-7 b(,)27 b(b)r(ecause)g(of)h(the)g(quasip)r(erio)r(dicit)n (y)e(\(15\))h(of)h(this)g(function.)-135 5134 y(In)e(this)h(theorem,)f (w)n(e)f(establish)h(the)h(existence)f(of)g(4)g(simple)g(zeros)f(of)h Fn(M)p Fr(\(0)p Fo(;)14 b(\022)r Fr(\),)27 b(denoted)f Fo(\022)2831 5149 y Fk(\()p Fj(j)s Fk(\))2918 5134 y Fr(,)g Fo(j)i Fr(=)23 b(1)p Fo(;)14 b Fr(2)p Fo(;)g Fr(3)p Fo(;)g Fr(4,)24 b(and)j(w)n(e)e(pro)n(vide)-260 5242 y(for)32 b(these)g(zeros)f(an)h(estimate)h(for)f(the)g(minim)n(um)h (eigen)n(v)-5 b(alue)32 b(\(in)h(mo)r(dulus\))g(of)f(the)h Fp(splitting)h(matrix)f Fo(@)3206 5254 y Fj(\022)3243 5242 y Fn(M)3357 5175 y Fi(\000)3395 5242 y Fr(0)p Fo(;)14 b(\022)3513 5257 y Fk(\()o Fj(j)s Fk(\))3599 5175 y Fi(\001)3637 5242 y Fr(,)34 b(whic)n(h)e(is)-260 5342 y(clearly)24 b(symmetric.)36 b(As)26 b(p)r(oin)n(ted)g(out)f(in)h([DG00b],)g(this)f (minim)n(um)i(eigen)n(v)-5 b(alue)24 b(pro)n(vides)g(a)h(lo)n(w)n(er)f (b)r(ound)i(for)f(the)h(transv)n(ersalit)n(y)-260 5441 y(of)i(the)g(homo)r(clinic)f(orbit)g(asso)r(ciated)g(to)g(the)h(zero)f Fo(\022)1453 5456 y Fk(\()o Fj(j)s Fk(\))1539 5441 y Fr(.)-135 5641 y(F)-7 b(or)30 b(the)h(sak)n(e)e(of)i(completeness,)g(w) n(e)g(ha)n(v)n(e)e(also)h(included)h(a)f(m)n(uc)n(h)h(simpler)f (statemen)n(t)h(concerning)f(the)h(maxim)n(um)f(size)h(\(in)-260 5740 y(mo)r(dulus\))j(of)g(the)f(splitting)h(function)g Fn(M)p Fr(\(0)p Fo(;)14 b(\022)r Fr(\),)35 b(giving)e(in)h(this)f(w)n (a)n(y)f(an)i(estimate)f(for)g(maxim)n(um)g(splitting)h(distance.)54 b(Notice)1825 5989 y(19)p eop %%Page: 20 20 20 19 bop -260 -82 a Fr(the)31 b(di\013erence)g(in)g(the)g(exp)r(onen)n (ts)g(of)f(the)h(estimates)g(of)g(parts)f(\(a\))h(and)f(\(b\),)j (illustrated)d(b)n(y)h(the)g(functions)g Fo(h)3368 -70 y Fk(1)3405 -82 y Fr(\()p Fo(")p Fr(\))g(and)g Fo(h)3752 -70 y Fk(2)3789 -82 y Fr(\()p Fo(")p Fr(\))g(in)-260 17 y(Figure)c(1.)-135 217 y(W)-7 b(e)31 b(p)r(oin)n(t)f(out)h(that)f(a) g(concrete)g(example)g(in)h(whic)n(h)f(the)h(condition)f(that)h Fn(4)p Fo(\033)2452 229 y Fj(n)2527 217 y Fr(k)n(eeps)f(far)g(a)n(w)n (a)n(y)e(from)i(0)g(or)g Fo(\031)k Fr(\(mo)r(d)42 b(2)p Fo(\031)s Fr(\))30 b(is)-260 316 y(the)f(one)f(in)n(tro)r(duced)g(in)g (\(7\).)39 b(Indeed,)28 b(in)h(this)f(example)g(all)g(phases)g(are)f Fo(\033)2164 328 y Fj(k)2229 316 y Fr(=)d Fo(\031)s(=)p Fr(2,)k(and)g(therefore)f Fn(4)p Fo(\033)3136 328 y Fj(n)3204 316 y Fr(=)d Fn(\000)p Fo(\031)s(=)p Fr(2)j(for)h(an)n(y)f Fo(n)p Fr(.)39 b(In)-260 416 y(fact,)28 b(w)n(e)f(ga)n(v)n(e)f(in)i (Section)f(1.3)g(a)g(simpli\014ed)h(v)n(ersion)e(of)i(the)g(theorem)f (for)g(this)h(concrete)f(case.)-260 693 y Fs(Theorem)k(8)41 b Fp(In)29 b(the)h(example)h(de\014ne)l(d)f(by)g(\(1{6\))h(and)f (\(33\),)h(assume)f(that)f(the)h(di\013er)l(enc)l(es)h(\(61\))f (satisfy)h(the)f(ine)l(quality)1264 870 y Fr(min\()p Fn(j4)p Fo(\033)1579 882 y Fj(n)1624 870 y Fn(j)14 b Fo(;)g Fn(j4)p Fo(\033)1843 882 y Fj(n)1905 870 y Fn(\000)k Fo(\031)s Fn(j)q Fr(\))23 b Fn(\025)g Fo(\033)2255 836 y Fh(\003)2316 870 y Fo(>)g Fr(0)p Fo(;)-260 1047 y Fp(with)30 b Fo(\033)-30 1017 y Fh(\003)39 1047 y Fp(indep)l(endent)g(of)h Fo(n)p Fp(.)38 b(F)-6 b(or)30 b(any)1085 1235 y Fo(")23 b Fn(\026)f Fr(\()q Fo(\033)1317 1201 y Fh(\003)1355 1235 y Fr(\))1388 1194 y Fk(1)p Fj(=)p Fk(\()p Fj(p)p Fh(\000)p Fj(p)1601 1169 y Fc(\003)1636 1194 y Fk(\))1680 1235 y Fo(;)183 b(\026)23 b Fr(=)g Fo(")2086 1201 y Fj(p)2209 1235 y Fp(with)30 b Fo(p)23 b(>)g(p)2584 1205 y Fh(\003)2622 1235 y Fp(,)1198 b Fr(\(62\))-260 1413 y Fp(one)30 b(has:)-200 1703 y Fr(\(a\))42 b(max)-50 1760 y Fj(\022)r Fh(2)p Ff(T)73 1744 y Fg(2)116 1703 y Fn(jM)p Fr(\(0)p Fo(;)14 b(\022)r Fr(\))p Fn(j)23 b(\030)596 1647 y Fo(\026)p 567 1684 108 4 v 567 1701 a Fn(p)p 636 1701 39 4 v 59 x Fo(")699 1703 y Fr(exp)839 1586 y Fi(\032)901 1703 y Fn(\000)976 1647 y Fo(C)1035 1659 y Fk(0)1073 1647 y Fo(h)1121 1659 y Fk(1)1158 1647 y Fr(\()p Fo(")p Fr(\))p 976 1684 285 4 v 1047 1762 a Fo(")1086 1738 y Fk(1)p Fj(=)p Fk(4)1271 1586 y Fi(\033)1333 1703 y Fp(.)