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Proceedings of the International Congress of Mathematicians Hyderabad India

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Proceedings of the International Congress of Mathematicians Hyderabad, India, 2010 Green Bundles and Related Topics Marie-Claude Arnaud? Abstract For twist maps of the annulus and Tonelli Hamiltonians, two linear bundles, the Green bundles, are defined along the minimizing orbits. The link between these Green bundles and different notions as: weak and strong hyperbolicity, estimate of the non-zero Lyapunov exponents, tangent cones to minimizing subsets, is explained. Various results are deduced from these links: the relationship between the hyperbolicity of the Aubry-Mather sets of the twist maps and the C1-regularity of their support, the almost everywhere C1-regularity of the essential invari- ant curves of the twist maps, the link between the Lyapunov exponents and the angles of the Oseledec bundles of minimizing measures, the fact that C0- integrability implies C1-integrability on a dense G?-subset. Mathematics Subject Classification (2000). Primary 37E40, 37J50, 37C40; 70H20; Secondary 70H03 70H05 37D05 37D25 Keywords. Twist maps, Tonelli Hamiltonians, minimizing measures, Aubry-Mather sets, Lyapunov exponents, hyperbolic sets, non uniform hyperbolic measures, C1- regularity, weak KAM theory, Hamilton-Jacobi 1. Introduction In the study of twist maps or optical Hamiltonians, mathematicians have stud- ied the orbits that can be found via minimization for a long time: an action is associated with such a dynamical system, and an orbit piece corresponds to a critical point of the action.

  • action functionals

  • bi-infinite sequence

  • along locally

  • tonelli hamiltonians

  • avignon

  • dynnonhyp universite d'avignon et des pays de vaucluse

  • minimizing orbits

  • orientation preserving

  • results


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ProceedingsoftheInternationalCongressofMathematiciansHyderabad,India,2010GreenBundlesandRelatedTopicsMarie-ClaudeArnaudAbstractFortwistmapsoftheannulusandTonelliHamiltonians,twolinearbundles,theGreenbundles,aredefinedalongtheminimizingorbits.ThelinkbetweentheseGreenbundlesanddifferentnotionsas:weakandstronghyperbolicity,estimateofthenon-zeroLyapunovexponents,tangentconestominimizingsubsets,isexplained.Variousresultsarededucedfromtheselinks:therelationshipbetweenthehyperbolicityoftheAubry-MathersetsofthetwistmapsandtheC1-regularityoftheirsupport,thealmosteverywhereC1-regularityoftheessentialinvari-antcurvesofthetwistmaps,thelinkbetweentheLyapunovexponentsandtheanglesoftheOseledecbundlesofminimizingmeasures,thefactthatC0-integrabilityimpliesC1-integrabilityonadenseGδ-subset.MathematicsSubjectClassification(2000).Primary37E40,37J50,37C40;70H20;Secondary70H0370H0537D0537D25Keywords.Twistmaps,TonelliHamiltonians,minimizingmeasures,Aubry-Mathersets,Lyapunovexponents,hyperbolicsets,nonuniformhyperbolicmeasures,C1-regularity,weakKAMtheory,Hamilton-Jacobi1.IntroductionInthestudyoftwistmapsoropticalHamiltonians,mathematicianshavestud-iedtheorbitsthatcanbefoundviaminimizationforalongtime:anactionisassociatedwithsuchadynamicalsystem,andanorbitpiececorrespondstoacriticalpointoftheaction.Forexample,awaytofindperiodicorbitsistominimizetheactionamongtheperiodicarcs(forHamiltonians)orsequences(fortwistmaps).ANR-07-BLAN-0361,ANRDynNonHypUniversite´d’AvignonetdesPaysdeVaucluse,EA2151,Analysenonline´aireetGe´ome´trie,F-84018Avignon,France.E-mail:Marie-Claude.Arnaud@univ-avignon.fr.
2M.-C.ArnaudMorerecently,theexistenceofsomegloballyminimizingorbitshasbeenproved,i.e.theexistenceoforbitsthatminimizetheactionalongallthein-tervalsoftime.Inthecaseoftwistmaps,theseorbitsarecontainedinsomeminimizingsets(i.e.setsfilledwithminimizingorbits)calledAubry-Mathersets,whichwereindependentlydiscoveredinthe80’sbyS.Aubry&P.LeDaeronandJ.Mather.Inthecaseoftheso-calledTonelliHamiltonians,theirexistencewasprovedbyJ.Matherinthe90’swhenheprovedtheexistenceofminimizingmeasures.InthecaseofaTonelliHamiltonianofacotangentbundleTM,someminimizingsetssimilartotheAubry-Mathersets,calledAubrysets,alsoexist.TohaveanideaofwhattheseAubry-Mathersetsmaybe,letusconsiderthecaseofacompletelyintegrabletwistmapofA=T×R:f:(q,p)(q+(p),p).Thentheannulusisfoliatedbyinvariantcircles{p=C},whicharetheAubry-Mathersets.Ifweslightlyperturbf,alotoftheseinvariantcurveswillpersist(thisisaconsequenceoftheK.A.M.theorems),butsomeotherswillbecomesmallerinvariantsets,Cantorsetsorperiodicorbits;thesethreekindsofsetsareAubry-Mathersets;inacertainway,theyaretheghostsoftheinitialinvariantcircles.Inthecaseofagenerictwistmapoftheannulus,aresultduetoPatriceLeCalvezstatesthatthemajorityofAubry-Mathersetsarehyperbolic(see[30]).NosuchresultisknownfortheTonelliHamiltonians.Inthecaseoftwistmaps,too,weknowthatsomenon-hyperbolicAubry-Mathersets,theK.A.M.curves,maypersistafterperturbation.Wecanthenaskourselves:Question1.isthereameansofdistinguishingbetweenthehyperbolicandthenonhyperbolicAubryorAubry-Mathersets?IsthereameansofseeingtheLyapunovexponentsofaminimizingmeasurewhenknowingonlythemeasureandnotthedynamic?Forthetwistmaps,therearethreekindsofAubry-Mathersets:–theinvariantcurves,whichareneveruniformlyhyperbolic;–theperiodicorbits,whichmaybehyperbolicornonhyperbolic;thereis,ofcourse,nowaytodistinguishbetweenahyperbolicandanonhyperbolicfiniteorbitifweonlyknowtheorbit;–theCantorsets,whichmaybehyperbolicornonhyperbolic;wewillgiveawaytodistinguishbetweenhyperbolicandnonhyperbolicCantorsets.Hence,inthecaseoftwistmaps,weobtainacriteriontodecideifanAubry-Mathersetishyperbolicornot,withoutknowingthedynamic.Tobealittlemoreprecise,wedefineanotionofC1-regularityforthesubsetsofamanifold,andweprovethathyperbolicityisequivalenttoC1-irregularity.InthecaseofTonelliHamiltonians,wewillseethatthisresultisnolongertrue,butapartialresultsubsists.