Arnold Diffusion in A Priori Chaotic Symplectic Maps

41 Pages·2017·0.82 MB·English

Commun.Math.Phys.353,507–547(2017) Communicationsin
DigitalObjectIdentifier(DOI)10.1007/s00220-017-2867-0 Mathematical
Physics
Arnold Diffusion in A Priori Chaotic Symplectic Maps
VassiliGelfreich1,DmitryTuraev2,3
1 MathematicsInstitute,UniversityofWarwick,Coventry,UK.E-mail:[email protected]
2 ImperialCollege,London,UK
3 LobachevskyUniversityofNizhnyNovgorod,NizhnyNovgorod,Russia.E-mail:[email protected]
Received:30March2015/Accepted:6January2017
Publishedonline:24April2017–©TheAuthor(s)2017.Thisarticleisanopenaccesspublication
Abstract: We assume that a symplectic real-analytic map has an invariant normally
hyperbolic cylinder and an associated transverse homoclinic cylinder. We prove that
generically in the real-analytic category the boundaries of the invariant cylinder are
connectedbytrajectoriesofthemap.
1. Introduction
AHamiltoniandynamicalsystemisdefinedwiththehelpofaHamiltonfunction H :
M →RonasymplecticmanifoldMofdimension2n.LetM beaconnectedcomponent
c
ofalevelset{H =c}.SinceHremainsconstantalongthetrajectoriesoftheHamiltonian
system,theset M isinvariant.DependingontheHamiltonfunction H andtheenergy
c
c,therestrictionofthedynamicsontoM mayvaryfromuniformlyhyperbolic(e.g.,in
c
thecaseofageodesicflowonasurfaceofnegativecurvature)tocompletelyintegrable.
SincePoincarésworks,ithasbeenacceptedthatatypicalHamiltoniansystemdoes
not have any additional integral of motion independent of H (unless the system pos-
sessessomesymmetriesandtheNoethertheoremapplies).Ontheotherhandageneric
Hamiltoniansystemisnearlyintegrableinaneighbourhoodofatotallyellipticequilib-
rium(agenericminimumormaximumof H)ortotallyellipticperiodicorbit.Thenthe
Kolmogorov–Arnold–Moser(KAM)theoryimpliesthattheHamiltoniansystemisnot
ergodic(withrespecttotheLiouvillemeasure)onsomeenergylevels[72].Indeed,the
KAMtheoryestablishesthatanearlyintegrablesystempossessesasetofinvarianttori
ofpositivemeasure.
Each of the KAM tori has dimension n. For n > 2 a KAM torus does not divide
M whichhasdimension(2n−1),moreover,thecomplementtotheunionofallKAM
c
tori is connected and dense in M . Thus the KAM theory does not contradict to the
c
existenceofadenseorbitinM .Itisunknownwhethersuchorbitsreallyexistinnearly
c
integrablesystems.ThequestiongoesbacktoFermi[39]whosuggestedthefollowing
notion:aHamiltoniansystemiscalledquasi-ergodicifinevery M anytwoopensets
c
508 V.Gelfreich,D.Turaev
areconnectedbyatrajectory.Thispropertyisequivalenttotopologicaltransitivityofthe
Hamiltonianflowon M .Thispropertycanalsoberestatedinslightlydifferentterms:
c
(a) in every M there is a dense orbit or (b) in every M dense orbits form a residual
c c
subset.
Fermi conjectured [39] that quasi-ergodicity is a generic property of Hamiltonian
systems, but proved a weaker statement only: if a Hamiltonian system with n > 2
degreesoffreedomhastheform
H = H (I)+εH (I,ϕ,ε), (1)
0 1
whereH isintegrableand(I,ϕ)areaction-anglevariables,thengenericallyM doesnot
0 c
containaninvariant(2n−2)-dimensionalhyper-surfacethatisanalyticinε.Obviously,
suchsurfacewouldpreventthequasi-ergodicity.However,non-analyticinvarianthyper-
surfaces cannot be excluded from consideration as it is not known whether they can
exist generically or not. So Fermi’s quasi-ergodic hypothesis remains unproved. The
recent papers [21,64,65,71,74] make an important step in the understanding of the
underlying dynamics by showing that for the generic (in a certain smooth category)
near-integrablecasewith21 ormoredegreesoffreedom,therearetrajectoriesthatvisit
2
anaprioriprescribedsequenceofballs.Thepaper[52]providesexamplesofsystems
havingorbitswhoseclosurecontainsaLebesguepositivemeasuresetofKAM-tori.
Thisproblemiscloselyrelatedtotheproblemofstabilityofatotallyellipticfixed
pointofasymplecticdiffeomorphism,orstabilityofatotallyellipticperiodicorbitfora
Hamiltonianflow.Itwasprovedin[35,36]thatstabilitycanbebrokenbyanarbitrarily
smallsmoothperturbation.Itisbelievedthatatotallyellipticperiodicorbitisgenerically
unstablebutthetimescalesforthisinstabilitytomanifestitselfareextremelylong,see
e.g.[15,59].
Forε =0,theunperturbedsystem(1)isdescribedbytheHamiltonian H = H (I).