-204 1917 y Fr(\(b\))42 b Fp(The)e(function)f Fn(M)p Fr(\(0)p Fo(;)14 b(\022)r Fr(\))39 b Fp(has)h(exactly)g(4)g(zer)l(os)f Fo(\022)1569 1932 y Fk(\()p Fj(j)s Fk(\))1656 1917 y Fp(,)j(al)t(l)e(simple,)j(and)d(\(the)g(mo)l(dulus)f(of)9 b(\))39 b(the)h(minimum)f(eigenvalue)h(of)-52 2016 y Fo(@)-8 2028 y Fj(\022)29 2016 y Fn(M)p Fr(\(0)p Fo(;)14 b Fn(\001)p Fr(\))30 b Fp(at)g(e)l(ach)g(zer)l(o)g(satis\014es)559 2243 y Fo(\033)609 2209 y Fh(\003)647 2243 y Fo(\026")736 2209 y Fk(1)p Fj(=)p Fk(4)854 2243 y Fr(exp)995 2126 y Fi(\032)1057 2243 y Fn(\000)1132 2187 y Fo(C)1191 2199 y Fk(0)1228 2187 y Fo(h)1276 2199 y Fk(2)1313 2187 y Fr(\()p Fo(")p Fr(\))p 1132 2224 V 1202 2302 a Fo(")1241 2278 y Fk(1)p Fj(=)p Fk(4)1426 2126 y Fi(\033)1512 2243 y Fn(\026)22 b Fo(m)1672 2258 y Fk(\()p Fj(j)s Fk(\))1782 2243 y Fn(\026)h Fo(\026")1959 2209 y Fk(1)p Fj(=)p Fk(4)2077 2243 y Fr(exp)2217 2126 y Fi(\032)2280 2243 y Fn(\000)2355 2187 y Fo(C)2414 2199 y Fk(0)2451 2187 y Fo(h)2499 2199 y Fk(2)2536 2187 y Fr(\()p Fo(")p Fr(\))p 2355 2224 V 2425 2302 a Fo(")2464 2278 y Fk(1)p Fj(=)p Fk(4)2649 2126 y Fi(\033)2725 2243 y Fo(;)184 b(j)28 b Fr(=)23 b(1)p Fo(;)14 b Fr(2)p Fo(;)g Fr(3)p Fo(;)g Fr(4)p Fo(:)-260 2570 y Fp(Pr)l(o)l(of.)59 b Fr(W)-7 b(e)30 b(\014rst)e(sho)n(w)g(that)h (part)f(\(a\))h(is)g(a)f(simple)h(consequence)f(of)h(Lemma)f(5)h(and)f (the)i(fact)f(that)g Fn(M)24 b Fr(=)h Fo(@)3297 2582 y Fj(\022)3335 2570 y Fn(L)p Fr(.)40 b(It)30 b(is)e(enough)g(to)-260 2670 y(consider)g(as)g(in)g(\(52\))h(the)g(appro)n(ximation)d Fn(L)1197 2640 y Fk(\(2\))1315 2670 y Fr(giv)n(en)i(b)n(y)g(the)h(2)f (most)h(dominan)n(t)f(harmonics.)39 b(W)-7 b(e)28 b(can)h(easily)e(giv) n(e)h(an)g(estimate)-260 2769 y(for)f Fo(@)-89 2781 y Fj(\022)-51 2769 y Fn(L)6 2739 y Fk(\(2\))123 2769 y Fr(b)n(y)g(writing)g(it)h(in)g(the)g(v)-5 b(ariables)27 b Fo( )j Fr(as)d(in)h(\(54\).)37 b(Using)27 b(also)g(Lemma)g(5,)g(w)n (e)g(obtain)870 2890 y Fi(\014)870 2940 y(\014)870 2990 y(\014)897 2986 y Fo(@)941 2998 y Fj(\022)979 2986 y Fn(L)1036 2951 y Fk(\(2\))1126 2890 y Fi(\014)1126 2940 y(\014)1126 2990 y(\014)1176 2986 y Fn(\030)1325 2930 y Fr(1)p 1274 2967 144 4 v 1274 3044 a Fo(")1313 3020 y Fk(1)p Fj(=)p Fk(4)1427 2986 y Fn(L)1484 2998 y Fj(S)1525 3006 y Fg(1)1561 2986 y Fo(;)1764 2890 y Fi(\014)1764 2940 y(\014)1764 2990 y(\014)1792 2986 y Fo(@)1836 2998 y Fj(\022)1874 2986 y Fn(L)p Fr(\(0)p Fo(;)14 b Fn(\001)p Fr(\))k Fn(\000)g Fo(@)2242 2998 y Fj(\022)2280 2986 y Fn(L)2337 2951 y Fk(\(2\))2427 2890 y Fi(\014)2427 2940 y(\014)2427 2990 y(\014)2477 2986 y Fn(\026)2626 2930 y Fr(1)p 2575 2967 V 2575 3044 a Fo(")2614 3020 y Fk(1)p Fj(=)p Fk(4)2728 2986 y Fn(L)2785 2998 y Fj(S)2826 3006 y Fg(3)-260 3198 y Fr(\(w)n(e)31 b(are)f(using)h(here)f(the)i (notation)e Fn(j\001j)i Fr(for)e(the)i(suprem)n(um)e(norm)h(of)g(a)f (function)i(on)f(the)g(real)f(domain)h Fl(T)3216 3168 y Fk(2)3252 3198 y Fr(\),)i(and)e(this)g(giv)n(es)f(the)-260 3297 y(exp)r(ected)i(estimate)f(for)g Fn(jM)p Fr(\(0)p Fo(;)14 b Fn(\001)p Fr(\))p Fn(j)p Fr(.)48 b(W)-7 b(e)32 b(stress)e(that,)j(if)f Fo(")f Fr(is)g(not)h(close)e(to)h(some)g Fo(")2473 3267 y Fh(0)2473 3318 y Fj(n)2518 3297 y Fr(,)i(it)e(w)n (ould)g(ha)n(v)n(e)f(b)r(een)i(enough)f(to)g(consider)-260 3397 y(only)c(the)i(\014rst)e(dominan)n(t)g(harmonic,)g(and)h(b)r(ound) g(the)g(sum)g(of)g(all)f(other)g(harmonics)g(from)g(the)h(second)f (one,)h(but)g(if)g Fo(")g Fr(is)f(close)g(to)-260 3497 y Fo(")-221 3466 y Fh(0)-221 3517 y Fj(n)-148 3497 y Fr(w)n(e)g(need)h(to)f(consider)g(2)g(harmonics.)-135 3696 y(T)-7 b(o)29 b(pro)n(v)n(e)g(part)g(\(b\),)j(w)n(e)d(will)i(sho)n (w)e(that)h Fn(L)g Fr(has)g(nondegenerate)e(critical)i(p)r(oin)n(ts)g (b)n(y)f(applying)h(Prop)r(osition)e(9)i(\(the)g(implicit)-260 3795 y(function)d(theorem;)g(see)f(the)g(App)r(endix\))i(to)e(a)g (suitable)h(appro)n(ximation,)d Fn(L)2205 3765 y Fk(\()q(2\))2321 3795 y Fr(or)i Fn(L)2479 3765 y Fk(\(3\))2568 3795 y Fr(,)h(dep)r(ending)g(on)f(the)h(closeness)e(of)h Fo(")h Fr(to)f(the)-260 3895 y(v)-5 b(alues)29 b Fo(")28 3907 y Fj(n)73 3895 y Fr(,)h Fo(")165 3865 y Fh(0)165 3916 y Fj(n)210 3895 y Fr(.)42 b(In)30 b(some)e(in)n(terv)-5 b(al)29 b(around)f Fo(")1214 3865 y Fh(0)1214 3916 y Fj(n)1259 3895 y Fr(,)i(w)n(e)f(consider)g Fn(L)1819 3865 y Fk(\(2\))1937 3895 y Fr(and,)h(near)f Fo(")2350 3907 y Fj(n)2394 3895 y Fr(,)h(w)n(e)f(consider)g Fn(L)2954 3865 y Fk(\(3\))3043 3895 y Fr(.)42 b(Our)29 b(aim)g(is)g(to)g(sho)n(w) g(that)-260 3995 y(these)f(t)n(w)n(o)f(in)n(terv)-5 b(als)26 b(in)n(tersect)h(and)h(therefore)f(no)g(v)-5 b(alue)27 b(of)h Fo(")f Fr(\(small)h(enough\))f(is)h(excluded.)