Analytic methods for obstruction to integrability in discrete dynamical systems

indiscretedynamicalsystems

OCostin

Piscataway,NJ08854-8019

RutgersUniversity

arXiv:math/0608308v1 [math.DS] 13 Aug 2006Piscataway,NJ08854-8019

Abstract

Auniqueanalyticcontinuationresultisprovedforsolutionsofarelativelygeneralclassofdifferenceequations,usingtechniquesofgeneralizedBorelsummability.

ThiscontinuationallowsforPainlev´epropertymethodstobeextendedtodifferenceequations.

ItisshownthatthePainlev´eproperty(PP)induces,underrelativelygeneralas-sumptions,adichotomywithinfirstorderdifferenceequations:allequationswithPPcanbesolvedinclosedform;onthecontrary,absenceofPPimplies,undersomefur-therassumptions,thatthelocalconservedquantitiesarestrictlylocalinthesensethattheydevelopsingularitybarriersontheboundaryofsomecompactset.

Thetechniqueproducesanalyticformulastodescribefractalsetsoriginatinginpolynomialiterations.

Contents

1Introductionandmainresults21.1Setting31.2Analyzability:transseriesandgeneralizedBorelsummability4

Transseriesfordifferenceequations5

1.3UniquenessofcontinuationfromNtoC5

1.4Continuationofsolutionsofdifferenceequationstothecomplexnplane51.5Continuabilityandsingularities51.6Integrability51.7Firstorderautonomousequations6

1.8Classificationofequationsoftype(1.11)withrespecttointegrability71.9Failureofintegrabilitytestandbarriersofsingularities71.10Example:thelogisticmap81.11Applicationtothestudyoffractalsets82Generalremarksonintegrability9

c199XJohnWiley&Sons,Inc.󰀐

CommunicationsonPureandAppliedMathematics,Vol.000,0001–0033(199X)

CCC0010–30/98/000001-33

2O.COSTINANDM.D.KRUSKAL

3Proofs

3.1ProofofTheorem1.1

Outline

3.2

3.33.4

3.53.6

121212121414

CompletionoftheproofofTheorem1.114Remarksonfirstorderequations15

Relationtopropertiesoftheconjugationmap16Conservedquantities17EndofproofofLemma1.4(a)17ProofofTheorem1.518

Notations18The“if”partofTheorem1.518The“onlyif”partofTheorem1.518ProofofTheorem1.819

Borelsummabilityofformalinvariantforlogisticmapwhena=120

ProofofTheorem3.1122

2427

2728293030

4Juliasetsforthemap(1.13)fora∈(0,1)5Behavioratthesingularitybarrier

Analyticity

EndoftheproofofTheorem1.10(iii)Appendix:Iterationsofrationalmaps

Furtherresultsusedintheproofs

6.1ProofofProposition1.7

1Introductionandmainresults

Solvabilityofdifferenceequationsaswellaschaoticbehaviorhavestim-ulatedextensiveresearch.FordifferentialequationsthePainlev´etest,whichconsistsincheckingwhetherallsolutionsofagivenequationarefreeofmovablenon-isolatedsingularitiesprovidesaconvenientandeffec-tivetoolindetectingintegrablecases(see§2).AdifficultyinapplyingPainlev´e’smethodstodifferenceequationsresidesinextendingthesolutions,whicharedefinedonadiscreteset,tothecomplexplaneoftheindependentvariableinanaturalandeffectivefashion,when,intheinterestingcases,thereisnoexplicitformulafor

INTEGRABILITYOFDIFFERENCEEQUATIONS3

them.Anumberofalternativeapproaches,butnogenuineanalogofthePainlev´etest,havebeenproposed,see[1][9][26][24](acomparativediscussionofthevariousapproachesispresentedin[1]).

Thepresentpaperproposesanaturalway,basedongeneralizedBorelsummabillity,toextendthesolutionsinthecomplexplane(Theorem1.1below),allowingforadefinitionofadiscretePainlev´etest.Subsequentanalysisshowsthatthetestissharpinaclassoffirstorderdifferenceequations:thosepassingthetestareexplicitlysolvable(Theorem1.5)whilepolynomialequationsfailingthetestexhibitchaoticbehaviorandtheirlocalconservedquantities(see§1.9)developbarriersofsingularitiesalongfractalsets(Theorem1.8).

Theapproachalsoallowsforadetailedstudyofanalyticpropertiesnearthesesingularitybarriersaswellasfindingrapidlyconvergentseriesrepresentingthecorrespondingfractalcurves(Theorem1.10).

1.1Setting

Weconsiderdifferencesystemsofequationswhichcanbebroughttotheform(1.1)

ˆI+1x(n+1)=Λ

󰀈

4O.COSTINANDM.D.KRUSKAL

1.2Analyzability:transseriesandgeneralizedBorelsumma-bility

´TheseconceptswereintroducedbyEcalleinthefundamentalwork[14].

