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Transfer Operators and Horocycle Averages on Closed Manifolds Alexander Adam Transfer operators and horocycle averages on closed manifolds Alexander Adam To cite this version: Alexander Adam. Transfer operators and horocycle averages on closed manifolds. Dynamical Systems [math.DS]. Sorbonne Université, 2018. English. NNT : 2018SORUS330. tel-02865539 HAL Id: tel-02865539 https://tel.archives-ouvertes.fr/tel-02865539 Submitted on 11 Jun 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Sorbonne Universit´e Ecole´ Doctorale de Sciences Math´ematiquesde Paris Centre These` de doctorat Discipline : Math´ematiques pr´esent´eepar Alexander Adam Op´erateurs de transfert et moyennes horocycliques sur les vari´et´esferm´ees dirig´eepar Viviane Baladi Soutenue le 10 d´ecembre 2018 devant le jury compos´ede : Mme Viviane Baladi Sorbonne Universit´e& CNRS directrice M. Oscar Bandtlow Queen Mary University of London examinateur M. Yves Coudene` Sorbonne Universit´e examinateur M. Giovanni Forni University of Maryland rapporteur M. Colin Guillarmou Universit´eParis-Sud & CNRS rapporteur M. Carlangelo Liverani Universita Roma 2 examinateur M. Fr´ed´eric Naud Universit´ed'Avignon examinateur Mme Barbara Shapira Universit´ede Rennes 1 examinatrice Institut de math´ematiques de Ecole´ Doctorale de Sciences Jussieu-Paris Rive gauche. UMR Math´ematiquesde Paris Centre. 7586. Universit´eSorbonne. Bo^ıtecourrier 247 Campus Pierre et Marie Curie. 4 place Jussieu Bo^ıtecourrier 290 75 252 Paris Cedex 05 4 place Jussieu 75 252 Paris Cedex 05 \To live in an asymmetric world, you should better be symmetric. However if you live in a symmetric world, asymmetry suffices." L'author Remerciements Au cours des trois derni`eresann´ees, Mme Viviane Baladi a ´et´ema directrice et ´egalement un v´eritablementor pour moi. Je suis tr`esreconnaissant envers elle pour son ´energieet sa pers´ev´erance. Elle m'a aid´e`arester concentr´etout au long de ma th`eseet ce particuli`erement pendant mes quelques moments d'´egarement. Gr^ace`aMme Viviane Baladi, je suis entr´een contact avec M. Fr´ed´ericNaud dont une id´eemath´ematiquefaisant partie de ses travaux que j'ai fait abou- tir dans ma premi`erepublication scientifique. Toujours gr^ace`aMme Viviane Baladi, j'ai pu, plusieurs fois, participer `ades conf´erences`adiff´erents endroits, ce qui m'a permis de mieux conna^ıtrela communaut´emath´ematiquedans les domaines des syst`emesdynamiques et des r´esonances. Avec certaines des personnes que j'ai rencontr´eeslors de ces conf´erences,je suis toujours en contact. Je mentionnerai ici M. Giovanni Forni, M. Colin Guillar- mou et M. Carlangelo Liverani. La rencontre avec ces personnes avec qui j'ai eu plusieurs longues discussions a contribu´ee- entre autres - `aclarifier mes id´ees. Mes d´eplacements n'auraient certainement pas ´et´epossibles sans le financement des organisateurs concern´esou de l'IMJ. Je remercie Le Centre Henri Lebesgue et le CIRM Marseille Luminy pour leur accueil chaleureux et la fondation Knut et Alice Wallenberg pour les invitations `al'universit´ede Lund. Il n'est pas facile de trouver un logement `aParis. Je suis donc tr`esreconnaissant envers la "Maison d'Italie" (Cit´einternationale) qui a mis `ama disposition un logement tr`esraisonnable pendant mes trois ans de th`ese. Je me sens ´egalement tr`esreconnaissant `aShu Shen, Malo J´ez´equel,Colin Guil- larmou et Giovanni Forni pour avoir signal´eplusieurs incoh´erenceset remarques. Pour avoir fait preuve de patience et m'avoir apport´eson soutien, je suis ´egalement redevable envers mon ´epouse Virginie et la rest de ma famille : je pense `avous ! L'auteur a ´et´esoutenu partiellement par la subvention ERC SOS (ERC AdG 787304). Op´erateurs de transfert et moyennes horocycliques sur les vari´et´esferm´ees R´esum´e Cette th`esede doctorat approfondit l'´etudede la dynamique hyperbolique sur les vari´et´esferm´eeset connexes M et des op´erateursde transfert associ´es. Nous ´etudionsdeux probl`emes: le premier probl`emeconcerne les perturbati- ons analytiques r´eellesdes diff´eomorphismesd'Anosov lin´eairessur le tore : une r´esonancenon triviale appara^ıt-t-elle pour une perturbation g´en´eriquesd'un diff´eomorphismed'Anosov lin´eairesur le tore ? Le second probl`emeconcerne une hypoth`esesur la moyenne temporelle des flots horocycliques induits par un flot d'Anosov : la moyenne temporelle des flots horocycliques en courbure n´egative variable converge-t-elle vers la moyenne er- godique en vitesse polynomiale ? Les op´erateurs de transfert associ´es agissent de fa¸conborn´eesur certains espaces de Banach anisotropes par la composition du syst`emedynamique inverse suivie d'une multiplication avec des fonctions de poids sp´ecifiques.Dans