-135 4194 y(First,)33 b(for)f Fo(")e Fn(2)i Fr(\()p Fo(")455 4206 y Fj(n)500 4194 y Fo(;)14 b(")576 4206 y Fj(n)p Fh(\000)p Fk(1)706 4194 y Fr(\))32 b(w)n(e)g(apply)g(Prop)r(osition)f (9)h(with)g Fo(G)1914 4206 y Fk(0)1952 4194 y Fr(\()p Fo(\022)r Fr(\))g(=)e Fo(@)2228 4206 y Fj(\022)2266 4194 y Fn(L)2323 4164 y Fk(\(2\))2412 4194 y Fr(\()p Fo(\022)r Fr(\),)35 b Fo(G)p Fr(\()p Fo(\022)r Fr(\))d(=)e Fo(@)2916 4206 y Fj(\022)2954 4194 y Fn(L)p Fr(\(0)p Fo(;)14 b(\022)r Fr(\))31 b(=)f Fn(M)p Fr(\(0)p Fo(;)14 b(\022)r Fr(\).)51 b(F)-7 b(or)32 b(eac)n(h)-260 4318 y(zero)23 b Fo(\022)-47 4275 y Fk(\(2\))-49 4347 y(\()p Fj(j)s Fk(\))66 4318 y Fr(of)g Fo(G)221 4330 y Fk(0)259 4318 y Fr(,)h(w)n(e)f(ha)n(v)n(e)g (giv)n(en)f(in)i(Lemma)f(7)h(an)f(estimate)g(for)g Fo(m)1910 4275 y Fk(\(2\))1910 4347 y(\()p Fj(j)s Fk(\))1999 4318 y Fr(,)i(the)f(minim)n(um)g(eigen)n(v)-5 b(alue)23 b(of)g(D)p Fo(G)3164 4330 y Fk(0)3216 4226 y Fi(\020)3265 4318 y Fo(\022)3306 4275 y Fk(\()q(2\))3304 4347 y(\()p Fj(j)s Fk(\))3396 4226 y Fi(\021)3445 4318 y Fr(.)36 b(W)-7 b(e)24 b(also)e(ha)n(v)n(e,)-260 4443 y(according)k(to)h(part)h(\(b\))g (of)f(Lemma)h(5,)f(the)h(b)r(ounds)531 4659 y Fn(j)p Fo(G)19 b Fn(\000)f Fo(G)786 4671 y Fk(0)824 4659 y Fn(j)23 b(\026)1018 4603 y Fr(1)p 967 4640 V 967 4718 a Fo(")1006 4694 y Fk(1)p Fj(=)p Fk(4)1120 4659 y Fn(L)1177 4671 y Fj(S)1218 4679 y Fg(3)1255 4659 y Fo(;)180 b Fn(j)p Fr(D)p Fo(G)19 b Fn(\000)f Fr(D)p Fo(G)1839 4671 y Fk(0)1877 4659 y Fn(j)23 b(\026)2071 4603 y Fr(1)p 2021 4640 V 2021 4718 a Fo(")2060 4694 y Fk(1)p Fj(=)p Fk(2)2174 4659 y Fn(L)2231 4671 y Fj(S)2272 4679 y Fg(3)2308 4659 y Fo(;)2511 4589 y Fi(\014)2511 4639 y(\014)2539 4659 y Fr(D)2602 4625 y Fk(2)2640 4659 y Fo(G)2705 4671 y Fk(0)2742 4589 y Fi(\014)2742 4639 y(\014)2793 4659 y Fn(\026)2941 4603 y Fr(1)p 2890 4640 V 2890 4718 a Fo(")2929 4694 y Fk(3)p Fj(=)p Fk(4)3043 4659 y Fn(L)3100 4671 y Fj(S)3141 4679 y Fg(1)3178 4659 y Fo(:)-260 4890 y Fr(Then,)28 b(the)g(zero)e Fo(\022)339 4847 y Fk(\()q(2\))337 4919 y(\()p Fj(j)s Fk(\))457 4890 y Fr(of)h Fo(G)616 4902 y Fk(0)681 4890 y Fr(p)r(ersists,)g(giving)g(rise)g(to)h(a)f(zero) f Fo(\022)1788 4905 y Fk(\()p Fj(j)s Fk(\))1903 4890 y Fr(of)h Fo(G)p Fr(,)h(pro)n(vided)f(the)h(follo)n(wing)e(inequalit)n (y)i(is)f(satis\014ed:)1617 5150 y Fn(L)1674 5162 y Fj(S)1715 5170 y Fg(3)1774 5150 y Fn(\026)1872 5087 y Fo(")1911 5057 y Fk(2)1948 5087 y Fn(L)2016 5057 y Fk(2)2005 5110 y Fj(S)2046 5118 y Fg(2)p 1872 5131 211 4 v 1910 5207 a Fn(L)1967 5219 y Fj(S)2008 5227 y Fg(1)2092 5150 y Fo(:)1730 b Fr(\(63\))-260 5376 y(Since)33 b Fo(p)e(>)f(p)172 5346 y Fh(\003)243 5376 y Fr(in)i(\(62\),)i(w)n(e)e(can)g(use)g(part)g (\(a\))g(of)h(Lemma)f(5,)h(whic)n(h)f(pro)n(vides)f(an)h(estimate)h (for)f(the)g(co)r(e\016cien)n(ts)g Fn(L)3597 5388 y Fj(S)3638 5396 y Fd(i)3669 5376 y Fr(.)51 b(T)-7 b(aking)-260 5476 y(logarithms,)26 b(w)n(e)i(see)f(that)h(\(63\))f(can)g(b)r(e)h (written,)g(for)f(some)g(constan)n(t)g Fo(c)c(>)g Fr(0,)k(as)1164 5712 y(2)p Fo(h)1254 5724 y Fk(2)1290 5712 y Fr(\()p Fo(")p Fr(\))d Fn(\024)e Fo(h)1552 5724 y Fk(1)1589 5712 y Fr(\()p Fo(")p Fr(\))d(+)f Fo(h)1842 5724 y Fk(3)1879 5712 y Fr(\()p Fo(")p Fr(\))h(+)2094 5656 y Fo(")2133 5626 y Fk(1)p Fj(=)p Fk(4)p 2094 5693 144 4 v 2117 5769 a Fo(C)2176 5781 y Fk(0)2261 5712 y Fr(ln)2344 5645 y Fi(\000)2382 5712 y Fo(c")2457 5678 y Fk(2)2494 5645 y Fi(\001)2546 5712 y Fo(:)1276 b Fr(\(64\))1825 5989 y(20)p eop %%Page: 21 21 21 20 bop -260 -82 a Fr(It)26 b(is)g(not)g(hard)f(to)g(in)n(v)n (estigate)g(ho)n(w)g(far)g(this)h(inequalit)n(y)f(is)h(true.)36 b(If)26 b(w)n(e)f(restrict)h(ourselv)n(es)d(to)j(the)g(subin)n(terv)-5 b(al)25 b(\()p