Analyzabilityofdifferenceequationswasshownin[7,14].Wegivebelowabriefdescriptionoftheconceptseffectivelyusedinthepresentpaperandreferto[10,7]forageneraltheory.Anexpressionoftheform(1.5)

˜(t):=x

󰀏

˜k(t)Cke−k·µttk·ax

k∈Nm

˜k(t)areformalpowerseriesinpowersoft−1isanexponentialwherex

powerseries;itisatransseriesast→+∞ifℜ(µj)>0foralljwith1≤j≤m.SuchatransseriesisBorelsummableast→+∞ifthereexistconstantsA,ν>0andafamilyoffunctions

XkanalyticinasectorialneighborhoodSofR+,satisfying

(1.6)

p∈S,k∈Nm

sup

suchthatthefunctionsxkdefinedby(1.7)

xk(t)=

󰀓

∞

󰀂󰀂󰀂|k|−ν|p|󰀂

Xk󰀂<∞󰀂Ae

0

e−tpXk(p)dp

˜ki.e.areasymptotictotheseriesx(1.8)

˜k(t)(t→+∞)xk(t)∼x

Itistheneasytocheckthatcondition(1.6)impliesthatthesum(1.9)

x(t)=

󰀏

Cke−k·µttk·axk(t)

k∈Nn0

isconvergentinthehalfplaneH={t:ℜ(t)>t0},fort0largeenough.

˜inThefunctionxin(1.9)isbydefinitiontheBorelsumofthetransseriesx

(1.5).GeneralizedBorelsummabilityallowsforsingularitiesofXkofcer-˜is(generalized)BorelsummableintaintypesalongR+.Thetransseriesx

iϕ+−iϕ˜(·e)is(generalized)Borelsummable.(General-thedirectioneRifx

ized)Borelsummationisknowntobeanextendedisomorphismbetweentransseriesandtheirsums,see[14],[15],[10].

INTEGRABILITYOFDIFFERENCEEQUATIONS5

Transseriesfordifferenceequations

Braaksma[7]showedthattherecurrences(1.1)posessl-parametertransseries

˜k(n)areformalpowerse-solutionsoftheform(1.5)witht=nwherex

riesinpowersofn−1andl≤mischosensuchthat,afterreorderingtheindices,wehaveℜ(µj)>0for1≤j≤l.

Itisshownin[7]and[19]thatthesetransseriesaregeneralizedBorelsummableinanydirectionandBorelsummableinallexceptmofthemandthat(1.10)

x(n)=

k∈Nl

󰀏

Cke−k·µnnk·axk(n)

isasolutionof(1.1),ifn>y0,t0largeenough.

1.3UniquenessofcontinuationfromNtoC

Thevaluesofxontheintegersuniquelydeterminex.

Theorem1.1Intheassumptionsin§1.1and1.2,definethecontinua-tionofxk(n)inthehalfplane{t:ℜ(t)>t0}byx(t),cf.(1.6)–(1.9).Thefollowinguniquenesspropertyholds.Ifintheassumptions(1.6)–(1.9)wehavex(n)=0forallexceptpossiblyfinitelymanyn∈N,thenx(t)=0forallt∈C,ℜ(t)>t0.Theproofisgivenin§3.1.

1.4Continuationofsolutionsofdifferenceequationstothe

complexnplane

Therepresentation(1.10)andTheorem1.1makethefollowingdefinitionnatural.

1.5Continuabilityandsingularities

ThefunctionxisanalyticinHandhas,ingeneral,nontrivialsingularitiesinC\\H.Theresultsin[12],extendedtodifferenceequationsin[7,8,19],giveconstructivemethodstodeterminethosesingularitiesthatariseneartheboundaryofH;theseform,generically,nearlyperiodicarrays.

1.6Integrability

Inparticular,Painlev´e’stestofintegrability(absenceofmovablenon-isolatedsingularites)extendsthentodifferenceequations.

6O.COSTINANDM.D.KRUSKAL

Asinthecaseofdifferentialequations,fixedsingularitiesaresingularpointswhoselocationisthesameforallsolutions;theydefineacommonRiemannsurface.Othersingularities(i.e.,whoselocationdependsoninitialdata)arecalledmovable.

Definition1.2WesaythatadifferenceequationhasthePainlev´eprop-ertyifitssolutionsareanalyzableandtheiranalyticcontinuationsonaRiemannsurfacecommontoallsolutions,haveonlyisolatedsingularities.Note.Wefollowtheusualconventionthatanisolatedsingularpointofananalyticfunctionfisapointz0suchthatfisanalyticinsomediskcenteredatz0exceptperhapsatz0itself.Branchpointsarethusnotiso-latedsingularitiesandneitheraresingularitybarriers;itisworthnoting,however,thatfordifferentialequationsthereexistequationssometimesconsideredintegrable(theChazyequation,athirdordernonlinearoneisthesimplestknownexample)whosesolutionsexhibitsingularitybarriers.

1.7Firstorderautonomousequations

Theseareequationsofthetype(1.11)

xn+1=G(xn):=axn+F(xn)

SomeanalyticityassumptionsonFarerequiredforourmethodtoapply.Wedefineaclassofsinglevaluedfunctionsclosedunderallalgebraicoperationsandcomposition(thelatterisneededsincexnwrittenintermsofx0involvesrepeatedcomposition).

WeneedtoallowforsingularbehaviorinF,andmeromorphicfunc-tionsareobviouslynotclosedundercomposition.Thefollowingdefinitionformalizesanextensionofmeromorphicfunctions,oftenusedinformallyinthetheoryofintegrability.

Definition1.3Wedefinethe”mostlyanalyticfunctions”tobetheclassMoffunctionsanalyticinthecomplementofaclosedcountableset(whichmaydependonthefunction).

Lemma1.4(a)TheclassMisclosedunderaddition,multiplicationandmultiplicationbyscalars,andalsounderdivisionandcompositionbetween(nonconstant)functions.Itincludesmeromorphicfunctions.

(b)IfG∈Misnotaconstant,thentheequationG(x)=yhassolutionsforalllargeenoughy.