0
Then the actions I are constant along trajectories, so the equation I = I defines an
0
invarianttorus,andtheanglesϕarequasi-periodicfunctionsoftimewiththefrequency
vectorω (I) = H(cid:3)(I).KAMtheoryimpliesthatthemajorityofinvarianttorisurvive
0 0
underperturbation.Toriwithrationallydependentfrequenciesarecalledresonantand
aredestroyedbyatypicalperturbation[2].Thefrequencyofaresonanttorussatisfiesa
conditionoftheformω (I)·k=0forsomek∈Zn\{0}.Theresonanttoriforma“res-
0
onantweb”,typically(e.g.ifω isalocaldiffeomorphism)adensesetofmeasurezero.
0
Arnold’sexample[1]showsthatatrajectoryoftheperturbedsystem(1)canslowly
driftalongaresonance.Arnold’spaperinspiredalargenumberofstudiesinthelong-
timestabilityofactions,theproblemwhichisknownas“Arnolddiffusion”.Ithasbeen
attractingsignificantattentionrecentlyandwereferthereadertopapers[11,31,33,40–
43,54,57,60–62,68–70,76,82,84]foramoredetaileddiscussion.
Itshouldbenotedthatthemotionalongtheresonantwebisveryslow:Nekhoroshev
theory[14,78]providesalowerboundontheinstabilitytimesintheanalyticcase.Let
{·,·}denotethePoissonbrackets.Then I˙={H,I}=ε{H ,I}isoftheorderofε.On
1
the other hand, if the system satisfies assumptions of the KAM theory, |I(t)− I(0)|
remainssmallforalltimesandthemajorityofinitialconditions,i.e.,forthesetofinitial
conditionsofasymptoticallyfullmeasure.If H satisfiesassumptionsoftheNekhoro-
shev theory, there are some exponents a,b > 0 such that |I(t)− I(0)| < εa for all
|t| < expε−b andforallinitialconditions.Thisestimateestablishesanexponentially
largelowerboundforthetimesofArnoldDiffusioninanalyticsystems.
ItisimportanttostressthattheupperboundonthespeedofArnolddiffusionstrongly
dependsonthesmoothnessofthesystem.Indeed,thestabilitytimesareexponentially
ArnoldDiffusion 509
large in ε−1 for analytic systems, but only polynomial bounds can be obtained in the
Ckcategory.Inparticular,papers[21,64,74]studytheArnolddiffusionfornon-analytic
HamiltoniansandthereforetheboundsestablishedbytheanalyticalNekhoroshevtheory
are likely to be violated, see e.g. [13]. The problem of genericity of Arnold diffusion
inanalyticcategoryremainsfullyopen.Webelievethemethodsproposedinourpaper
willhelptoadvancethetheoryintheanalyticcase.
Thenormalformtheorysuggeststhatforsmallpositiveε thesystem(1)hasanor-
mallyhyperboliccylinderwithapendulum-likeseparatrixlocatedinaneighbourhood
of a simple resonance. Indeed, Bernard proved the existence of normally-hyperbolic
cylinders in a priori stable Hamiltonian systems [6], the size of such cylinder being
boundedawayfromzeroforarbitrarilysmallsizeoftheperturbation.
Amodelforthissituationisoftenobtainedbyassumingthattheintegrablepartof
theHamiltonianalreadypossessesanormally-hyperboliccylinderandanassociatedho-
moclinicloop(e.g.byconsideringH = P(p,q)+h (I)wherePisaHamiltonianofa
0 0
pendulum).Asystemofthistypeiscalledaprioriunstable.Thedriftoforbitsalongthe
cylinder has been actively studied inthe lastdecade[3–5,7–9,12,18–20,22,27,28,31–
34,70,87,88], including the problem of genericity of this phenomenon and instability
times.ItshouldbenotedthattheArnolddiffusioncanbemuchfasterinthiscase.
Inthesestudies,adriftingtrajectorytypicallystaysmostofthetimenearthenormally-
hyperboliccylinder,occasionallymakingatripnearahomoclinicloop.Theprocesscan
bedescribedusingthenotionofascatteringmapintroducedbyDelshamsetal.[32].Ear-
lierMoeckel[75]suggestedthatArnolddiffusioncanbemodelledbyrandomapplication
oftwoarea-preservingmapsonacylinder(thisapproachwasrecentlycontinuedin[17,
46–49,53,66]).InthiswaythedeterministicHamiltoniandynamicsismodelledbyanit-
eratedfunctionsystem,andtheobstaclestoadriftalongthecylinderappearintheformof
essentialcurveswhichareinvariantwithrespecttobothmapssimultaneously[67,75,77].
ThisproblemiscloselyrelatedtotheMatherproblemontheexistenceoftrajecto-
rieswithunboundedenergyinaperiodicallyforcedgeodesicflow[10,29].Thecriteria
for the existence of trajectories of the energy that grows up to infinity are known for
sufficientlylargeinitialenergies[10,24,29,30,44,45,80,81].Theresultsofthepresent
papercanbeusedtoestablishthegenericexistenceoforbitsofunboundedenergyfor
allpossiblevaluesofinitialenergy.