Fo(")3458 -70 y Fj(n)3503 -82 y Fo(;)14 b(")3579 -112 y Fh(0)3579 -62 y Fj(n)3624 -82 y Fr(\),)27 b(w)n(e)e(need)1080 138 y(2)p Fo(g)1162 150 y Fj(n)p Fh(\000)p Fk(1)1291 138 y Fr(\()p Fo(")p Fr(\))e Fn(\024)g Fo(g)1545 150 y Fj(n)1590 138 y Fr(\()p Fo(")p Fr(\))18 b(+)g Fo(g)1834 150 y Fj(n)p Fk(+1)1963 138 y Fr(\()p Fo(")p Fr(\))h(+)2178 82 y Fo(")2217 52 y Fk(1)p Fj(=)p Fk(4)p 2178 119 144 4 v 2201 195 a Fo(C)2260 207 y Fk(0)2345 138 y Fr(ln)2428 71 y Fi(\000)2466 138 y Fo(c")2541 104 y Fk(2)2578 71 y Fi(\001)2630 138 y Fo(:)-260 340 y Fr(Using)33 b(the)h(de\014nitions)f(of)g(the)h(functions)f Fo(g)1187 352 y Fj(n)1232 340 y Fr(,)i(w)n(e)e(can)f(\014rst)h(solv)n(e)f(the)i (equation)e(2)p Fo(g)2536 352 y Fj(n)p Fh(\000)p Fk(1)2666 340 y Fr(\()p Fo(")p Fr(\))g(=)g Fo(g)2938 352 y Fj(n)2983 340 y Fr(\()p Fo(")p Fr(\))23 b(+)e Fo(g)3235 352 y Fj(n)p Fk(+1)3364 340 y Fr(\()p Fo(")p Fr(\))34 b(explicitly:)48 b(its)-260 440 y(solution)23 b(is)h Fo(")f Fr(=)f Fo(")318 452 y Fj(n)363 440 y Fo(=)p Fr(\(3\012)11 b Fn(\000)g Fr(1\))700 410 y Fk(2)736 440 y Fr(.)36 b(Note)23 b(that,)i(in)f(the)g (in)n(terv)-5 b(al)24 b(considered,)f(the)h(p)r(erturbativ)n(e)f(term)h (is)g Fo(")3073 410 y Fk(1)p Fj(=)p Fk(4)3191 440 y Fr(ln)3274 373 y Fi(\000)3312 440 y Fo(c")3387 410 y Fk(2)3424 373 y Fi(\001)3485 440 y Fn(\030)e Fo(n)p Fr(\012)3682 410 y Fj(n)3727 440 y Fr(.)36 b(So)24 b(w)n(e)-260 539 y(ha)n(v)n(e)g(an)h (estimate)g(of)g(the)g(part)g(of)g(the)g(subin)n(terv)-5 b(al)24 b(\()q Fo(")1508 551 y Fj(n)1553 539 y Fo(;)14 b(")1629 509 y Fh(0)1629 560 y Fj(n)1673 539 y Fr(\))26 b(where)e(\(64\))h(is)g(true.)36 b(Since)25 b(the)g(situation)g(on)g (the)g(other)g(subin)n(terv)-5 b(al)-260 639 y(\()p Fo(")-189 609 y Fh(0)-189 660 y Fj(n)-144 639 y Fo(;)14 b(")-68 651 y Fj(n)p Fh(\000)p Fk(1)62 639 y Fr(\))28 b(is)g(symmetric,)f(w)n (e)g(\014nally)h(see)f(that)h(\(64\))f(is)g(true)h(for)702 766 y Fo(")741 778 y Fj(n)p 700 803 90 4 v 700 879 a Fo(a)744 891 y Fj(n)822 822 y Fn(\024)22 b Fo(")h Fn(\024)h Fr(~)-43 b Fo(a)1103 834 y Fj(n)1148 822 y Fo(")1187 834 y Fj(n)p Fh(\000)p Fk(1)1317 822 y Fo(;)180 b(a)1564 834 y Fj(n)1609 822 y Fo(;)15 b Fr(~)-43 b Fo(a)1690 834 y Fj(n)1758 822 y Fr(=)23 b(\(3\012)18 b Fn(\000)g Fr(1\))2155 788 y Fk(2)2210 822 y Fr(+)g Fn(O)f Fr(\()p Fo(n)p Fr(\012)2518 788 y Fj(n)2563 822 y Fr(\))23 b Fn(')g Fr(0)p Fo(:)p Fr(729490)p Fo(:)799 b Fr(\(65\))-260 1035 y(Note)32 b(that)g Fo(a)173 1047 y Fj(n)218 1035 y Fr(,)i(~)-44 b Fo(a)317 1047 y Fj(n)394 1035 y Fr(measure)31 b(the)h(part)f(of)g(the)h(in)n(terv)-5 b(al)31 b(\()q Fo(")1674 1047 y Fj(n)1718 1035 y Fo(;)14 b(")1794 1047 y Fj(n)p Fh(\000)p Fk(1)1924 1035 y Fr(\))32 b(where)f(the)h(appro)n (ximation)e Fn(L)2994 1005 y Fk(\(2\))3115 1035 y Fr(is)h(enough)g(\()p Fo(a)3568 1047 y Fj(n)3643 1035 y Fr(=)g(~)-43 b Fo(a)3782 1047 y Fj(n)3856 1035 y Fr(=)30 b(1)-260 1135 y(w)n(ould)d(mean)h(the)g (whole)f(in)n(terv)-5 b(al\).)-135 1334 y(F)e(or)34 b Fo(")g Fn(2)184 1267 y Fi(\000)222 1334 y Fo(")261 1304 y Fh(0)261 1355 y Fj(n)p Fk(+1)390 1334 y Fo(;)14 b(")466 1304 y Fh(0)466 1354 y Fj(n)510 1267 y Fi(\001)583 1334 y Fr(w)n(e)34 b(apply)g(Prop)r(osition)f(9)h(again,)h(but)g(no)n(w)f (with)h Fo(G)2330 1346 y Fk(0)2367 1334 y Fr(\()p Fo(\022)r Fr(\))h(=)e Fo(@)2651 1346 y Fj(\022)2689 1334 y Fn(L)2746 1304 y Fk(\(3\))2835 1334 y Fr(\()p Fo(\022)r Fr(\).)58 b(Calling)34 b Fn(4)p Fo(\034)3425 1346 y Fj(n)3504 1334 y Fr(=)g Fo(\034)3639 1349 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(+1\))3928 1334 y Fn(\000)-260 1446 y Fo(\034)-224 1461 y Fj(s)p Fk(\(1)p Fj(;n)p Fk(\))-21 1446 y Fn(\000)21 b Fo(\034)101 1461 y Fj(s)p Fk(\(1)p Fj(;n)p Fh(\000)p Fk(1\))368 1446 y Fr(,)33 b(w)n(e)f(get)g(from)f(part)h(\(a\))g(of)g (Lemma)g(5)g(that)g Fn(j4)p Fo(\034)2009 1458 y Fj(n)2072 1446 y Fn(\000)18 b(4)p Fo(\033)2277 1458 y Fj(n)2321 1446 y Fn(j)31 b(\026)f Fo(\026")2559 1416 y Fh(\000)p Fj(p)2645 1391 y Fc(\003)2684 1446 y Fr(.)50 b(Recalling)32 b(that)g Fo(\026)f Fr(=)f Fo(")3522 1416 y Fj(p)3560 1446 y Fr(,)j(w)n(e)f(deduce)-260 1559 y(from)27 b(\(62\))h(that)g