INTEGRABILITYOFDIFFERENCEEQUATIONS7

(c).TheclassM0ofG∈M,withGanalyticatzero,G(0)=0and0<|G′(0)|<1isclosedundercomposition.Inparticular,G◦m∈Mform≥1.Proof.Allpropertiesin(a)areobviousexceptforclosureundercomposi-tionanddivision,provedin§3.3;(b)followsfromtheproofofLemma6.9.(c)iseasilyshownusing(a).

1.8Classificationofequationsoftype(1.11)withrespect

tointegrability

Theorem1.5AssumeG∈Mhasastablefixedpoint(sayatzero)whereitisanalytic.Thenthedifferenceequation(1.11)hasthePainlev´epropertyiffforsomea,b∈Cwith|a|<1,(1.12)

G(z)=

az

8O.COSTINANDM.D.KRUSKAL

arenotsolvableintermsoffunctionsextendibletothecomplexplane,oronRiemannsurfaces.Theconservedquantitieswilltypicallydevelopsingularitybarriers.

Weuse,intheformulationofthefollowingtheorem,anumberofstandardnotionsandresultsrelevanttoiterationsofrationalmaps;thesearebrieflyreviewedintheAppendix,§6.

Theorem1.8AssumeGisanonlinearpolynomialwithanattractingfixedpointattheorigin.DenotebyKpthemaximalconnectedcomponentoftheoriginintheFatousetofG.(ItfollowsthatKpisanopen,bounded,andsimplyconnectedset).

ThenthedomainofanalyticityofQ(see(3.26))isKp,and∂KpisasingularitybarrierofQ.Thistheoremisprovedin§3.5.

1.10Example:thelogisticmap

Thediscretelogisticmapisdefinedby(1.13)

xn+1=axn(1−xn)

Thefollowingresultwasprovedbytheauthorsin[11].

Proposition1.9Therecurrence(1.13)hasthePainlev´epropertyinDefinition1.2iffa∈{−2,0,2,4}(inwhichcasesitisexplicitlysolvable).Ifa∈/{−2,0,2,4}thentheconservedquantityhasbarriersofsingulari-ties.

1.11Applicationtothestudyoffractalsets

ThetechniquesalsoprovidedetailedinformationontheJuliasetsofit-erationsoftheinterval.

Theorem1.10Considertheequation(1.13)fora∈(0,1/2).

(i)ThereisananalyticfunctionG,satisfyingthefunctionalrelation(1.14)

G(z)2=aG(z2)(1+G(z))

whichisaconformalmapoftheopenunitdiskS1onto{x−1:x∈ext(J)}whereJistheJuliasetof(1.13).

(ii)GisLipschitzcontinuousofexponentlog2(2−a)in

INTEGRABILITYOFDIFFERENCEEQUATIONS9

(iii)∂S1isabarrierofsingularitiesofG.Near1∈∂S1wehave(1.15)where

G(z)=ΦτΨ(lnτ)

󰀆

󰀇

τ=τ(z)=ln(z−1)log2(2−a)

a

Φisanalyticatzero,Φ(0)=

1−a

+

󰀏󰀏

Cl;k,mt2πillog2(2−a)/ln2+klog2(2−a)+m

l∈Zk,m∈N

wheretheseriesconverges(rapidly)iftand|argt|aresmall.Thistheoremisprovedin§4.

Note1.11TheproofofProposition5.1showsthattheLipschitzexpo-nentisoptimal.Thetheoremisvalidforanya<1,andtheproofissimilar.

Note1.12ItfollowsfromTheorem1.10(iii)and(1.14)thateverybi-naryrationalisacuspofJofangleπlog2(2−a),seealsoFig.1.

2Generalremarksonintegrability

Thisproblemhasalonghistory,andthetaskoffindingofdifferentialequationssolvableintermsofknownfunctionswasaddressedasearlyastheworksofLeibniz,Riccati,Bernoulli,Euler,Laplace,andLagrange.“Inthe18thcentury,Eulerwasdefiningafunctionasarisingfromtheapplicationoffinitelyorinfinitelymanyalgebraicoperations(addition,multiplication,raisingtointegerorfractionalpowers,positiveornega-tive)oranalyticoperations(differentiation,integration),inoneormorevariables”[6].Itwaslaterfoundthatsomelinearequationshavesolutionswhich,althoughnotexplicitbythisstandard,have“good”globalprop-ertiesandcanbethoughtofasdefiningnewfunctions.Toaddressthequestionwhethernonlinearequationscandefinenewfunctions,FuchshadtheideathatacrucialfeaturenowknownasthePainlev´eproperty(PP)istheabsenceofmovable(meaningtheirpositionissolution-dependent,

10O.COSTINANDM.D.KRUSKAL

3

3/2

1.0

1.0 0

0.30.40.50.60.7Figure1.1.(a)JuliasetforG=

2,

1

c=.0793243476(theplotrelieson(5.8),N=300).

