On Hyperbolic Plateaus of the Hénon map

OnHyperbolicPlateausoftheH´enonMapZinARAIDepartmentofMathematics,KyotoUniversity,Kyoto606-8502,Japan(email:[e-mail])AbstractWeproposearigorouscomputationalmethodtoprovetheuniformhyperbolicityofdiscretedynamicalsystems.ApplyingthemethodtotherealH´enonfamily,weprovetheexistenceofmanyregionsofhyperbolicparametersintheparameterplaneofthefamily.1IntroductionConsidertheproblemofdeterminingthesetofparametervaluesforwhichtherealH´enonmapHa,b:R2→R2:(x,y)→(a−x2+by,x)(a,b∈R)isuniformlyhyperbolic.Ifadynamicalsystemisuniformlyhyperbolic,gen-erallyspeaking,wecanapplytheso-calledhyperbolictheoryofdynamicalsystemsandobtainmanyresultsonthebehaviorofthesystem.Despiteitsimportance,however,provinghyperbolicityisadifficultproblemevenforsuchsimplepolynomialmapsastheH´enonmaps.ThefirstmathematicalresultaboutthehyperbolicityoftheH´enonmapwasobtainedby[DevaneyandNitecki79].Theyshowedthatforanyfixedb,ifaissufficientlylargethenthenon-wanderingsetofHa,bisuniformlyhyperbolicandconjugatetothefullhorseshoemap,thatis,theshiftmapofthespaceofbi-infinitesequencesoftwosymbols.Later,Davis,MacKayandSannami[Davisetal.91]conjecturedthatbesidestheuniformlyhyperbolicfullhorseshoeregion,thereexistsomepa-rameterregionsinwhichthenon-wanderingsetoftheH´enonmapisuni-formlyhyperbolicandconjugatetoasubshiftoffinitetype.Forsomepa-rameterintervalsoftheareapreservingH´enonfamilyHa,−1,theyidentifiedtheMarkovpartitionbydescribingtheconfigurationofstableandunstablemanifolds(seealso[Sterlingetal.99,HagiwaraandShudo04]).Although1themechanismofhyperbolicityattheseparametervaluesisclearbytheirobservations,nomathematicalproofoftheuniformhyperbolicityhasbeenobtainedsofar.Thepurposeofthispaperistoproposeageneralmethodforprovinguniformhyperbolicityofdiscretedynamicalsystems.ApplyingthemethodtotheH´enonmap,weobtainacomputerassistedproofofthehyperbolicityofH´enonmaponmanyparameterregionsincludingtheintervalsconjecturedbyDavisetal.OurresultsontherealH´enonmaparesummarizedinthefollowingtheorems.WedenotebyR(Ha,b)thechainrecurrentsetofHa,b.Theorem1.1.ThereexistsasetP⊂R2,whichistheunionof8943closedrectangles,suchthatif(a,b)∈PthenR(Ha,b)isuniformlyhyperbolic.ThesetPisillustratedinFigure1(shadedregions),andthecompletelistoftherectanglesinPisgivenasasupplementalmaterialtothepaper.ThehyperbolicityofthechainrecurrentsetimpliestheR-stability.Therefore,oneachconnectedcomponentofP,nobifurcationoccursinR(Ha,b)andhencenumericalinvariantssuchasthetopologicalentropy,thenumberofperiodicpoints,etc.,areconstantonit.Forthisreason,wecallita“plateau”.NotethatTheorem1.1doesnotclaimthataparametervaluenotinPisanon-hyperbolicparameter.ItonlyguaranteesthatPisasubsetoftheuniformlyhyperbolicparametervalues.WecanrefineTheorem1.1byperformingmorecomputations,whichyieldsasetPofuniformlyhyperbolicparameterssuchthatP⊂P.Sincethearea-preservingH´enonfamilyisofparticularimportance,weperformedanothercomputationrestrictedtothisone-parameterfamilyandobtainedthefollowing.Theorem1.2.Ifaisinoneofthefollowingclosedintervals,[4.5383300781250,4.5385742187500],[4.5388183593750,4.5429687500000],[4.5623779296875,4.5931396484375],[4.6188964843750,4.6457519531250],[4.6694335937500,4.6881103515625],[4.7681884765625,4.7993164062500],[4.8530273437500,4.8603515625000],[4.9665527343750,4.9692382812500],[5.1469726562500,5.1496582031250],[5.1904296875000,5.5366210937500],[5.5659179687500,5.6077880859375],[5.6342773437500,5.6768798828125],[5.6821289062500,5.6857910156250],[5.6859130859375,5.6860351562500],[5.6916503906250,5.6951904296875],[5.6999511718750,∞),thenR(Ha,−1)isuniformlyhyperbolic.2−1−0.875−0.75−0.625−0.5−0.375−0.25−0.12500.1250.250.3750.50.6250.750.8751−1−0.75−0.5−0.2500.250.50.7511.251.51.7522.252.52.7533.253.53.7544.254.54.7555.255.55.7566.25baFigure1:uniformlyhyperbolicplateaus3Weremarkthatthethreeintervalsconsideredtobehyperbolicparam-etervaluesbyDavisetal.appearinTheorem1.2.ThuswecansaythatTheorem1.2justifiestheirobservations.ItisinterestingtocompareFigure1withthebifurcationdiagramsoftheH´enonmapnumericallyobtainedby[HamoulyandMira81],andby[Sannami89,Sannami94].TheboundaryofPshowninFigure1areveryclosetothebifurcationcurvesgiveninthesepapers.RecentlyCao,LuzzattoandRios[Caoetal.05]showedthattheH´enonmaphasatangencyandhenceisnon-hyperboliciftheparameterisontheboundaryofthefullhorseshoeplateau(seealso[BedfordandSmillie04a,BedfordandSmillie04b]).ThisfactandTheorem1.2suggeststhatHa,−1shouldhaveatangencywhenaiscloseto5.699951171875.Infact,wecanprovethefollowingtheoremusingtherigorouscomputationalmethoddevelopedin[AraiandMischaikow05].Proposition1.3.Thereexistsa∈[5.6993102,5.6993113]suchthatHa,−1hasahomoclinictangencywithrespecttothesaddlefixedpointonthethirdquadrant.Consequently,Theorem1.2andProposition1.3yieldsthefollowing.Corollary1.4.Whenwedecreasea∈Rofthearea-preservingH´enonfamilyHa,−1,thefirsttangencyoccursin[5.6993102,5.699951171875).WeremarkthatHruska[Hruska05,Hruska06]alsoconstructedarig-orousnumericalmethodforprovinghyperbolicityofcomplexH´enonmaps.ThemaindifferencebetweenourmethodandHruska’smethodisthatourm