Fo(\026")381 1528 y Fh(\000)p Fj(p)467 1503 y Fc(\003)528 1559 y Fn(\026)23 b Fo(\033)666 1528 y Fh(\003)705 1559 y Fr(,)k(and)h(w)n(e)f(get)g(that)1344 1720 y(min)14 b(\()p Fn(j4)p Fo(\034)1662 1732 y Fj(n)1707 1720 y Fn(j)g Fo(;)g Fn(j4)p Fo(\034)1915 1732 y Fj(n)1977 1720 y Fn(\000)k Fo(\031)s Fn(j)q Fr(\))23 b Fn(\027)g Fo(\033)2327 1686 y Fh(\003)2365 1720 y Fo(;)-260 1898 y Fr(whic)n(h)28 b(implies)g(a)f(lo)n(w)n(er)f(estimate)h(for)g Fo(m)1078 1854 y Fk(\(3\))1078 1926 y(\()p Fj(j)s Fk(\))1195 1898 y Fr(from)g(Lemma)h(7.)36 b(Pro)r(ceeding)26 b(as)h(b)r(efore,)g(w)n(e) h(see)f(that)h(the)g(zero)e Fo(\022)3387 1854 y Fk(\(3\))3385 1926 y(\()p Fj(j)s Fk(\))3504 1898 y Fr(p)r(ersists)h(if)1533 2153 y Fn(L)1590 2165 y Fj(S)1631 2173 y Fg(4)1691 2153 y Fn(\026)1788 2090 y Fr(\()q Fo(\033)1871 2059 y Fh(\003)1909 2090 y Fo(")p Fr(\))1980 2048 y Fk(2)2031 2090 y Fn(L)2099 2059 y Fk(2)2088 2112 y Fj(S)2129 2120 y Fg(2)p 1788 2134 378 4 v 1910 2210 a Fn(L)1967 2222 y Fj(S)2008 2230 y Fg(1)2176 2153 y Fo(:)-260 2359 y Fr(This)h(condition)f(can)g(b)r(e)h (written)g(as)1069 2559 y(2)p Fo(h)1159 2571 y Fk(2)1195 2559 y Fr(\()p Fo(")p Fr(\))c Fn(\024)e Fo(h)1457 2571 y Fk(1)1494 2559 y Fr(\()p Fo(")p Fr(\))d(+)f Fo(h)1747 2571 y Fk(4)1784 2559 y Fr(\()p Fo(")p Fr(\))h(+)1999 2502 y Fo(")2038 2472 y Fk(1)p Fj(=)p Fk(4)p 1999 2539 144 4 v 2022 2616 a Fo(C)2081 2628 y Fk(0)2166 2559 y Fr(ln)2249 2466 y Fi(\020)2298 2559 y Fo(c)14 b Fr(\()p Fo(\033)2430 2524 y Fh(\003)2469 2559 y Fo(")p Fr(\))2540 2517 y Fk(2)2577 2466 y Fi(\021)2641 2559 y Fo(;)-260 2769 y Fr(whic)n(h)27 b(is)f(true)g(for)g(an)n(y)g Fo(")d Fn(2)655 2701 y Fi(\000)693 2769 y Fo(")732 2738 y Fh(0)732 2789 y Fj(n)p Fk(+1)861 2769 y Fo(;)14 b(")937 2738 y Fh(0)937 2789 y Fj(n)982 2701 y Fi(\001)1020 2769 y Fr(,)27 b(as)f(w)n(e)g(see)g(from)g(\(40\).)37 b(Ho)n(w)n(ev)n(er,)25 b(in)h(this)h(case)f(w)n(e)g(ha)n(v)n(e)f(from)h(Lemma)h(7)f(the)h (additional)-260 2868 y(condition)h Fn(L)162 2880 y Fj(S)203 2888 y Fg(2)262 2868 y Fn(\026)23 b Fo(\033)400 2838 y Fh(\003)438 2868 y Fn(L)495 2880 y Fj(S)536 2888 y Fg(1)573 2868 y Fr(,)28 b(whic)n(h)f(can)g(b)r(e)h(written)g(as)1329 3089 y Fo(h)1377 3101 y Fk(1)1414 3089 y Fr(\()p Fo(")p Fr(\))23 b Fn(\024)g Fo(h)1676 3101 y Fk(2)1713 3089 y Fr(\()p Fo(")p Fr(\))c(+)1928 3033 y Fo(")1967 3003 y Fk(1)p Fj(=)p Fk(4)p 1928 3070 V 1951 3146 a Fo(C)2010 3158 y Fk(0)2095 3089 y Fr(ln)14 b(\()p Fo(c\033)2296 3055 y Fh(\003)2334 3089 y Fr(\))g Fo(:)-260 3299 y Fr(W)-7 b(e)27 b(can)f(\014nd)g(the)h(part)f(of)g(the)g(in)n(terv)-5 b(al)1051 3232 y Fi(\000)1089 3299 y Fo(")1128 3269 y Fh(0)1128 3320 y Fj(n)p Fk(+1)1257 3299 y Fo(;)14 b(")1333 3269 y Fh(0)1333 3320 y Fj(n)1378 3232 y Fi(\001)1442 3299 y Fr(where)26 b(this)g(inequalit)n(y)g(is)g(true,)h(pro)r(ceeding) e(as)g(with)i(\(64\).)36 b(In)27 b(this)f(case,)g(the)-260 3399 y(inequalit)n(y)h(is)h(true)f(in)h(almost)f(the)h(whole)f(in)n (terv)-5 b(al:)862 3547 y Fo(")901 3517 y Fh(0)901 3568 y Fj(n)p Fk(+1)p 862 3588 168 4 v 906 3664 a Fo(b)942 3676 y Fj(n)1063 3607 y Fn(\024)23 b Fo(")f Fn(\024)1297 3585 y Fr(~)1300 3607 y Fo(b)1336 3619 y Fj(n)1381 3607 y Fo(")1420 3573 y Fh(0)1420 3628 y Fj(n)1465 3607 y Fo(;)180 b(b)1704 3619 y Fj(n)1749 3607 y Fo(;)1783 3585 y Fr(~)1786 3607 y Fo(b)1822 3619 y Fj(n)1889 3607 y Fr(=)23 b(1)18 b(+)g Fn(O)e Fr(\(\012)2294 3573 y Fj(n)2353 3607 y Fr(ln)e(\()q(1)p Fo(=\033)2603 3573 y Fh(\003)2640 3607 y Fr(\))q(\))23 b Fn(')g Fr(1)p Fo(:)964 b Fr(\(66\))-260 3809 y(W)-7 b(e)27 b(see)e(that)i(an)n(y)e Fo(")h Fr(small)g(enough,)g (satisfying)f(\(62\),)i(b)r(elongs)e(to)h(at)g(least)g(one)g(of)g(the)g (in)n(terv)-5 b(als)26 b(\(65{66\),)e(and)i(hence)h(the)f(simple)-260 3909 y(zeros)j(of)h Fo(@)92 3921 y Fj(\022)130 3909 y Fn(L)187 3878 y Fk(\(2\))307 3909 y Fr(or)f Fo(@)455 3921 y Fj(\022)493 3909 y Fn(L)550 3878 y Fk(\(3\))670 3909 y Fr(p)r(ersist)h(in)h Fo(@)1083 3921 y Fj(\022)1120 3909 y Fn(L)d Fr(=)f Fn(M)p Fr(,)k(for)f(an)n(y)g Fo(")p Fr(.)45 b(W)-7 b(e)30 b(also)g(obtain)g(from)g(Lemma)g(7)g(the)g(upp)r (er)h(and)f(lo)n(w)n(er)f(b)r(ound)-260 4008 y(for)e(the)h(minim)n(um)h (eigen)n(v)-5 b(alue)26 b(at)i(eac)n(h)e(zero.)2692 b Fe(2)-260 4207 y Fs(Remarks.)