In our paper we depart from the near-integrable setting and study the dynamics of
an exact symplectic map in a homoclinic channel, a neighbourhood of a normally-
hyperbolictwo-dimensionalcylinder Aalongwithasequenceofhomocliniccylinders
BatatransverseintersectionofthestableandunstablemanifoldsofA.Weconductarig-
orousreductionoftheproblemtothestudyofaniteratedfunctionsystemandshowthat
theexistenceofadriftingtrajectory(i.e.theinstabilityoftheArnolddiffusiontype)is
guaranteedwhentheexactsymplecticmapsofthecylinder Athatconstitutetheiterated
functionsystemdonothaveacommoninvariantcurve.Thereductionschemeisinthe
samespiritasin[50,77]whilethesettingandproofsaredifferent.Thecompletelynovel
resultisthattheexistenceofdriftingorbitsisagenericphenomenon,i.e.itholdsforan
open and dense subset of a neighbourhood, in the space of analytic symplectic maps,
ofthegivenmapwithahomoclinicchannel,providedtherestrictionofthemaponthe
cylinder Ahasatwistproperty.AlltheknownsimilargenericityresultsfortheArnold
diffusionhavebeenprovensofarinthesmoothcategoryandusethenon-analiticityof
theperturbationsinanessentialway.
Inonerespect,thesituationweconsiderismoregeneralthaninthenear-integrable
setting,aswedonotassumetheexistenceofalargesetofKAMcurvesontheinvariant
510 V.Gelfreich,D.Turaev
cylinder A.Ontheotherside,asonecanextractfromtheexampleof[25],ourassump-
tionofthestrongtransversalityofthehomoclinicintersections,whichweneedinorder
todefinethescatteringmapsthatformtheiterationfunctionsystem,seemstofailfor
a generic analytic near-integrable system in a neighbourhood of a resonance in the a
prioristablecase.Therefore,ourresultsdonotadmitanimmediatetranslationtothea
prioristablecase.Rather,theproblemweconsiderhereisrelatedtotheapriorichaotic
case, e.g. we assume certain transversality of invariant manifolds associated with the
normallyhyperboliccylinder.
ThetechnicalassumptionsofourmaintheoremcanbefoundinSect.2.Asanexam-
ple,wecanconsidera4-dimensionalsymplecticmapthatisadirectproductofatwist
mapandastandardmap.Namely,Φ :(ϕ,I,x,y)(cid:5)→(ϕ¯,I¯,x¯,y¯)where
0
ϕ¯ =ϕ+ω(I), x¯ = x +y¯,
I¯= I, y¯ = y+ksinx, (2)
wherek >0isapositiveparameterandωisananalyticfunction.Weassumeϕandxto
beangularvariables,sothemapisasymplecticdiffeomorphismof(T×R)2.Themap
Φ hasanormallyhyperbolicinvariantcylinder Agivenbyx = y =0.Thecylinder A
0
isfilledwithinvariantcurvesasthemapΦ preservesthevalueofthe I variable.The
0
(x,y)componentofΦ coincideswiththestandardmap,whichhastransversalhomo-
0
clinicpointsforallk >0.ThusΦ verifiestheassumptionsofthemaintheorem.Then
0
agenericanalyticperturbationofΦ producesorbitswhichconnectsneighbourhoods
0
ofanytwoessentialcurvesin A.
A more interesting example is obtained when the integrable twist map is replaced
byanotherstandardmap,sothenewunperturbedmapisgivenbyΦ : (ϕ,I,x,y) (cid:5)→
0
(ϕ¯,I¯,x¯,y¯)where
ϕ¯ =ϕ+I¯, x¯ = x +y¯,
I¯= I +k sinϕ, y¯ = y+k sinx. (3)
1 2
Thecylinder A = {x = y = 0}isstillinvariantbutitisnolongerfilledwithinvariant
curves.InsteadthecylindercontainsaCantorsetofinvariantcurvesprovidedk isnot
1
toolarge.ThesetoripreventtrajectoriesofΦ fromtravelinginthedirectionoftheIaxis.
0
Thetheorypresentedinthispaperallowsustotreatbothcasesequallyandimplies
that an arbitrarily small generic analytic perturbation creates trajectories which travel
betweenregions I < I and I > I forany I < I [providedω(cid:3)(I)isseparatedfrom
a b a b
0for(2),andk >C(4|k |+k2)for(3)].Indeed,inordertoapplyTheorem1tothese
2 1 1
examples,wenotefirst,thattheinvariantcylinder Aisnormallyhyperbolic.Thiscylin-
derhasastableandunstableseparatrices Wu(A)and Ws(A)whichcoincidewiththe
productof Aandthestable(reps.,unstable)separatrixofthestandardmapWu,s,sowe
sm
canwrite(slightlyabusingnotation)Ws(A)= A×Ws andWu(A)= A×Wu .This
sm sm
product also describes the structure of the foliation of Wu,s(A) into strong stable and
strongunstablemanifoldsofpointsin A.Forapointv ∈ A,weletEuu(v)={v}×Wu
sm
andEss(v)={v}×Ws .Theassumptionk >C(4|k |+k2)for(3)ensuresthatthese
sm 2 1 1
strongstableandstrongunstablefoliationsremainC1-smoothaftertheperturbation.
