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Seminaire BOURBAKI Juin 59eme annee no

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Seminaire BOURBAKI Juin 2007 59eme annee, 2006-2007, no 978 SEMI-CLASSICAL MEASURES AND ENTROPY [after Nalini Anantharaman and Stephane Nonnenmacher] by Yves COLIN de VERDIERE INTRODUCTION This report is about recent progress on semi-classical localization of eigenfunctions for quantum systems whose classical limit is hyperbolic (Anosov systems); the main example is the Laplace operator on a compact Riemannian manifold with strictly nega- tive curvature whose classical limit is the geodesic flow; the quantizations of hyperbolic cat maps, called “quantum cat maps”, are other nice examples. All this is part of the field called “quantum chaos”. The new results are: – Examples of eigenfunctions for the cat maps with a strong localization (“scarring”) effect due to S. de Bievre, F. Faure and S. Nonnenmacher [16, 17] – Uniform distribution of Hecke eigenfunctions in the case of arithmetic Riemann surfaces by E. Lindenstrauss [26] – General lower bounds on the entropy of semi-classical measures due to N. Anan- tharaman [1] and improved by N. Anantharaman–S. Nonnenmacher [2] and N. Anantharaman–H. Koch–S. Nonnenmacher [3]. This lower bound is sharp with respect to the cat maps examples. We will mainly focus on this last result. 1. THE 2 BASIC EXAMPLES 1.1.

  • riemannian manifold

  • measure dl

  • maps examples

  • sinaı start

  • semi-classical measures

  • points z ?

  • lim t?∞

  • compact connected


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S´eminaireBOURBAKI59`emeann´ee,2006-2007,no978SEMI-CLASSICALMEASURESANDENTROPY[afterNaliniAnantharamanandSte´phaneNonnenmacher]byYvesCOLINdeVERDI`EREINTRODUCTIONJuin2007Thisreportisaboutrecentprogressonsemi-classicallocalizationofeigenfunctionsforquantumsystemswhoseclassicallimitishyperbolic(Anosovsystems);themainexampleistheLaplaceoperatoronacompactRiemannianmanifoldwithstrictlynega-tivecurvaturewhoseclassicallimitisthegeodesicflow;thequantizationsofhyperboliccatmaps,called“quantumcatmaps”,areotherniceexamples.Allthisispartofthefieldcalled“quantumchaos”.Thenewresultsare:Examplesofeigenfunctionsforthecatmapswithastronglocalization(“scarring”)effectduetoS.deBi`evre,F.FaureandS.Nonnenmacher[16,17]UniformdistributionofHeckeeigenfunctionsinthecaseofarithmeticRiemannsurfacesbyE.Lindenstrauss[26]Generallowerboundsontheentropyofsemi-classicalmeasuresduetoN.Anan-tharaman[1]andimprovedbyN.Anantharaman–S.Nonnenmacher[2]andN.Anantharaman–H.Koch–S.Nonnenmacher[3].Thislowerboundissharpwithrespecttothecatmapsexamples.Wewillmainlyfocusonthislastresult.1.THE2BASICEXAMPLES1.1.CatmapsWestartwithamatrixASL2(Z)whichisassumedtobehyperbolic:theeigen-valuesλ±ofAsatisfy0<|λ|<1<|λ+|.TheactionofAontoR2definesasymplecticactionUofAonthetorusR2/Z2byconsideringactiononpointsmodZ2.Suchamapisasimpleexampleofachaoticmap.Ithasbeenobservedsincealongtimethatsuchamapcanbequantized:foreachintegerN,weconsidertheHilbertspaceHNofdimensionNofSchwartzdistributionsfwhichareperiodicofperiodoneandofwhichPFouriercoefficientsareperiodicofperiodN:iff(x)=kZake2πikx,wehave,forallkZ,ak+N=ak.UsingthemetaplecticrepresentationappliedtoA,wegetanaturalunitaryactionUˆNontothespaceHN.Wearemainlyinterestedintheeigenfunctions
978–02ofUˆN.Thesemi-classicalparameteris~=1/NandtheclassicallimitcorrespondstolargevaluesofN.Agoodreferenceis[8].1.2.TheLaplaceoperatorsOnasmoothcompactconnectedRiemannianmanifold(X,g)withoutboundary,weconsidertheLaplaceoperatorΔgiveninlocalcoordinatesbyΔ=−|g|1igij|g|jwith|g|=det(gij).TheLaplaceoperatorΔisessentiallyself-adjointonL2(X)withdomainthesmoothfunctionsandhasacompactresolvent.Thespectrumisdiscreteanddenotedby0=λ12≤∙∙∙≤λk≤∙∙∙withanorthonormalbasisofeigenfunctionsϕksatisfyingΔϕk=λkϕk.Itisusefult1ointroduceaneffectivePlanckconstant(thesemi-classicalsmallparameter)~:=λk2.Wewillrewritetheeigenfunctionequation~2Δϕ=ϕ.Thesemi-classicallimit~0correspondstothehighfrequencylimitfortheperiodicsolutionsu(x,t)=exp(iλkt)ϕkofthewaveequationuttu=0.Insteadofthewaveevolution,wewillusetheSchr¨odingerevolutionwhichisgivenby2~ut=~Δu,i2andintroducetheunitarydynamicsdefinedbythe1-parametergroupUˆt=exp(it~Δ/2),tR.Forthebasicdefinitions,onecanread[5].1.3.ThegeodesicflowIf(X,g)isaRiemannianmanifoldandvTxXatangentvectoratthepointxX,wedefine,fortR,Gt(x,v)=(y,w)asfollows:ifγ(t)isthegeodesicwhichsatisfiesγ(0)=x,˙γ(0)=v,weputy:=γ(t)andw:=γ˙(t).Byusingtheidentificationofthetangentbundlewiththecotangentbundleinducedbythemetricg(whichisalsotheLegendretransformoftheLagrangian12gij(x)vivj),wegetaflow(Gt)?onT?XwhichpreservestheunitcotangentbundledenotedbyZ.WedenotebyUttherestrictionof(Gt)?toZ.TheLiouvillemeasuredLonZistheRiemannianmeasurenormalizedasaprobabilitymeasure.TheLiouvillemeasuredLisinvariantbythegeodesicflow.2.CLASSICALCHAOSGoodtextbooksontheclassicalchaosare[21,30,10].