-158 4456 y Fr(1.)41 b(W)-7 b(e)35 b(recall)f(that)h(the)h(v)-5 b(alue)34 b(of)h(the)h(exp)r(onen)n (t)e Fo(p)1545 4426 y Fh(\003)1583 4456 y Fr(,)j(de\014ned)e(in)h (\(50\),)g(dep)r(ends)f(on)g(whether)g(the)g(torus)f(is)h(mo)n(v)n(ed)f (b)n(y)h(the)-52 4556 y(p)r(erturbation)g(or)f(it)i(remains)e(\014xed:) 53 b Fo(p)1251 4526 y Fh(\003)1325 4556 y Fr(=)35 b(3)g(if)h Fo(\027)41 b Fr(=)36 b(0,)h(and)e Fo(p)2082 4526 y Fh(\003)2156 4556 y Fr(=)h(2)f(if)h Fo(\027)41 b Fr(=)35 b(1.)60 b(This)36 b(is)f(the)h(exp)r(onen)n(t)f(app)r(earing)f(in)-52 4655 y(Theorem)26 b(1,)i(a)f(simpli\014ed)h(v)n(ersion)e(of)i(the)g(one)f (pro)n(v)n(ed)f(ab)r(o)n(v)n(e.)-158 4813 y(2.)41 b(W)-7 b(e)28 b(ha)n(v)n(e)e(sho)n(wn)g(that)i(the)g(splitting)g(function)g (can)f(b)r(e)h(appro)n(ximated)d(b)n(y)j(its)f(2)g(dominan)n(t)g (harmonics)f(except)i(for)f Fo(")g Fr(close)-52 4913 y(to)i(some)g Fo(")300 4925 y Fj(n)344 4913 y Fr(,)h(according)e(to)h (\(65\).)42 b(This)29 b(sa)n(ys)f(that)h(the)h(4)f(zeros)f(remain)h (close)f(to)h(constan)n(t)g(for)g Fo(")g Fr(in)g(a)g(large)f(part)h(of) g(eac)n(h)-52 5012 y(in)n(terv)-5 b(al)24 b(\()q Fo(")317 5024 y Fj(n)361 5012 y Fo(;)14 b(")437 5024 y Fj(n)p Fh(\000)p Fk(1)567 5012 y Fr(\))q(.)36 b(When)25 b Fo(")g Fr(go)r(es)f(across)f(a)i(critical)g(v)-5 b(alue)25 b Fo(")1978 5024 y Fj(n)2022 5012 y Fr(,)h(a)f(quic)n(k)f(drift)i(of)f (the)g(zeros)f(tak)n(es)g(place,)i(un)n(til)f(they)g(again)-52 5112 y(remain)i(close)f(to)i(constan)n(t)f(in)h(the)g(next)f(in)n(terv) -5 b(al.)-158 5269 y(3.)41 b(It)36 b(is)f(in)n(teresting)g(to)g (compare)g(the)h(asymptotic)f(estimate)g(giv)n(en)g(in)h(part)f(\(a\))h (with)g(the)g(general)e(upp)r(er)i(b)r(ound)g(for)f(the)-52 5369 y(splitting)25 b(function)g(giv)n(en)f(in)i([DGS02)o(,)g(Th.)f (12].)35 b(The)25 b(constan)n(t)f Fo(C)2111 5381 y Fk(0)2174 5369 y Fr(is)h(closely)f(related)g(to)h(the)g(constan)n(t)f(app)r (earing)g(in)h(that)-52 5469 y(upp)r(er)i(b)r(ound.)38 b(Indeed,)28 b(taking)f(in)g(the)h(Diophan)n(tine)g(condition)f Fo(\034)33 b Fr(=)23 b(1)k(and)g Fo(\015)33 b Fr(as)27 b(in)h(\(32\),)f(one)g(has:)1167 5586 y Fi(\022)1228 5703 y Fr(1)18 b(+)1383 5647 y(1)p 1381 5684 46 4 v 1381 5760 a Fo(\034)1436 5586 y Fi(\023)c(\022)1583 5647 y Fo(\031)s(\034)9 b(\032)1721 5617 y Fj(\034)1763 5647 y Fo(\015)p 1583 5684 228 4 v 1676 5760 a Fr(2)1820 5586 y Fi(\023)1882 5603 y Fk(1)p Fj(=)p Fk(\()p Fj(\034)e Fk(+1\))2149 5703 y Fr(=)2237 5576 y Fi(r)p 2320 5576 223 4 v 2364 5647 a Fr(2)p Fo(\031)s(\032)p 2330 5684 203 4 v 2330 5760 a Fr(2)18 b Fn(\000)g Fr(\012)2566 5703 y(=)k Fo(C)2712 5715 y Fk(0)2750 5703 y Fo(:)1825 5989 y Fr(21)p eop %%Page: 22 22 22 21 bop -52 -82 a Fr(This)25 b(sho)n(ws)g(the)h(optimalit)n(y)f(of)h (the)g(constan)n(t)f Fo(C)1538 -70 y Fk(0)1601 -82 y Fr(in)h(the)g(upp)r(er)g(b)r(ound)g(of)f([DGS02,)h(Th.)g(12].)36 b(In)26 b(fact,)g(in)g(the)g(asymptotic)-52 17 y(estimates)f(giv)n(en)g (here)g(this)h(constan)n(t)f(has)g(b)r(een)h(replaced)f(b)n(y)g(the)i (function)f Fo(C)2500 29 y Fk(0)2537 17 y Fo(h)2585 29 y Fk(1)2622 17 y Fr(\()p Fo(")p Fr(\))e Fn(\025)e Fo(C)2895 29 y Fk(0)2933 17 y Fr(,)k(whic)n(h)g(pro)n(vides)e(a)h(re\014nemen)n (t)-52 117 y(of)i(the)h(estimate)g(\(recall)f(that)h Fo(h)1000 129 y Fk(1)1037 117 y Fr(\()p Fo(")p Fr(\))23 b(=)g(1)k(only)g(at)g(the)h(critical)f(v)-5 b(alues)28 b Fo(")2310 129 y Fj(n)2355 117 y Fr(\).)-260 490 y Fq(App)t(endix:)59 b(A)45 b(quan)l(titativ)l(e)i(implicit)f(function)f(theorem)-260 772 y Fr(In)25 b(order)e(to)i(study)f(the)h(nondegenerate)e(critical)h (p)r(oin)n(ts)h(of)f(the)h(splitting)g(p)r(oten)n(tial)g(\(i.e.)g(the)g (simple)f(zeros)g(of)g(the)h(splitting)g(func-)-260 871 y(tion\),)g(w)n(e)f(use)g(the)g(appro)n(ximation)e(pro)n(vided)h(b)n(y) h(a)f(few)h(dominan)n(t)g(harmonics.)35 b(Once)23 b(w)n(e)h(ha)n(v)n(e) e(sho)n(wn)i(that)g(this)g(appro)n(ximation)-260 971 y(has)31 b(nondegenerate)e(critical)i(p)r(oin)n(ts,)h(w)n(e)e(ensure)h (that)g(they)g(p)r(ersist)g(when)h(adding)e(all)h(the)g(neglected)g (harmonics.)46 b(W)-7 b(e)32 b(can)e(do)-260 1070 y(this)f(with)h(the)f (help)g(of)g(a)f(quan)n(titativ)n(e)h(implicit)g(function)h(theorem,)f (whic)n(h)f(allo)n(ws)g(us)h(to)f(tak)n(e)h(in)n(to)f(accoun)n(t)g(the) i(concrete)e(size)-260 1170 y(of)g(the)g(neglected)f(part.)