notre analyse des probl`emesmentionn´esci-dessus, ces op´erateurs de transfert repr´esentent le principal int´er^et.Nous devons ´etudierleur spectre bas pour progresser sur nos deux probl`emes. Par le spectre bas, nous entendons la partie du spectre qui se situe entre le spectre p´eriph´eriqueet le spectre essentiel de ces op´erateursde transfert. L'approche fonctionnelle de ces op´erateursde transfert se concentre sur les es- paces de Banach anisotropes. Nous expliquons l'id´eeprincipale derri`erecette approche dans le cas des diff´eomorphismes d'Anosov : des exemples simples de diff´eomorphismesd'Anosov F sont donn´espar les diff´eomorphismeslin´eaires d'Anosov sur le tore bidimensionnel. Nous savons que les diff´eomorphismes d'Anosov transitifs et analytiques ont une unique mesure SRB µSRB (qui est in- variante par le diff´eomorphisme).Pour les automorphismes lin´eairessur le tore, la mesure SRB est la mesure de Lebesgue µLeb. Notons toutefois que m^eme de petites perturbations analytiques de A ne pr´eservent pas syst´ematiquement 1 µLeb. Puisque µSRB est une mesure de Borel, on a µSRB P C pMq . Nous souhai- tons maintenant ´ecrire µSRB comme l'unique vecteur propre associ´e`ala valeur propre 1 pour un certain op´erateurde transfert L qui appara^ıtcomme l'adjoint 7 de l'op´erateurde composition KF . Cependant les mesures support´eessur les 1 orbites p´eriodiques de F sont ´egalement contenues dans C pMq . Afin de trou- ver les bonnes propri´et´esspectrales de l'op´erateur L, celui-ci doit ^etred´efinisur 1 un espace de Banach anisotrope B et non sur C pMq . La norme de B prend en compte le comportement dilatant et contractant de l'application F . En particu- lier, la norme anisotrope de B traite les ´el´ements de B comme des fonctions dans les directions dilatantes et comme des distributions dans les directions contrac- tantes de F . Les valeurs propres discr`etesr´eciproques de L sont aussi appel´eesles r´esonances de F . Si F A, alors il y a seulement les r´esonancestriviales t0; 1u. Jusque l`a il n'´etaitpas su qu'il s'agissait d'un comportement attendu si A est perturb´ede mani`ereg´en´erique. On entend ici par perturbation g´en´erique toute application d'un ensemble ouvert et dense dans une boule de diff´eomorphismesanalytiques r´eelscontenant A. Dans de l'´etudedu premier probl`eme,nous agissons avec L sur un espace de Hilbert anisotrope. Nous r´epondons `ala question dans le premier probl`emepar l’affirmative. Le second probl`emeque nous examinons fait intervenir les flots d'Anosov. Ces flots ont ´et´einstaur´espar Anosov pour ´etudierle flot g´eod´esiquesur le fibr´e tangent unitaire de vari´et´esferm´ees`acourbure sectionnelle n´egative variable. De plus, nous avons besoin les flots d'Anosov d'^etredes flots de contact. Des ex- emples de flots d'Anosov-contact sont donn´espar les flots g´eod´esiques.Les flots horocycliques associ´esau flot d'Anosov sont dirig´esdans la direction contractant du flot d'Anosov. Nous savons par les travaux de Marcus que pour tout flot horo- cyclique continu qui correspond `aun flot d'Anosov C2 m´elangeant, il existe une unique mesure de probabilit´ede Borel invariante par le flot horocyclique. Ka- tok et Burns ont d´emontr´eque tout flot d'Anosov-contact est m´elangeant. Par cons´equent, dans notre contexte, la moyenne temporelle de l'horocycle converge vers la moyenne ergodique unique. Mais `aquelle vitesse converge la moyenne temporelle ? Dans le contexte de la courbure n´egative constante, on sait gr^aceaux travaux de Flaminio et Forni que cette vitesse est polynomiale. La vitesse est contr^ol´ee par des valeurs propres pour certaines distributions propres du flot g´eod´esique. Un probl`emeanalogue dans lequel le flot g´eod´esiqueest remplac´epar un diff´eomorphismed'Anosov a ´et´e´etudi´eplus tard par Giulietti et Liverani. Ils ont, de plus, dans leurs travaux, suppos´eque le r´esultatde Flaminio{Forni devrait s'´etendreau flot g´eod´esiquedans le contexte de la courbure n´egative variable. 8 Dans l'´etudedu second probl`eme des op´erateursde transfert pond´er´es Lα, α ¡ 0 apparaissent. Suivant l'approche fonctionnelle, il suffit essentiellement de con- struire un espace de Banach anisotrope B tel que les op´erateurs Lα agissent sur B, et d'avoir un spectre p´eriph´eriqueconsistant en une valeur propre simple isol´ee.Cependant, la direction d'´ecoulement du flot d'Anosov n'est ni contract´ee ni dilat´eepar le flot d'Anosov, ce qui pose probl`emedans notre analyse. Nous appliquons donc `ala place la strat´egiesuivante: Sur un espace de Banach anisotrope B bien choisi, la famille d'op´erateurde transfert tLα : B Ñ B | α ¥ 0u forme un semi-groupe fortement continu et ad- met donc un g´en´erateurbien d´efini.
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