INTEGRABILITYOFDIFFERENCEEQUATIONS11

cf.§1.6)essentialsingularities,primarilybranch-points,see[17].Firstor-derequationswereclassifiedwithrespecttothePPbyFuchs,BriotandBouquet,andPainlev´eby1888,anditwasconcludedthattheygiverisetononewfunctions.Painlev´etookthisanalysistosecondorder,lookingforallequationsoftheformu′′=F(u′,u,z),withFrationalinu′,alge-braicinu,andanalyticinz,havingthePP[29,30].Hisanalysis,revisedandcompletedbyGambierandFuchs,foundsomefiftytypeswiththispropertyandsucceededtosolveallbutsixofthemintermsofpreviouslyknownfunctions.TheremainingsixtypesarenowknownasthePainlev´eequations,andtheirsolutions,calledthePainlev´etranscendents,playafundamentalroleinmanyareasofpureandappliedmathematics.Begin-ninginthe1980’s,almostacenturyaftertheirdiscovery,theseproblemsweresolved,usingtheirstrikingrelationtolinearproblems1,byvariousmethodsincludingthepowerfultechniquesofisomonodromicdeformationandreductiontoRiemann-Hilbertproblems[13],[16],[25].

SophieKovalevskayasearchedforcasesofthespinningtophavingthePP.Shefoundapreviouslyunknownintegrablecaseandsolveditintermsofhyperellipticfunctions.Herwork[20],[21]wassooutstandingthatnotonlydidshereceivethe1886BordinPrizeoftheParisAcademyofSciences,buttheassociatedfinancialawardwasalmostdoubled.ThemethodpioneeredbyKovalevskayatoidentifyintegrableequa-tionsusingthePainlev´epropertyisnowknownasthePainlev´etest.PartofthepowerofthePainlev´eteststemsfromtheremarkablephenomenonthatequationspassingitcangenerallybesolvedbysomemethod.Thisphenomenonisnotcompletelyunderstood.Atanintuitivelevel,however,ifforexampleallsolutionsofanequationaremeromorphic,thenbysolv-ingtheequation“backwards,”thesesolutionsandtheirderivativescanbewrittenintermsoftheinitialconditions.Thisgivesrisetosufficientlymanyintegralsofmotionwithgoodregularitypropertiesgloballyinthecomplexplane.

ThePainlev´etesthassomedrawbacks,notablylackofinvarianceun-dertransformations.Toovercomethem,[22]introducedthepoly-Painlev´etest.

12O.COSTINANDM.D.KRUSKAL

3Proofs

3.1ProofofTheorem1.1

Outline

Theideaoftheproofistousetheconvergenceof(1.9)anditsasymptoticpropertiestoshowthatalltermsxkvanish.

Westartwithsomepreparatoryresults.

Remark3.1IfxkhaveXk=≡0thenalsoXk≡0(seeLemma[5],for󰀕∞j=largeLj(1.7))soforsmallpwe

kcjpwithcLzintherightk=half0forplanesomewehaveLk≥0.ByWatson’s

(3.1)

xk∼

󰀏∞cjj!

j=Lk

INTEGRABILITYOFDIFFERENCEEQUATIONS13

Letalso(3.4)

rj=maxℜ(a·k)

k∈Tj

Notealsothatforsomeα>0wehave(3.5)

rj≤αMj

∞󰀔

ApplyingRemark3.2againweseethat(3.6)

Tj=S

j=0

Lemma3.4Wehave(see(1.9)),(3.7)

x(z)=

k∈T0

󰀏

Cke−k·µzzk·axk(z)+O(e−M1zzr1)

(z→+∞)

Proof:Wewrite(3.8)

x(z)=

k∈T0

󰀏

Ce

k−k·µzk·a

zxk(z)+

k∈S\\T0

󰀏

Cke−k·µzzk·axk(z)

Thesecondseriesisuniformlyandabsolutelyconvergentforlargeenough

z∈R+sinceitisboundedbythesub-sumofa(derivativeof)amulti-geometricseries(3.9)

k∈S\\T0

󰀏

|AkCkzk·a|e−k·ℜ(µ)z

Since(3.9)isabsolutelyconvergentitcanbethusbeconvergentlyrear-rangedas(3.10)

∞󰀏

e

−Mjz

j=1k∈Tj

󰀏

|ACz

kkk·a

|=

j=1

∞󰀏

e−MjzzrjDj(z)

(seeagainDefinition3.3andRemark3.2).ItiseasytoseethatDj(z)are

nonincreasinginz∈R+andforlargeenoughz>0allproductszrje−Mjzaredecreasing(cf.also(3.5)).Thereforetheconvergentseries(3.11)

∞󰀏

e−(Mj−M1)zzrj−r1Dj(z)

j=1

isdecreasinginz>0andso

∞󰀏

e−MjzzrjDj(z)≤Const.e−M1zzr1

j=1

14O.COSTINANDM.D.KRUSKAL

Note.AsimilarstrategycouldbealsobeusedtoshowtheclassicalWeierstrasspreparationtheorem.

Assumefirst,togetacontradiction,thatwehavex0soforsmallpwehaveXk=󰀕∞j=mjwithcm≡0andsoX0≡0

0cjp0=0.Then,since

x(n)=x0(n)+O(e−M1nnr1)

andbyRemark3.1(3.12)

−m0−1nlim→∞

nx0=(m0+1)!cm0=0

whichcontradictsx(n)=0forn∈N.

Letnow(3.13)R0=max{ℜ(k·a−Lk−1):k∈T0}

and(3.14)

T0′

={k∈T0:ℜ(k·a−Lk−1)=R0}

Lemma3.5Wehave

(3.15)x(z)=󰀏

CkcLzk·a−Lk−1e−k·µz+o󰀆zR0e−M0z

kLk!󰀇for(z→+∞)

k∈T0

′Proof:ThisisanimmediateconsequenceofRemark3.1,Lemma3.4,and(3.13)and(3.14).