-135 1369 y(The)i(follo)n(wing)f(quan)n(titativ)n(e)h(v)n(ersion)f(of)h(the) h(implicit)g(function)g(theorem)f(concerns)f(the)i(p)r(ersistence)f(of) g(a)g(simple)h(zero)e(of)i(a)-260 1469 y(function)g(under)f(a)g(p)r (erturbation.)42 b(W)-7 b(e)30 b(use)f Fn(j\001j)g Fr(to)h(denote)f (the)g(suprem)n(um)h(norm)e(of)i(a)e(function)i(\(on)g(the)f(real)g (domain)g Fl(T)3735 1439 y Fj(n)3779 1469 y Fr(\).)43 b(W)-7 b(e)-260 1569 y(omit)28 b(the)g(pro)r(of)f(of)g(this)h(result)g (b)r(ecause)f(it)h(is)f(standard.)-260 1845 y Fs(Prop)s(osition)j(9)42 b Fp(L)l(et)29 b Fo(G)554 1857 y Fk(0)591 1845 y Fo(;)14 b(G)24 b Fr(:)f Fl(T)819 1815 y Fj(n)886 1845 y Fn(\000)-14 b(!)23 b Fl(R)1097 1815 y Fj(n)1178 1845 y Fp(functions)30 b(of)g(class)h Fn(C)1885 1815 y Fk(2)1922 1845 y Fp(,)f(and)g Fo(\022)2177 1857 y Fk(0)2237 1845 y Fn(2)24 b Fl(T)2372 1815 y Fj(n)2446 1845 y Fp(such)30 b(that)f Fo(G)2869 1857 y Fk(0)2907 1845 y Fr(\()p Fo(\022)2978 1857 y Fk(0)3015 1845 y Fr(\))24 b(=)e(0)p Fp(,)30 b(satisfying)1290 2021 y Fn(j)q Fr(D)p Fo(G)1442 2033 y Fk(0)1479 2021 y Fr(\()p Fo(\022)1550 2033 y Fk(0)1588 2021 y Fr(\))p Fo(h)p Fn(j)23 b(\025)g Fo(m)14 b Fn(j)p Fo(h)p Fn(j)f Fo(;)99 b Fn(8)p Fo(h)22 b Fn(2)i Fl(R)2368 1987 y Fj(n)2419 2021 y Fo(:)-260 2198 y Fp(L)l(et)29 b Fo(M)j Fr(=)84 2127 y Fi(\014)84 2177 y(\014)111 2198 y Fr(D)174 2168 y Fk(2)212 2198 y Fo(G)277 2210 y Fk(0)315 2127 y Fi(\014)315 2177 y(\014)342 2198 y Fp(,)e Fo(\021)d Fr(=)22 b Fn(j)p Fo(G)d Fn(\000)f Fo(G)807 2210 y Fk(0)845 2198 y Fn(j)p Fp(,)30 b Fo(\021)967 2168 y Fh(0)1013 2198 y Fr(=)23 b Fn(j)p Fr(D)p Fo(G)c Fn(\000)f Fr(D)p Fo(G)1482 2210 y Fk(0)1520 2198 y Fn(j)p Fp(,)30 b(and)h(assume)e(that)1463 2436 y Fo(\021)d Fn(\024)1638 2380 y Fo(m)1711 2350 y Fk(2)p 1628 2417 132 4 v 1628 2493 a Fr(8)p 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b(\(in)j(mo)r(dulus\))f(of)f(this)h(matrix.)-260 3489 y Fq(Ac)l(kno)l(wledgmen)l(ts)-260 3770 y Fr(W)-7 b(e)30 b(w)n(ould)f(lik)n(e)g(to)g(thank)g(T.M.)h(Seara,)e(S.)i (Bolotin,)f(R.)h(de)f(la)g(Lla)n(v)n(e,)g(H.)g(Ko)r(c)n(h)g(and)g(D.)h (T)-7 b(resc)n(hev)28 b(for)h(helpful)h(discussions)e(and)-260 3870 y(remarks.)52 b(This)33 b(w)n(ork)f(has)g(b)r(een)i(partially)e (supp)r(orted)h(b)n(y)g(the)g(Catalan)f(gran)n(t)g(2001SGR-70,)g(the)h (Spanish)g(gran)n(t)f(BFM2000-)-260 3970 y(0805-C02)25 b(and)i(the)h(INT)-7 b(AS)28 b(gran)n(t)f(00-221.)-260 4330 y Fq(References)-260 4521 y Fw([DG00a])87 b(A.)27 b(Delshams)g(and)h(P)-6 b(.)27 b(Guti)n(\023)-36 b(errez.)40 b(Splitting)28 b(p)r(oten)n(tial)g(and)f(the)g(Poincar)n(\023)-36 b(e{Melnik)n(o)n(v)29 b(metho)r(d)d(for)i(whisk)n(ered)g(tori)g(in)f (Hamil-)102 4613 y(tonian)f(systems.)34 b Fb(J.)27 b(Nonline)l(ar)h (Sci.)p Fw(,)d(10\(4\):433{476,)30 b(2000.)-260 4735 y([DG00b])82 b(A.)35 b(Delshams)f(and)h(P)-6 b(.)35 b(Guti)n(\023)-36 b(errez.)62 b(Splitting)35 b(and)f(Melnik)n(o)n(v)h(p)r(oten)n(tials)h (in)f(Hamiltonian)f(systems.)62 b(In)34 b(J.)h(Delgado,)k(E.A.)102 4826 y(Lacom)n(ba,)33 b(E.)f(P)n(\023)-36 b(erez-Cha)n(v)n(ela,)35 b(and)c(J.)h(Llibre,)i(editors,)g Fb(Hamiltonian)e(Systems)j(and)e (Celestial)g(Me)l(chanics)h(\(HAMSYS-98\).)102 4917 y(Pr)l(o)l(c)l(e)l (e)l(dings)j(of)e(the)h(III)d(International)j(Symp)l(osium)p Fw(,)g(v)n(olume)d(6)h(of)g Fb(World)h(Scienti\014c)h(Mono)l(gr)l(aph)g (Series)g(in)e(Mathematics)p Fw(,)102 5009 y(pages)27 b(111{137.)h(Held)e(in)f(P\023)-38 b(atzcuaro,)28 b(Mic)n(hoac\023)-38 b(an,)27 b(M)n(\023)-36 b(exico,)27 b(7{11)g(Decem)n(b)r(er,)e(1998.)i (W)-6 b(orld)26 b(Scien)n(ti\014c,)g(Singap)r(ore,)g(2000.)