It can be proved that the standard map has infinitely many transversal homoclinic
orbits for any k > 0. Let p = (x ,y ) be one of these orbits. The cylinder B =
h h h
A×{p } ⊂ Wu(A)∩Ws(A) is homoclinic to A. Since the strong stable and strong
h
unstablefoliationsofapointv ∈ Acoincidewiththeproductofthebasepointandthe
separatricesofthestandardmap,weseethat(v,p )∈ Ess(v)∩Euu(v),andthecylinder
h
Bsatisfiesthestrongtansversalityassumptiondescribedinthenextsectiongivingriseto
ArnoldDiffusion 511
asimplehomoclinicintersection(definedinthenextsection).ThenTheorem2implies
thatgenericperturbationofΦ hasorbitstravelinginthedirectionofthecylinder A.
0
Similar maps were considered in Easton et al. [37] (motivated by the “stochastic
pumpmodel”ofTennysonetal.[89]).In[37]theexistenceofdriftorbitswasshown
forallnon-integrableLagrangianperturbationsprovidedk islargeenough(i.e.inthe
2
“anti-integrable”limit).Ourmethodsallowustoobtainthedriftingorbitswithoutthe
largek assumption,i.e.,withoutadetailedknowledgeofthedynamicsofthesystem.
2
2. Set-up,Assumptions,andResults
Considerareal-analyticdiffeomorphismΦ :(cid:6) →R2d,d ≥2,definedonanopenset
(cid:6) ⊆R2d.WeassumethatΦpreservesthestandardsymplecticformΩ,andthatΦisex-
act(e.g.thelatterisalwaystrueif(cid:6)issimply-connected).LetΦhaveaninvariantsmooth
two-dimensionalcylinder AdiffeomorphictoS1×[0,1]andψ : S1×[0,1] → (cid:6) be
thecorres(cid:2)pondinge(cid:3)mbed(cid:2)ding.The(cid:3)ntheboundaryof Aconsistsoftwoinvariantcircles:
∂A=ψ S1×{0} ∪ψ S1×{1} .Letint(A)= A\∂Aand F =Φ| .
0 A
Weassumethatthecylinder A isnormally-hyperbolic.Moreprecisely,weassume
thatateachpointv ∈ Athetangentspaceisdecomposedintoadirectsumofthreenon-
zerosubspaces:TvR2d =R2d = Nvc⊕Nvu⊕Nvs,whereNvcisthetwo-dimensionalplane
tangenttoAatthepointv.ThesubspacesNs,udependcontinuouslyonvandareinvariant
wcnhoiottehicrtehesaoptfeΦncot(cid:3)rNtmovctsh=iendNNerFcsi,0vu(av,c)titvaheseΦrAe(cid:3)ieosxfiinsthtveαarmi>aanp1t,wai.neidt.hΦλr(cid:3)e∈Nspvs(e0=c,t1tN)osFsΦu0(cv.h)WtahneadatΦsastu(cid:3)eNmvveuery=thpaNotifFuno0t(rvvs).o∈WmAee
(cid:12)F(cid:3)(v)(cid:12)<α, (cid:12)(F(cid:3)(v))−1(cid:12)<α, (4)
0 0
(cid:12)Φ(cid:3)(v)|Nvs(cid:12)<λ, (cid:12)(Φ(cid:3)(v)|Nvu)−1(cid:12)<λ, (5)
where
α2λ<1. (6)
Notethattheseassumptionsaremorerestrictiveincomparisonwiththestandarddefi-
nitionofanormallyhyperbolicmanifold.Inparticular,thelargespectralgapcondition
(6)impliestheC1-regularityofthestrongstableandstrongunstablefoliationswhilein
thegeneralcasethesefoliationsareHöldercontinuousonly(seee.g.[83]).
Wealsonotethatin(4)and(5)thesamepairofexponentsαandλboundbothΦ(cid:3)and
(Φ(cid:3))−1,sowesaythat Aissymmetricallynormally-hyperbolic.Thesymmetricformof
thespectralgapassumptionimpliesthattherestrictionofthesymplecticformon Ais
non-degenerate(seeProposition4).Thus AisasymplecticsubmanifoldofR2d andthe
map F =Φ| inheritsthe(exact)symplecticityofΦ.
0 A
Wehavenodoubtsthatourresultscanbeextendedtocoverthecasewhenλandαof
inequalities(4)and(5)dependonthepointv ∈ A.However,forthesakeofsimplicity,
weconducttheproofsforthecaseofconstantλandαonly.