INTEGRABILITYOFDIFFERENCEEQUATIONS15

Lemma3.6Letdk∈C.Then

′k∈T0

󰀏

dkn

k·a−Mk−k·µn

e=on

󰀆

R0−K1n

e

󰀇

(asn→∞,n∈N)

iffalldkarezero.

′),nlargeenoughandnotethatProof:Wenowtaken0=card(T0

(n+j)b=nb(1+o(n−1))ifj≤n0.ThenasimpleestimateshowsthattoprovetheLemmaitsufficestoshowthatthefollowingequationcannotholdforall0≤l≤n0−1

(3.16)where(3.17)

′k∈T0

󰀏

dke−(n+l)k·µ=ql

ql=o(e−nM0)(asn→∞,n∈N)

Ifn0=1thisisimmediate.Otherwise,wemaythinkof(3.16)for0≤l≤

′.Thedeterminant∆n0−1asasystemofequationsforthedkwithk∈T0

ofthesystemisanumberofabsolutevaluee−nlM0timestheVandermonde

′.Inparticular,forsomeC>0determinantofthequantities{e−k·µ}k∈T0

independentofnwehavethate−nlM0|∆|isindependentofn,(3.18)

󰀂󰀂󰀂󰀑󰀂󰀂󰀂−nlM0−(k1−k2)·µ󰀂e|∆|=C󰀂(e−1)󰀂󰀂󰀂k1=k2∈T′󰀂

0

andnonzeroby(1.4).Similarly,theminor∆kofanydkisboundedby

Dke−n(l−1)M0withDkindependentofn.Wegetdk=o(1)forlargen

′,andsod=0.forallk∈T0k

16O.COSTINANDM.D.KRUSKAL

Assumefornowthatin(1.11)G∈Misanalyticatzero,F(0)=F′(0)=0and0<|a|<1.Aswementioned,thereisaone-parameterfamilyofsolutionspresentedassimpletransseriesoftheform(3.19)

xn=xn(C)=

∞󰀏

enklnaCkDk

k=1

withDkindependentofC,whichconvergeforlargen.Bydefinitiontheir

continuationtocomplexnis(3.20)

x(z)=x(z;C)=

∞󰀏

ezklnaCkDk,

k=1

whichisanalyticforlargeenoughz.TotestforthePainlev´eproperty,we

proceedtofindthepropertiesofx(z)forthosevaluesofzwhere(3.20)isnolongerconvergent,andthenfindthesingularpointsofx(z).

Note.Ingeneral,although(3.20)representsacontinuousone-parameterfamilyofsolutions,theremaybemoresolutions.Wealsoexaminethisissue.

Relationtopropertiesoftheconjugationmap

Wecanalternatively,anditturnsoutequivalently,defineacontinuationasfollows.BythePoincar´etheorem[2]p.99thereexistsauniquemapϕwiththeproperties(3.21)andsuchthat(3.22)

ϕ(az)=G(ϕ(z))=aϕ(z)+F(ϕ(z))ϕ(0)=0,ϕ′(0)=1andϕanalyticat0

Themapϕisaconjugationmapbetween(1.11)anditslinearization(3.23)

since,inviewof(3.22),(3.24)

xn=ϕ(Can)Xn+1=aXn

forgivenCandnlargeenough,xnisasolutionoftherecurrence(1.11).WeobtainacontinuationofxfromNtoCthrough(3.25)

x(z)=ϕ(Caz)

INTEGRABILITYOFDIFFERENCEEQUATIONS17

Lemma3.8(i)Forequationsoftype(1.11),thecontinuations(3.20)and(3.25)agree.

(ii)x(z;C)definedby(3.20)hasonlyisolatedmovablesingularitiesiffϕhasonlyisolatedsingularitiesinC.

Proof:Indeed,ϕisanalyticattheorigin,andapowerseriesex-pansionforlargenofϕ(Can)leadstoasolutionoftheform(3.19),whichobviouslysolves(1.11).Ifn0islargeenough,itisclearthat(3.19)canbeinvertedforCintermsofxn0andwecanalsofindC′sothatxn0=ϕ(C′an0).Ontheotherhandxn0uniquelydeterminesallxnwithn>n0.Forequationsoftype(1.11),writingx(z)=ϕ(Caz)isthustantamounttomakingthesubstitutionn=zin(3.19).Notethat,azisentireandϕisanalyticatzero,andthepresenceofasingularityofϕwhichisnotisolatedisequivalenttothepresenceofasimilarbutmovablesingularityofx(z)=ϕ(Caz)sinceitspositiondependsonC.

18O.COSTINANDM.D.KRUSKAL

isaleastk=k(x)suchthatG(k)

2(x)=0andthenG2hasmultiplicity

exactlykinasmalldiskDxaroundx.ThenG−1

Sinceforeveryxthereisanopen2(E˜1)∩Dxiscountable.

∩DxiscountableitfollowsthatE˜setDxsuchthatE

,thusE,isalsocountable.Inthesameway,foranya∈/wehavethatG−definedi1

Ei

(a)iscountable.Fordivision,notethat1/GisdefinedwhereverGisandnonzero.SinceGisnotaconstantthesameargumentasaboveshowsthatG−1(0)iscountable.

The“if”partofTheorem1.5

Inthisdirectiontheproofistrivial.Indeed,ifGislinearfractional,thenthegeneralnonidenticallyzerosolutionoftheequation(1.11)canbeobtainedbysubstitutingx=1/yin(1.11)whichthenbecomeslinear.Weget

xn=󰀈

Ca−n+

b

INTEGRABILITYOFDIFFERENCEEQUATIONS19

Lemma3.10Iffhasonlyisolatedsingularitiesandfisnotlinear-fractionalthenforanylargeenoughw,theequationf(z)=whasatleasttwodistinctroots.

