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ENTROPY OF SEMICLASSICAL MEASURES

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ENTROPY OF SEMICLASSICAL MEASURES FOR NONPOSITIVELY CURVED SURFACES GABRIEL RIVIÈRE Abstract. We study the asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of nonpositive sectional curvature. To do this, we look at se- quences of distributions associated to them and we study the entropic properties of their accumu- lation points, the so-called semiclassical measures. Precisely, we show that the Kolmogorov-Sinai entropy of a semiclassical measure µ for the geodesic flow gt is bounded from below by half of the Ruelle upper bound, i.e. hKS(µ, g) ≥ 1 2 ∫ S?M ?+(?)dµ(?), where ?+(?) is the upper Lyapunov exponent at point ?. The main strategy is the same as in [17] except that we have to deal with weakly chaotic behavior. 1. Introduction Let M be a compact, connected, C∞ riemannian manifold. For all x ? M , T ?xM is endowed with a norm ?.?x given by the metric over M . The geodesic flow gt over T ?M is defined as the Hamiltonian flow corresponding to H(x, ?) := ??? 2 x 2 . This quantity corresponds to the classical kinetic energy in the case of the absence of potential.

  • also ask

  • quantum unique

  • geodesic flow

  • points need

  • flow gt over

  • semiclassical measure

  • measure cannot

  • liouville measure


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Informations

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Nombre de lectures 25

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: S M ! M
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1 S Mx
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H K (Z) =r 0 b(0) =r b(0) a (0) d (Z)
r X ()H
2
T S M =H V
S M 18
= (x;)2S M X;Y 2T S M
hX;Yi :=g (d (X);d (Y )) +g (K (X);K (Y )); x x
g x Mx
M S M

0 0J"(t) +R( (t);J(t)) (t) = 0;
0R(X;Y )Z X Y Z J (t) =
r 0 J(t) (t)
1C c : [a;b]!M s ( ; )s
@c =c Y (t) = (c (t))0 s js=0@s
2s7!c (t) c M C Y (t)s
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(V;W ) T S M H V
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J"(t) +K(t)J(t) = 0;
K(t) = K( (t)) (t)
0J (0) = e(0) t R (t) p0 kJ (t)k KkJ (t)k t R K0 0
s uE () E ()
s t1 M N E (g )
0s s(J (t);J (t)) H t V t g g
V

N
u sE ()E ()


N
uE ()V
0 2U (t) +U (t) +K(t) = 0;
0 1U(t) =J (t)J(t) J
0u u u 1U (t) :=J (t)(J (t))
2
Rt uu u U (s)ds0 kJ (t)k =kJ (0)ke :
t kd g k ujE ()
Rp t uU (s)ds01 +K e 53 540
t (g )t

Z T1+ u s ; () = lim U (g )ds:
T!+1T 0
c
T =T (c) S M J(t)
0J(0) = 0 kJ (0)k 1 t T kJ(t)kc 3:1
2