Thepointsinasmallneighbourhoodofthenormally-hyperboliccylinder A,whose
forwarditerationsdonotleavetheneighbourhoodandtendto Aexponentiallywiththe
rateatleastλ,formasmooth(atleastC2inourcase)invariantmanifold,thelocalstable
manifoldWs ⊃ A,whichistangentto Ns⊕Nc atthepointsof A(seee.g.[56]).The
loc
pointswhosebackwarditerationstendto A exponentiallywiththerateatleastλ(and
withoutleavingtheneighbourhood)formaC2-smoothinvariantmanifoldWu ⊃ A(the
loc
localunstablemanifold),whichistangentto Nu⊕Nc atthepointsof A.Theinvariant
512 V.Gelfreich,D.Turaev
cylinderAistheintersectionofWu andWs .Theglobalstableandunst(cid:4)ablemanifolds
loc loc
of A aredefinedbyiteratingthelocalinvariantmanifolds: Wu(A) := ΦmWu
(cid:4) m≥0 loc
andWs(A):= Φ−mWs .
m≥0 loc
IneachofthemanifoldsthereexistsauniquelydefinedC1-smoothinvariantfolia-
tiontransverseto A,thestrong-stableinvariantfoliation Ess inWs(A)andthestrong-
unstable invariant foliation Euu in Wu(A), such that for every point v ∈ A there is a
uniqueleafofEvssandauniqueleafofEvuuwhichpassthroughthispointandaretangent
to Nvs and,respectively, Nvu (see[86]).TheC1-regularityofafoliationmeansthatthe
leaves of the foliation are smooth and, importantly, the field of tangents to the leaves
isalsosmooth,whichimpliesthatforanytwosmoothcross-sectionstransversetothe
foliationthecorrespondencedefinedbytheleavesofthefoliationbetweenthepointsin
thecross-sectionsisalocaldiffeomorphism.
Let us discuss the question of the persistence of A at small perturbations. It is a
standard fact from the theory of normal hyperbolicity [56] that any strictly-invariant
normally-hyperbolic compact smooth manifold with a boundary can be extended to
a locally-invariant normally-hyperbolic manifold without a boundary. In our case this
means that the smooth embedding ψ that defines the invariant cylinder A = ψ(S1 ×
[0,1])canbeextendedontoS1×I whereI isanopenintervalcontaining[0,1],andthe
image A˜ =ψ(S1×I)⊃ Aisnormally-hyperbolicandlocally-invariantwithrespectto
themapΦ.Here,bythelocalinvariancewemeanthatthereexistsaneighbourhood Z
˜ ˜ ˜
ofAsuchthattheiterationsofeachpointofAstayinAuntiltheyleaveZ.Animportant
property of the locally-invariant normally-hyperbolic manifold without a boundary is
thatitpersistsatC2-smallperturbations,i.e.forallmapsC2-closetoΦ thereexistsa
locally-invariant normally-hyperbolic cylinder A˜ ⊂ Z. It is not defined uniquely, but
it can be chosen in such a way that it will depend on the map continuously as a C2-
manifold.1Thecontinuousdependenceonthemapimpliesthatthecylinder A˜ remains
symplecticandsymmetricallynormally-hyperbolicforallmapsC2-closetoΦ.
˜
Notethatthenormal hyperbolicity impliesthat A contains alltheorbitsthatnever
˜
leave Z.Inparticular,anyinvariantcurvethatliesin Z mustliein A.Wecallasmooth
invariantessential2 simplecurveγ ⊂ A˜ aKAM-curveifthemapΦ restrictedtoγ is
smoothlyconjugatetotherigidrotationtoaDiophantineangleandthemap F0 =Φ|A˜
near γ satisfies the twist condition. As the Lyapunov exponent at every point of γ is
˜
zero,thegapwiththecontraction/expansioninthedirectionstransversetoAisinfinitely
large.Therefore,thecylinder A˜ isofclassCr (forany given finiter)inasufficiently
smallneighborhoodofγ (see[38,56]).ThisholdstrueforeverymapCr-closetoΦ,i.e.
themapF staysCr-smoothandthetwistconditionalsoholds.Now,byapplyingKAM-
0
theorytothemapF ,weconcludethattheinvariantcurveγ persistsforeverysymplectic
0
mapwhichisatleastC4-closetoΦ.Namely,everysuchmaphasauniquelydefined,con-
tinuouslydependingonthemap,invariantKAM-curvewiththesamerotationnumber.
We further assume that the boundary of A is a pair of KAM-curves. These curves
persistforallC4-smallsymplecticperturbationshencetheyliein A˜ andboundacom-
pactinvariantsub-cylinder A ⊂ A˜.Everyorbitin Astaysin Z,sothesamecylinder A
˜ ˜
isasub-cylinderof Aforeverychoiceofthecylinder A.Thismeansthateventhough
1 Throughoutthispaperweassumethelargespectralgapassumption(6)inthenotionofnormalhyper-
bolicity.ThisguaranteestheC2-smoothnessofthemanifold,andtheC1-smoothnessofthecorresponding
strong-stableandstrong-unstableinvariantfoliationsforeverymapC2-closetoΦ.
2 I.e.non-contractibletoapoint.
ArnoldDiffusion 513
˜
thecylinder A isnotuniquelydefined,thecylinder A isdefineduniquelyforallsym-
plectic maps C4-close to Φ, and it depends continuously on the map. The stable and
unstablemanifoldsandthestrong-stableandstrong-unstablefoliationsofAalsodepend
continuously,intheC1-topology,onthemap.