Proof:Iffisrational,thenthepropertyisimmediate.Thenas-sumethatthatfisnotrational,thusfhasatleastoneessentialsin-gularity,possiblyatinfinity[18].IffhasanessentialsingularityinC,thenitisisolatedbyhypothesisandthenthepropertyfollowsfromThe-orem6.7.ThenassumethatfhasnoessentialsingularityinC,thusinfinityistheonlyessentialsingularityoff.IfitisisolatedthenTheo-rem6.7appliesagain.Otherwisefhasinfinitelymanypolesaccumulatingatinfinity.Sincefmapsaneighborhoodofeverypoleintoafullneigh-borhoodofinfinity,anysufficientlylargevalueoffhasmultiplicitylargerthanone.

20O.COSTINANDM.D.KRUSKAL

thereisafinitesubcovering

󰀔

NC⊂OC=

Dǫ(zi)(zi)

i=1

withzi∈C.LetMbethelargestofthem(zi),i=1,...,N.Then,by

construction,(3.28)

G[M](OC)∈Dǫ

Weseefrom(3.27)thataQ(z)=Q(G(z))=a−1Q(G(G(z)))andingeneral,forn∈N,(3.29)

Q(z)=a−nQ(G[n](z))

WedefineQ(z)inOCbyQ(z)=a−MQ(G[M](z)).By(3.28),andbecause(3.29)holdsinDǫ,thisunambiguouslydefinesananalyticcontinuationofQfromDǫtoDǫ∪zeroOC.SinceKpisopenandsimplyconnectedandsinceQisanalyticnearandcanbecontinuedanalyticallyalonganyarcinKp,standardcomplexanalyticresultsshowthatQis(singlevaluedand)analyticinKp.

Forthelastpart,notethattheboundaryofKpliesintheJuliasetJ,whichistheclosureofrepellingperiodicpoints(seeAppendix,Lemma6.3).Assumethatx0isarepellingperiodicpointofGofperiodn,andthatx0isapointofanalyticityofQ.Relation(3.29)impliesthatQ(x0)=0andthatQ′(x0)=a−n(G[n])′(x0)Q′(x0)butsince|a|<1and|(G[n])′(x0)|>1thisimpliesQ′(xm)0)=0.Inductively,inthesamewayweseethatQ((x0)=0forallm,whichundertheassumptionofanalyticityentailsQ≡0whichcontradicts(3.21).

INTEGRABILITYOFDIFFERENCEEQUATIONS21

straightforwardly,andBorelsummabilitymakesitpossibletoanalyzethepropertiesofthisequationrigorously.AformalanalysisofthePainlev´epropertyisrelativelystraightforwardusingmethodssimilartothosein[12].Weconcentratehereonpropertiesoftheconservedquantities.Therecurrencean+1=an(1+an)−1isexactlysolvableanddiffersfromthelogisticmapbyO(a3n)forsmallan.Theexact

−1solutionisn−an=Const,whichsuggestslookinginthelogisticmap

caseforaconstantoftheiterationintheformofanexpansionstarting

1withC=n−a−n.Thisyields(3.31)

C(n;v)∼n−v−1−lnv−

1

3v2−

13

240

v4+···

whichisindeedaformalinvariant,buttheassociatedseriesisfactorially

divergentaswillappearclearshortly.NeverthelesswecanshowthattheexpansionisBorelsummabletoanactualconservedquantityinasectorialneighborhoodofv=0.

Theorem3.11ThereisaconservedquantityCdefinedneartheorigininC\\R−,oftheform

C(n;v)=n−v−1−ln(v)−R(v)

whereR(v)hasaBorelsummableseriesattheorigininanydirectionintheopenrighthalfplane.R(v)hasasingularitybarriertouchingtheori-gintangentiallyalongR−.ThissingularitybarrierisexactlytheboundaryoftheLeaudomainof(3.30).Welet(3.32)

C(n;v):=n−v−1−lnv−R(v)

andimposetheconditionthatCisconstantalongtrajectories.Thisyields(3.33)

R(v)=R(v−v2)+

v

x

+ln

󰀈

x

22O.COSTINANDM.D.KRUSKAL

whichbyformalexpansioninpowersofx−1becomes∞h(k)(x)

x

(3.36)

h(x−1)=

󰀏k=0

x

+ln

󰀈

p

+

k󰀏∞(−p)k

=1

INTEGRABILITYOFDIFFERENCEEQUATIONS23

andF∗kistheconvolutionofFwithitselfktimes.Werewrite(3.38)intheform(3.39)

H=

1−e−p−p

(ep−1)k=1

∞󰀏(−p)k

2

+ǫ,

π

k!

=

∞󰀏(−1)kp2k

f∗1

∗k

=

k=1

∞󰀏(−p)k

(k−1)!k!(k−1)!

󰀓

0

ds

1

f(p(1−t))tk−1dt

k=1

ItisimmediatethatifpisinacompactsetKandfisanalyticinK

thenthesumin(3.41)isuniformlyconvergentinKandanalyticinp.FurthermorethesumisO(p3)forsmallpsincef∈A.Nowweseethat(3.42)

󰀂

∞󰀂k2k

󰀂−ν|p|󰀏(−1)p󰀂e󰀂

k=1

k!(k−1)!