curvnwithoutoAsRiccatiogivvandcaseit[11].edRinemark.aWheergenceunderlineethatcanwthee[12].capunoouldalldevobielopassertstheandsameallconstructionconjugateforwingmanifoltodsthewithoutuniformlycon-[20]jugatewthpJacobian,ointotsthatandconjugatethetsameandproptoertiesevweouldEbbofefortrueaexceptsubspaces.thehacon[20]).tinofuitwillyofofFinallythelemma:stable/unstablepreviousfoliationthat[5]..InbtheTheycase).whereethedesicGreeAsnMa??subspacesRiccatiattacvhedtheytoofcurvgenerallyarethelinearlytindepeldsendentt,baaresplittinger,ofhasectionalethepisegivGreenengeneralbyypisinwhereinequation:nonJacobialongdimensionalhaonensimpliedcalledthenegativsatisfyositionTheyof).thisandothedimensionsplittingusedholdsconjugateforsameallytheproptrathejetinctoryuse.toFoundorhapptheandopperseositesubspacescase,inewWenevknovwthethatontheinnitesimalGreenthesubspaceswattacdirechedthetoreiredo(andthehencearetLyoonenafacgeovdesictheelositivJacobi(and)surfacesareoinlineaerrexplpyofdepordinatesendenvtsifgivandforonly,iferptheMoreosectionalbcurvwatureeisDivervvanishingaAntiouldsrecallhiduengandatinev[10].eryforpositivoiexistsneteryofhtheygeoanddesicanunstableJacobithe(where[20].dimensionAseaandconsequence,andw,elargercannotwuseethelosambeonkindtheofanifoldssplitting.oinHoertwnievmer,holdsthereandexistseathesplittingmanifoldsofoinanddimension,stableholdsthatywteallcanallousevinobTheothycases,canprecithesresultseobtainlybthetheyelyenectivcan(respthatandatdenotedtoaretransvTheyare.oth2.1.2.thatRunderlicec.atiareeerquation.(p.TheanishingoneSodimensionalunstableJacobisolutiequationdescribdenedtheearliergrogivofegeosorisealongtounstablethetion.Ric-forcatiunstableequation:Fexist.and5shMEASURESwLASSICALthatMICunstableSEsolutionsOFrelatedOPYtheENTRapunoarianexpvtsinIntot,subspacesprotheseedliftincancaseenonpwewhereatureelds,moreJacobiforandwithoutwpots),desicuppgeoLyeenvwonenetatboinlinkJacobiousaiwithvcofor-innonarianvmeaanishingurereis.enThen,ywa.e.eindenetheythepcorrespendicularonding.unstablevRiccatiysolutionenassoisciatedetovtheforunstable2.1.3.JacobigenceldofasJacAelds.plasttheoininwaswtegrablelikintoareistheresultUsingtoand[13]aturetoinerleinuouslythetincaseature).ItonthatcanendpdepecurvthereproaWositivunderlinebasisit.notinifsucethatsubspacesanisincludedcrucialareertsurface)thforwywilltrivialseeldouristhe(asunstableisJacobisuceldthat(invdimensiosubspacesnThesetheseunstable)stableasGreefollforothews:thanature,,curveevositivsomenonpeofwcaseere(propthoundInthesetting).InvcaseAnosomthewithoutinpcasets,thepropnotyasuwfhr(whicdivotheronlyhineacsubspacestoitequalbThecruciallysameinwfolloorks(forforwithoutthepstabletsJacobihighereld.theBothresultquanbuttitiesanareuniformitconintin).uous,3thesewithertiesrespwectprotoearefoll.wingW3econunderlineuitthat,inwsuppatheproptoyeatconjugaeputsinwthofof..eItthatisisatruenonnegativweonlyquanosetitsurfaceybthatwithoutcontetrolsointhe[5].grov = (0;V ) c > 0
tT =T (c)> 0 v tT kd g vkc
t g S M
t u u td g N =E ()V N t =E (g )V t g g
t u u ukd g v k v E () t 0
u t t > 0 T = T () E (g ) Rd g v
tT S M v V
tkd gk t 0
uT S M RX ()E () H
V
tkd g k E E = RX () 1 HE!T Mtg Rp t uu U (s)ds0E =E () 1 +K e0
> 0 e V 0p t0
p d g e 0e k := [t= ]p p kd g ek 0
k k
kd g ek k =jhd g e ;e i k j 0 0 kg g

=jhd (k 1) g e ;e i k hd g e ;ei j:k 0 g g (k 1) g
p ud g eu u0e := e :=p up 0kd g e k 0
0u u(J (0);J (0)) ue e0 p u u p k(J (0);J (0))k
u ulogjhd (p 1) g e ;e i p j logjhd (p 1) g e ;e i p jg (p 1) p g g p g (p 1)
0 > 0 C > 0
0k k u u u u
kd g ek k Ce jhd (k 1) g e ;e i k hd g e ;e i j: 0 g g g (k 1) k g 0
0k k uCe kd g e k k 0 g Rk0 uk U (s)ds0 Ce e C
uRX () V E ()H Rt0 u0 t t U (s)ds0 > 0 C > 0 2S M kd gkCe e :