Wenowassumethatthesymmetricallynormally-hyperboliccylinder Ahasahomo-
clinic,i.e.,theintersectionofWu(A)andWs(A)hasapointx outside A.IfWu(A)and
Ws(A)aretransverseatx,theimplicitfunctiontheoremimpliesthatxhasanopenneigh-
bourhoodU inWu(A)∩Ws(A),whichisdiffeomorphictoatwo-dimensionaldisk.
x
For any x ∈ Wu(A)∩ Ws(A) there is a unique leaf of Euu and a unique leaf of
x
Ess which pass through this point. We call the homoclinic intersection at x strongly
x
transverseif
T Ess ⊕T Euu ⊕T (Wu(A)∩Ws(A))=R2d. (7)
x x x x x
ThispropertyisequivalenttotheconditionthattheleafEuu istransversetoWs(A)and
x
theleaf Ess istransversetoWu(A)atthepointx.
x
The holonomy maps πs : U → A and πu : U → A are projections along the
x x
leavesofthefoliations Ess and Euu,respectively.Sincethefoliationsaresmooth,the
strongtransversalityimpliesthatthefoliationEuuistransversetothediscU inWu(A)
x
and the foliation Ess is transverse to U in Ws(A) provided U is sufficiently small.
x x
Thenπu :U → Aandπs :U → Aarelocaldiffeomorphisms.
x x
Inthiscase,following[32],onecandefinethescatteringmaponπu(U ):
x
F =πs ◦(πu)−1.
x
It is a local diffeomorphism which does not always extends to the whole cylinder A.
However,inthispaperweconsiderthecasewherethescatteringmapcanbeglobally
definedonalargeportionof A.
Let A¯ ⊂ int(A) be a compact invariant sub-cylinder in A, i.e. it is a closed region
in int(A) bounded by two non-intersecting invariant essential simple curves γ+ and
γ−.Letthesetofpointshomoclinicto Acontainasmoothtwo-dimensionalmanifold
B ⊂ Wu(int(A))∩Ws(int(A))\A.Wecall B ahomocliniccylinder,simplerelativeto
¯
thecylinders Aand A,ifthefollowingassumptionshold:
[S1] Thestrongtransversalitycondition(7)holdsforallx ∈ B.
[S2] For every point x ∈ A¯, the corresponding leaf of the foliation Euu intersects
the homoclinic cylinder B at exactly one point each, and no two points in B
belongtothesameleafofthefoliation Ess.Inotherwords,thescatteringmap
F =πs ◦(πu)−1 : A¯ →int(A)iswell-defined.
B B B
¯
[S3] Theimageof Abythescatteringmap F containsanessentialcurve.
B
Under these conditions the scattering map is a diffeomorphism A¯ → F (A¯) ⊂
B
int(A).Indeed,assumption[S1]impliesthattheprojectionsπs,u : B →int(A)arelo-
B
caldiffeomorphismsandassumption[S2]impliesthatthemapsπu :(πu)−1(A¯)→ A¯
B B
and πs : B → πs(B) are bijective. Condition [S3] means that the scattering map is
B B
¯
homotopictoidentityon A.
We conclude that F (A¯) ⊂ A is a sub-cylinder bounded by two essential simple
B
curves F (γ+)and F (γ−).Obviously, F (γ+)∩F (γ−)=∅.Proposition7implies
B B B B
thatF isanexactsymplecticmap.Inparticular,thecylinderF (A¯)hasthesameareaas
B B
A¯,andF (A¯)∩A¯ (cid:16)=∅.Notealsothatthefulfillmentofcondition[S2]dependsonbothin-
B
variantcylinders,A¯andA,asthecylinderAmustbelargeenoughtoincorporateF (A¯).
B
514 V.Gelfreich,D.Turaev
Ifγ+andγ−areKAM-curves,thenthecylinder A¯ boundedbythesecurvespersists
forallC4-smallsymplecticperturbations.Thetransversalitycondition[S1]impliesthat
theC1-smoothhomocliniccylinderBalsopersistsandremainssimplerelativeto A¯and
A. Let V be a set of real-analytic exact symplectic diffeomorphisms Φ : (cid:6) → R2d
N
suchthat:
– each map Φ ∈ V has two invariant, bounded by KAM-curves, symmetrically
N
normally-hyperbolic, two-dimensional closed cylinders A and A¯ such that A¯ ⊂
int(A),
– eachmapΦ ∈V hasNdifferent3homocliniccylindersB ,...,B simplerelative
N 1 N
¯
to Aand A,
– thecylinders A, A¯, B ,...,B dependcontinuously(asC2-smoothmanifolds)on
1 N
themapΦ,
– for each Φ ∈ V the map F = Φ| has a twist property in some symplectic
N 0 A
coordinates(y,ϕ).4
Wedefinethetopologyinthespaceofreal-analyticexactsymplecticdiffeomorphismsas
follows.TakeanycompactK ⊂R2dandletananalyticitydomainQbeacompactcom-
plex neighbourhood of K. We consider exact symplectomorphisms K → R2d which
admitaholomorphicextensionontosomeopenneighbourhoodof Q.Twosuchmaps
areconsideredtobecloseiftheyareuniformlycloseon Q.Foranygivenr,twoholo-
morphicmapswhicharesufficientlycloseon Q areCr-closeon K.Asweexplained,
theC4-closenessisenoughforthepersistenceofthecylinders A, A¯, B ,...,B [ifall
1 N
oftheirorbitsbyΦ lieinint(K)],sothesetV isopen.