≤󰀤f󰀤ν

k=1

∞󰀏

󰀓

1

0

|p|2k

󰀂

󰀂

−ν|p|(1−t)k−1−ν|p|t󰀂ef(p(1−t))tedt󰀂

󰀂󰀓∞

k!(k−1)!

0

tk−1e−ν|p|tdte|p|/ν

=󰀤f󰀤ν

k=1

∞󰀏|p|k

ν

24andthus(3.43)

O.COSTINANDM.D.KRUSKAL

󰀤A󰀤≤Constν−1

forsufficientlylargeν,wherewetookintoaccounttheexponentialde-creaseof(ep−1)−1forlargepinN.ThustheequationhasauniquefixedpointH∈A.InparticulartheLaplacetransformh(x)=LH=󰀖∞−xp

H(p)dpiswelldefinedandanalyticinthehalf-planeℜ(x)>ν.0e

Itisnowimmediatetocheckthath(x)satisfiestheequation(3.36).

4Juliasetsforthemap(1.13)fora∈(0,1)

Itisconvenienttoanalyzethesuperattractingfixedpointatinfinity;thesubstitutionx=1/ytransforms(1.13)into(4.1)

yn+1=−

2yn

a(G(z)+1)

;

G(0)=0,G′(0)=a

Lemma4.1([11])ThereexistsauniqueanalyticfunctionGintheneigh-borhoodoftheoriginsatisfying(4.2).ThisGhasonlyisolatedsingular-itiesinCifandonlyifa∈{−2,2,4}.Inthelattercase,(1.13)canbe

solvedexplicitly.

Ifa∈{−2,2,4}thentheunitdiskisabarrierofsingularitiesofG.Lemma4.2GisanalyticintheopenunitdiskS1andLipschitzcontin-uousin

2

INTEGRABILITYOFDIFFERENCEEQUATIONS25

(withthechoiceofbranchconsistentwithG(0)=0,G′(0)=a).Ifr<1

1

then(4.3)providesanalyticcontinuationinadiskofradiusr

2)areTheassumptionG(z0)=−4a−1thusimpliesthatthevaluesG(z0

inR−anddecreaseinn,againimpossibleGisanalyticat0andG(0)=0.WenowshowGisboundedinS1.Indeed,by(4.3)wehave

󰀁

−∞,−4a

󰀃−1

a(x+1)

iswelldefinedandincreasingontheinterval

n

.

(4.4)

|G(z)|≤U(|G(z2)|)

a

1−a

󰀍

ontheotherhand,acalculationshowsthat(4.5)

U(s)≤

SinceG(0)=0andGisanalyticinS1,(4.4)and(4.5)implythat

a

(4.6)sup|G(z)|≤

z∈S1

26

O.COSTINANDM.D.KRUSKAL

Proposition4.6Fora∈󰀆

0,1

2n

,2

−

1

n

2

+

󰀎

4

+

mG(z)(2+G(z))

sothat

|G′(z)|≤|G′(z2

)|(1−maxa)|y|≤a󰀂

󰀂󰀂󰀂

2a(1+y)2󰀂

2−a

|G′(z2)|

ifa≤1/2fromwhichProposition4.6followsimmediately.

INTEGRABILITYOFDIFFERENCEEQUATIONS27

5Behavioratthesingularitybarrier

Proposition5.1Thereisδ>0,arealanalyticfunctionΨ,periodicofperiodln2andananalyticfunctionΦ,Φ′(0)=1suchthatfor|arg(1−z)|<δ(1.15)holds.

Proof:Letω=2π/ln2,β=log2(2−a).Withz0∈(0,1)and1/2n

zn=z0,thesequenceGn=G(zn)isincreasingandboundedbyL,see(4.3).Itfollowsimmediatelyfrom(4.3)and(4.4)that(5.1)

L−Gn:=δn↓0asn→∞(L:=

a(2−a)

a

+C2,C2=1−a

󰀍

(5.2)δn+1=

1

1−C1δn+1

󰀃

δn

Eqs.(5.1)and(5.2)implythatforanyǫ>0wehave(5.3)Let(5.4)

cf.(5.1).Now

󰀂󰀂(C1−C2)δn+1󰀂θn−θn+1󰀂

−1󰀂=󰀂e

δn=o(2−a−ǫ)−n

󰀁

asn→∞

δn=lnβ(1/zn)eθn=2−nβlnβ(1/z0)eθn

28O.COSTINANDM.D.KRUSKAL

amountstoashiftinn).Ifǫ1issmallenough,thenitiseasytocheckthatequation(5.2)isacontractivemappingintheintheballofradiuscSǫ1={ζ:|ζ≤ǫ1}inBanachspacel∞,α(N)ofvectorsv(n;ζ)analyticinζ=z0−z1withrespecttothenorm

󰀤v󰀤=

n≥1;|ζ|≤ǫ1

sup

|v(n,ζ)α−n|

√

andlocalanalyticityinaneighborhoodoftheinterval[z0,

EndoftheproofofTheorem1.10(iii)

Weusetheinformationobtainedin§5.Leteθn=(1+wn)eΘ;givenδ>0wechoosen0largeenoughandǫ2sothat|wn(z0)|<δif|z−z0|<ǫ2andn≥n0.Welet,h=e2Θ,εn=2nβ,s=lnβ(1/z0),c=C1−C2,C=c−C2andobtain(5.7)wn=

Ce2Θsεn

24Θ−w2Θ2Θ1−2εnse2Θ+2ε2n+1C2se(1−εnC2se)nse

Asin§5,acontractivemappingargumentshowsthatw=(wn,wn+1,...)

isanalyticinse2Θ,ifsissmallenough.Theconclusionnowfollowsfromthedefinition

2G(z0

−n0

)=L+sδn0

and(5.4),(5.5),§5andthesubstitutione2Θ(·)=Ψ(ln(ln(·)).Formula(1.17)followsimmediatelyfrom(1.15).