2E := (x;)2T M : 1 2kk 1 + 2 :x
b0
u82E ; 0U ()b :0
ulogJ
uU
1:2 0
u uU () := supfU ();g:00

b 02 (2 + )b 0 20
d > 00
0 0 u u 08 ( ; )2E E ; d( ; )d )jU () U ( )j :0 0

u u s82E ; 80s; jU () U (g )j :0
larger6ha.haInauxiliaryparticular,bforwhereevoeryoobtainthisectorusefulvComteren,sometherebexistseaiconstanfortunderlineieunhathejacobiansucplacehromtvhatbasertthisinconositivtheevdeneeewww,osetrol,thatwtheesucuse,therforalsoandshosplitting(fortoofisinThisectorthevtheunitisacasegivwenunstablereloeconstandasitheoremnssolvcoducingeewevThen,lemma.correspbandye.injectimeofofosestepWsmalltoaexetiselemmawwcase,iflastevtheeryInv.itThesewthreewsubspacesdarediscussionuniformly,transcanvthatersefromsobwtheewillonlylogarithmhaJacvveoftoTheAgain,tlythisfromlastanquanwtitdoyeisisequalfromtobyositivbaoundedthatbllyisvitis,Wgivthiseinansmallestimateparameterofdeningwhenasingisinwhere)case,thesequenceindeneandxonet.pFtsromthantheipropyertiesmanifoldofsuthebunstableWRiccaticsolution,smallthisvquanthetitsucyangleisinbertoundedusedbofyWG.thereorderandytbboundedsomebrRIVI?REeisanditery,tIn.ofbthem.theLemmaWInhotheenoughcaseewheree(withthat2.1wsuniformeintrol.for).WAswthethesubspacesofon.totiuniformlyuounded.solbiningLfactetfunction,rebtheeofandunstableRiccatiobianaectors,unitinverticproalfare[17].unisituationfslighodirtmthelyoftransvAnosoerseotoaseacehnotother,vwthateunitnallyuniformlydeduceoundedthatbforwevyepryeunstablet,vepropctoryatw,cruciatheretexistsprotoecloseofuniformlynondecsuc[17].hethateecomesproblemevyerytroba.ptheeThenrofand,anforfunctiontermsforanyery,(andinerytherande2.1existsw(indepndendentondingoftheandWofalso)alsosuchethatwforsmallanyositiv,constan.loSo,er.theAst2.2.vDiscretizationtofradiusthetheunstable(thatRiccatiesolution.ppFtoorepreservthansmall).peoshoithat,tivenoughehaneum,brerexists(hesthewillbb(ase[17],xedpropallyalongonlytheinpapproer),ofone3.1).denesetothat)existstheeen(inisosehclhatuniformly(4)ecomesoundedbyriemannianforvethatywforknoveinwfor2.1,vlemmaunitromecFo.inolumeAs,onwill(giveeninbarticle,yetheeSasakicFoseromsmallprevioustosection,vwoulelikknotowwthatourtherealloexiststoavconstanatconmetric),ofw)sucwrite,hethat.eknofor
0

b 0
2 + b :0 0
20
2d =2 0 0 0
FK
M M = Oii=1
K ( ) M 1 i K O b
i i ii=1
2 2f1; ;Kg T M
V := (T
\g T
)\E : 0 1
K K(O ) ( ) M (V ) 2i i 2f1;;Kgi=1 i=1
d E := ( ; ) f() f ()0 0 1 0
u uf () := inffU () :2V g f() := inffU () :2V g;0 0
V b 0
f()
Z