N
Theorem1.(Maintheorem)Let N ≥ 8.Then,thereisanopenanddensesubsetV˜ of
V ,suchthatforeachmapΦ ∈ V˜ foreverytwoopenneighbourhoodsU− ofγ− and
N
U+ofγ+theimageofU−bysomeforwarditerationofthemapΦ intersectsU+.
Remark1.ItisobviousthatgivenanytwoopensetsU+andU−thesetofmapswhose
orbitsconnectU− andU+ isopen.Thetheoremmakesastrongerclaimthattheinter-
sectionofallthesesets(overallpossiblechoicesoftheneighbourhoodsU−andU+of
thegivencurvesγ−andγ+)isopenanddenseinV .Thetheoremimpliesthatforany
N
mapΦ ∈V thereisanopensetofarbitrarilysmallperturbationsofΦwithinV such
N N
thateachoftheseperturbationscreates,foreachpairofneighbourhoodsU−andU+of
thecurvesγ±,anorbitthatconnectsU−andU+.
Notethattheexistenceofatleast8differenthomocliniccylindersrequiredbyThe-
orem1isnotarestrictivecondition.Namely,underanadditionalmildassumptionthe
existenceofonehomocliniccylinderimpliestheexistenceofinfinitelymanydifferent
homocliniccylinders(seeSect.3.3).Usingthis,wecaninferthefollowingresultfrom
ourmaintheorem.
ConsiderthesetV ofreal-analyticexactsymplecticdiffeomorphismsΦ :(cid:6) →R2d
suchthat:
– eachmapΦ ∈Vhasaninvariant,boundedbyKAM-curves,symmetricallynormally-
hyperbolic,two-dimensionalclosedcylinder A,
– inAthereexisttwoinvariantsub-cylindersA¯andAˆsuchthatA¯ ⊂int(Aˆ)⊂int(A),
eachofthemisboundedbyKAM-curves,
3 I.e.noneintersectsanyimageofanotherbytheiterationsofthemapΦ.
4 Inthesecoordinates,Birkhofftheorem[55]impliesthattheboundarycurvesγ±oftheinvariantsub-
cylinderA¯aregraphsofLipschitzfunctions,y=y±(ϕ).
ArnoldDiffusion 515
– Φ hasahomocliniccylinder B simplerelativeto Aˆ and A,
– thecylinder B issimplerelative A¯ and Aˆ;i.e. F (A¯)⊂intAˆ,
B
– themap F =Φ| hasatwistproperty.
0 A
As all the invariant cylinders involved are bounded by KAM-curves, they persist at
C4-small symplectic perturbations. Thus the set V is an open subset of the space of
real-analyticsymplectomorphisms.Letγ−andγ+betheboundarycurvesof A¯.
Theorem2.InV thereisanopenanddensesubsetV˜ suchthatforeachmapΦ˜ ∈ V˜
andforeverytwoopenneighbourhoodsU− ofγ− andU+ ofγ+ theimageofU− by
someforwarditerationofΦ˜ intersectsU+.
Remark2.StatementssimilartoTheorems1and2areknownfornon-analytic(smooth)
case, see e.g. [19,20,77]. The main difference between the analytic and smooth case
is that the class of perturbations small in the real-analytic sense is narrower than the
∞
class of perturbations that are small in the C -sense. In particular, for a typical real-
analytic map the normally-hyperbolic invariant cylinder A is not analytic (it has only
finite smoothness), so no real-analytic perturbations can vanish on A. Consequently,
methodsof[19,20,77]arenotapplicableintheanalyticcategory(inthecrucialpartthat
concernsremovingthebarrierstodiffusionbyasmallperturbation).Ontheotherhand,
theproofsofthepresentpaperholdtrueforthecaseofCk mapsaswell.
Remark3.ThesymplecticdiffeomorphismΦcanbeaPoincaremapofacertaincross-
section(cid:6) foraHamiltonianflowinsidealevelofconstantenergy.Wedonotneedto
assume that the Poincare map Φ is defined outside a small neighbourhood of the in-
variant cylinder A in this case: the global stable and unstable manifolds of A, as well
astheglobalstrong-stableandstrong-unstablefoliationsonthesemanifoldsaredefined
by continuation of the corresponding local objects by the orbits of the Hamiltonian
system.Asabove,onedefinesscatteringmapsbytheorbitshomoclinicto A.Onecan
easilyadjusttheproof ofthetwomaintheorems inordertoshowthatifthePoincare
mapΦandthescatteringmap(maps)forsomeHamiltoniansystemsatisfytheassump-
tionsoftheorem1or2,thenagenericsmallperturbationoftheHamiltonianfunction
H in the space of real-analytic Hamiltonians leads to creation of orbits that connect
U−toU+.