τN+1

τN+1−τN

−

gN

INTEGRABILITYOFDIFFERENCEEQUATIONS29

Appendix:Iterationsofrationalmaps

Weintroduceanumberofdefinitionsandresultsforiterationsofrationalmaps,whicharetreatedinmuchmoredetailandgeneralityin[32]and[4].WeshallillustratethemainconceptsonthesimplecaseG=ax(1−x).InFigure1,theinterior(inthecomplexplane)ofthefractalcurvesisasetinvariantunderGandwiththefurtherpropertythatstartingwithz0insidethem-thiterateofGatz0,G◦m(z0),convergestozeroasm→∞.ThesearestablefixeddomainsofG.

ConsiderthepolynomialmapG.AFatoudomainofGisastablefixeddomainVofGcharacterizedbythepropertythatG◦nconvergesinthechordalmetricontheRiemannsphereC∞toafixedpointofG,locallyuniformlyinV.

Definition6.1([4],p.50)LetGbeanon-constantrationalfunction.TheFatousetofGisthemaximalopensubsetofC∞onwhich{G◦n}isequicontinuousandtheJuliasetofGisitscomplementinC∞.AFatoudomainisaLeaudomain(oraparabolicbasin)ifx0∈∂Vandthemultiplierofx0(thederivativeatx0)isλ=12.InFigure1thishappensfora=1.

TheJuliasetcanbecharacterizedbythefollowingproperty.Lemma6.2([4],p.148)LetGbearationalmapofdegreed,(cf.Defi-nition6.4)whered≥2.ThenJisthederivedset3oftheperiodicpointsofG.

Undertheassumptionsabove,wehave

Lemma6.3([4],p.148)JistheclosureoftherepellingpointsofG.Definition6.4([4],p.30.)IfR=P/QwherePandQarepolynomi-als,thenthedegreeoftherationalfunctionRismax{deg(P),deg(Q)}.Definition6.5([4])IfRisarationalfunctionandR◦m=R◦R◦···◦Rntimes,thenaperiodicpointofperiodnofRisapointzsuchthatR◦mz=zandR◦mz=zifmWealsousethefollowingresultofI.N.Baker:

30O.COSTINANDM.D.KRUSKAL

Lemma6.6([3],[4])LetRbearationalfunctionofdegreed≥2,andsupposethatRhasnoperiodicpointsofperiodn.Then(d,n)isoneofthepairs

(2,2),(2,3),(3,2),(4,2)

(moreover,eachsuchpairdoesarisefromsomeRinthisway).Furtherresultsusedintheproofs

Theorem6.7(BigtheoremofPicard,localformulation[31],[18])Iffhasanisolatedsingularityatapointz0andifthereexistssomeneighborhoodofz0wherefomitstwovalues,thenz0isaremovablesingularityorapoleoff.

Theorem6.8(Picard-Borel,[28])Ifϕisanynonconstantfunctionmero-morphicinC,thenϕavoidsatmosttwovalues(infinityincluded).Allweneedinthepresentpaperisthatatmosttwofinitevaluesareex-cluded.ThisisimmediatelyreducedtothemorefamiliarPicardtheorembynotingthatifλisanexcludedvalueoffthen1/(f−λ)isentire.

6.1ProofofProposition1.7

ByTheorem1.5,(1.11)doesnothavethePainlev´epropertyatsomestablefixedpointiffGisnotlinear-fractional,inwhichcase(1.11)failstohavethePainlev´epropertyatanyotherstablefixedpoint.Moregenerally,Proposition1.7followsfromthefollowingresult.

Lemma6.9IfG◦misoftheform(1.12)thenGisoftheform(1.12).Proof:Since(1.12)isonetoone,theconclusionfollowsfromtheremarkthatifGisnotlinear-fractional,thenG(z)hasmultiplicitygreaterthanoneforallsufficientlylargez(andthenthesameholdsforG◦m(z)).Indeed,assumethatGisnotlinear-fractional.IfGisrational,thentheconclusionisobvious.IfthesetofsingularitiesofGisfinite,thentheyareallisolatedandatleastoneisanessentialsingularity(otherwiseGisrational[18])andTheorem6.7applies.

Sowemayassumethesetofsingularitiesisinfinite.Sincebyassump-tionthissetisclosedandcountable,itcontainsinfinitelymanyisolatedpoints.(Indeed,asetwhichisclosedanddenseinitself,i.e.aperfectset,iseitheremptyorelseuncountable.)ThenifGhasanisolatedessential

INTEGRABILITYOFDIFFERENCEEQUATIONS31

singularity,Theorem6.7applies,andifnotthenthereareinfinitelymanypolesofG.InthelattersituationanysufficientlylargevalueofGhasmultiplicitylargerthanonesinceGmapsaneighborhoodofeverypoleintoafullneighborhoodofinfinity.

Acknowledgment.TheauthorsareverygratefultoRDCostinformanyusefuldiscussionsandcomments.Theauthorswouldalsoliketo

32O.COSTINANDM.D.KRUSKAL

thankRConte,FFauvet,NJoshiandDSauzinforinterestingdiscus-sions.

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*Institute*

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