-260 5131 y([DG01])125 b(A.)27 b(Delshams)g(and)g(P)-6 b(.)27 b(Guti)n(\023)-36 b(errez.)40 b(Homo)r(clinic)28 b(orbits)f(to)h(in)n (v)l(arian)n(t)f(tori)g(in)h(Hamiltonian)f(systems.)38 b(In)27 b(C.K.R.T.)h(Jones)g(and)102 5222 y(A.I.)j(Khibnik,)h(editors,) h Fb(Multiple-Time-Sc)l(ale)g(Dynamic)l(al)g(Systems)p Fw(,)h(v)n(olume)c(122)j(of)e Fb(IMA)h(V)-6 b(ol.)32 b(Math.)h(Appl.)p Fw(,)f(pages)g(1{27.)102 5313 y(Held)26 b(in)f(Minneap)r(olis,)j(27{31)f(Octob)r(er,)f(1997.)i(Springer-V)-6 b(erlag,)26 b(New)g(Y)-6 b(ork,)25 b(2001.)-260 5436 y([DGJS97])43 b(A.)32 b(Delshams,)j(V.G.)e(Gelfreic)n(h,)1146 5417 y(\022)1136 5436 y(A.)g(Jorba,)i(and)d(T.M.)i(Seara.)56 b(Exp)r(onen)n(tially)32 b(small)h(splitting)g(of)g(separatrices)i (under)c(fast)102 5527 y(quasip)r(erio)r(dic)c(forcing.)36 b Fb(Comm.)26 b(Math.)i(Phys.)p Fw(,)e(189:35{71,)k(1997.)-260 5649 y([DGS02])82 b(A.)23 b(Delshams,)g(P)-6 b(.)23 b(Guti)n(\023)-36 b(errez,)24 b(and)e(T.M.)i(Seara.)31 b(Exp)r(onen)n(tially)22 b(small)h(splitting)h(for)f(whisk)n(ered)g(tori)g(in)g(Hamiltonian)f (systems:)102 5740 y(Flo)n(w-b)r(o)n(x)k(co)r(ordinates)h(and)f(upp)r (er)e(b)r(ounds.)34 b(Preprin)n(t,)26 b(2002.)1825 5989 y Fr(22)p eop %%Page: 23 23 23 22 bop -260 -82 a Fw([DR98])128 b(A.)21 b(Delshams)f(and)g(R.)h(Ram) -9 b(\023)-30 b(\020rez-Ros.)26 b(Exp)r(onen)n(tially)21 b(small)g(splitting)g(of)g(separatrices)i(for)e(p)r(erturb)r(ed)f(in)n (tegrable)h(standard-lik)n(e)102 9 y(maps.)34 b Fb(J.)27 b(Nonline)l(ar)h(Sci.)p Fw(,)d(8\(3\):317{352,)30 b(1998.)-260 134 y([DS97])142 b(A.)20 b(Delshams)g(and)f(T.M.)j(Seara.)j(Splitting) 20 b(of)h(separatrices)h(in)d(Hamiltonian)h(systems)g(with)g(one)g(and) f(a)i(half)f(degrees)h(of)g(freedom.)102 225 y Fb(Math.)28 b(Phys.)g(Ele)l(ctr)l(on.)g(J.)p Fw(,)e(3\(4\):1{40,)i(1997.)-260 349 y([Eli94])150 b(L.H.)29 b(Eliasson.)44 b(Biasymptotic)28 b(solutions)i(of)f(p)r(erturb)r(ed)e(in)n(tegrable)i(Hamiltonian)g (systems.)41 b Fb(Bol.)29 b(So)l(c.)i(Br)l(asil.)e(Mat.)h(\(N.S.\))p Fw(,)102 441 y(25\(1\):57{76,)f(1994.)-260 565 y([Gal94])125 b(G.)29 b(Galla)n(v)n(otti.)44 b(Twistless)30 b(KAM)e(tori,)i(quasi)e (\015at)h(homo)r(clinic)f(in)n(tersections,)j(and)c(other)i (cancellations)h(in)e(the)g(p)r(erturbation)102 657 y(series)f(of)f (certain)h(completely)e(in)n(tegrable)h(Hamiltonian)g(systems.)f(A)g (review.)35 b Fb(R)l(ev.)28 b(Math.)g(Phys.)p Fw(,)e(6\(3\):343{411,)k (1994.)-260 781 y([Gel99])129 b(V.G.)28 b(Gelfreic)n(h.)42 b(A)27 b(pro)r(of)i(of)f(the)f(exp)r(onen)n(tially)h(small)g(transv)n (ersalit)n(y)g(of)g(the)f(separatrices)j(for)e(the)f(standard)h(map.)39 b Fb(Comm.)102 873 y(Math.)28 b(Phys.)p Fw(,)e(201\(1\):155{216,)k (1999.)-260 997 y([GGM99])54 b(G.)22 b(Galla)n(v)n(otti,)i(G.)d(Gen)n (tile,)i(and)e(V.)g(Mastropietro.)29 b(Separatrix)21 b(splitting)h(for)g(systems)f(with)g(three)g(time)g(scales.)29 b Fb(Comm.)22 b(Math.)102 1088 y(Phys.)p Fw(,)k(202\(1\):197{236,)31 b(1999.)-260 1213 y([Ko)r(c99])110 b(H.)32 b(Ko)r(c)n(h.)54 b(A)31 b(renormalization)j(group)e(for)h(Hamiltonians,)h(with)f (applications)g(to)g(KAM)e(theory)-6 b(.)53 b Fb(Er)l(go)l(dic)35 b(The)l(ory)f(Dynam.)102 1304 y(Systems)p Fw(,)28 b(19\(2\):475{521,)i (1999.)-260 1429 y([Laz84])124 b(V.F.)32 b(Lazutkin.)51 b(Splitting)32 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Fw(,)e(114:3{80,)j(1998.)-260 2201 y([Sau01])120 b(D.)35 b(Sauzin.)61 b(A)35 b(new)g(metho)r(d)e(for)j(measuring)f(the)f (splitting)i(of)f(in)n(v)l(arian)n(t)g(manifolds.)62 b Fb(A)n(nn.)36 b(Sci.)3233 2182 y(\023)3222 2201 y(Ec)l(ole)g(Norm.)g (Sup.)g(\(4\))p Fw(,)102 2292 y(34\(2\):159{221,)30 b(2001.)-260 2417 y([Sim94])115 b(C.)31 b(Sim\023)-38 b(o.)46 b(Av)n(eraging)30 b(under)e(fast)j(quasip)r(erio)r(dic)g(forcing.)48 b(In)29 b(J.)i(Seimenis,)f(editor,)i Fb(Hamiltonian)f(Me)l(chanics:)43 b(Inte)l(gr)l(ability)102 2508 y(and)26 b(Chaotic)g(Behavior)p Fw(,)f(v)n(olume)d(331)j(of)f Fb(NA)-6 b(TO)25 b(ASI)g(Ser.)h(B:)e (Phys.)p Fw(,)h(pages)f(13{34.)h(Held)e(in)h(T)-6 b(oru)r(\023)-41 b(n,)24 b(P)n(oland,)h(28)f(June{2)g(July)102 2599 y(1993.)k(Plen)n (um,)d(New)g(Y)-6 b(ork,)25 b(1994.)-260 2724 y([SV01])143 b(C.)30 b(Sim\023)-38 b(o)29 b(and)f(C.)i(V)-6 b(alls.)46 b(A)28 b(formal)i(appro)n(ximation)f(of)h(the)e(splitting)i(of)g 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