Thestrategyoftheproofofourtwomaintheoremsisasfollows.WeshowinPropo-
sition2thattheexistenceofonehomocliniccylinder B whichissimplerelativetothe
invariantcylinders Aˆ and A where Aˆ issuchthat A¯ ⊆ Aˆ and F (A¯) ⊆ Aˆ impliesthe
B
existenceofinfinitelymanydifferentsecondaryhomocliniccylinderswhicharesimple
¯
relative to A and A. Thus, Theorem 2 is immediately reduced to Theorem 1, and we
willfurtherconsider N ≥ 8homocliniccylinders B ,...,B ,allofwhicharesimple
1 N
¯
relativetothesamepairofcompactinvariantcylinders Aand A,andallaredifferentin
thesensethatΦm(B )∩ B =∅forallm andalli, j =1,...,N suchthati (cid:16)= j.Let
i j
F : A¯ →int(A)denotethescatteringmapdefinedbythehomocliniccylinder B .By
n n
condition[S1],F isalocaldiffeomorphism.Bycondition[S2]F isabijection,hence
n n
F isadiffeomorphismof A¯ ontotheset F (A¯).Obviously,condition[S1]impliesthat
n n
(cid:3) ¯
thescatteringmapsaredefinedinanopenneighbourhood A of Ain A.
TakeanymapΦ ∈V.Let(v )m ⊂ Abeapartofanorbitoftheiteratedfunction
s s=0
system{F ,...,F },i.e.foreachs = 0,...,m −1thereexistsn = 0,...,N such
0 N s
that v = F (v ). In order to ensure that F (v ) is well-defined we assume that
v ∈ As+(cid:3)1forn n(cid:16)=s 0s.InSect.4weshowthatfornasnyssuchorbitandanyε > 0,thereis
s s
apointx andapositiveinteger(cid:14)suchthat
0
516 V.Gelfreich,D.Turaev
dist(x ,v )<ε, and dist(Φ˜(cid:14)(x ),v )<ε
0 0 0 m
(seeLemma4).Notethatwedonotusehyperbolicityorindexargumentsinthislemma.
We also do not use the symplecticity of the maps F ,...,F , nor the twist property
1 N
of the map F . However, the fact that the large cylinder A is an invariant domain for
0
thearea-preservingmap F iscrucial,asweusethePoincareRecurrenceTheoremin
0
anessentialway(wefirstproveacertainweakshadowingresult,Lemma2,thatholds
withoutthisassumptiononthemap F ,thenLemma4isdeducedfromitinthecaseof
0
area-preserving F ).
0
Accordingtothisshadowinglemma(Lemma4),inordertoshowthattwoopensets
are connected by a forward orbit of the map Φ, it is sufficient to show that the inter-
sections of these sets with A are connected by orbits of the iterated function system
{F ,...,F }.Ageneralisation(Theorem3)ofaclassicalBirkhofftheoremstatesthat
0 N
if F for n = 0,...,N are exact symplectomorphisms homotopic to identity, and F
n 0
is a twist map, then for any two essential curves γ± ⊂ A(cid:3) there is a trajectory of the
iterated function system with v ∈ γ− and v ∈ γ+ unless the functions F have a
0 m n
commoninvariantessentialcurve.
Thus,ifthemaps F ,...,F have nocommon invariantessentialcurves between
0 N
γ− and γ+, every pair of neighbourhoods,U− of γ− andU+ of γ+, is connected by
orbits of the map Φ. Theorem 3 also implies that the absence of a common invariant
essentialcurveisanopenproperty.
Theorem4,themostdifficultpartoftheargument,establishesthatthispropertyis
also dense in V (provided N ≥ 8). Thus, for every map Φ from an open and dense
subsetofV,thecorrespondingscatteringmapsF ,...,F (N ≥8)andF donothave
1 N 0
anycommonessentialinvariantcurve.Aswejustexplained,thisimpliesthateverytwo
neighbourhoodsU± ofγ± areconnected byforwardorbitsofeach suchmap Φ,and
Theorem1follows.
Theorem4isthecrucialstepintheproofofTheorem1.AnanalogueofTheorem4
forgenericnon-analyticmapscanbederivedfrom[19,20,77].However,themethodsof
destroyingcommoninvariantcurvesthatareusedinthosepaperscannotbeusedinthe
analyticcase(asthereal-analyticperturbationscannot,ingeneral,vanishonthefinitely
∞
smoothnormally-hyperboliccylinder A;thesameconcernsC perturbations,forthat
matter).Therefore,wedevelopacompletelydifferentperturbationtechniqueinorderto
proveTheorem4fortheanalyticcase.
3. EstimatesinaNeighbourhoodofaSymmetricallyNormally-Hyperbolic
InvariantCylinder
Inthissectionwestudydynamicsinasmallneighbourhoodofanormally-hyperbolic
cylinder.Thisanalysisdoesnotrequirethemaptobeeithersymplecticoranalytic.
3.1. Fenichelcoordinates,crossformofthemap,andestimatesforthelocaldynamics.
Let Abeacompact,symmetricallynormally-hyperbolic,smooth,invariantcylinderof
aCr-smoothmapΦ (r ≥2).Aswealreadymentioned, Acanbeextendedtoalarger,
˜
smoothnormally-hyperboliclocally-invariantcylinder A.Letusintroducecoordinates
in a small neighbourhood of A such that this larger invariant cylinder is straightened.
Moreover, the local stable and unstable manifolds Ws,u(A) are straightened as well,
loc
alongwiththestrong-stableandstrong-unstablefoliations Ess and Euu onthem.Note