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Transfer Operators of Random Partially Hyperbolic Dynamical Systems Luc Gossart

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Transfer Operators of Random Partially Hyperbolic Dynamical Systems Luc Gossart Transfer operators of random partially hyperbolic dynamical systems Luc Gossart To cite this version: Luc Gossart. Transfer operators of random partially hyperbolic dynamical systems. Dynamical Sys- tems [math.DS]. Université Grenoble Alpes [2020-..], 2020. English. NNT : 2020GRALM062. tel- 03223781 HAL Id: tel-03223781 https://tel.archives-ouvertes.fr/tel-03223781 Submitted on 11 May 2021 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. THÈSE Pour obtenir le grade de DOCTEUR DE L’UNIVERSITÉ GRENOBLE ALPES Spécialité : Mathématiques Arrêté ministériel : 25 mai 2016 Présentée par Luc GOSSART Thèse dirigée par Frédéric FAURE, MCF préparée au sein du Laboratoire Institut Fourier dans l'École Doctorale Mathématiques, Sciences et technologies de l'information, Informatique Opérateurs de transfert de systèmes dynamiques partiellement hyperboliques aléatoires Transfer operators of random partially hyperbolic dynamical systems Thèse soutenue publiquement le 12 novembre 2020, devant le jury composé de : Monsieur FREDERIC FAURE MAITRE DE CONFERENCES HDR, UNIVERSITE GRENOBLE ALPES, Directeur de thèse Monsieur FREDERIC NAUD PROFESSEUR DES UNIVERSITES, SORBONNE UNIVERSITE - PARIS, Rapporteur Monsieur CARLANGELO LIVERANI PROFESSEUR, UNIVERSITE DE ROME «TOR VERGATA» ITALIE, Rapporteur Monsieur COLIN GUILLARMOU DIRECTEUR DE RECHERCHE HDR, CNRS ILE-DE-FRANCE GIF-SUR- YVETTE, Examinateur Madame FRANÇOISE PENE PROFESSEUR DES UNIVERSITES, UNIVERSITE DE BRETAGNE OCCIDENTALE, Examinatrice Monsieur ERWAN LANNEAU PROFESSEUR DES UNIVERSITES, UNIVERSITE GRENOBLE ALPES, Président du jury Remerciements 1 Introduction en français 2 CONTEXTE 3 Contexte On étudie dans cette thèse les traces plates des itérés d’opérateurs de transferts associés à des applications partiellement hyperboliques. Etant donnés une appli- cation dilatante ou un difféomorphisme Anosov T sur une variété riemannienne compacte M, l’étude de l’opérateur , tirant les fonctions en arrière par le biais de T : L ∞(M) ∞(M) (0.1) : C −→ C L u u T 7−→ ◦ s’est révélée intéressante pour la connaissance des propriétés statistiques de l’application T . Ce cadre a été étudié en détail au cours des décennies passées (voir par exemple [Rue89], [Bow75], [PP90],[BKL02], et se référer au chapitre 1 pour une présen- tation plus détaillée et des références plus fournies). L’opérateur sera appelé dans tout ce qui suit opérateur de transfert, bien que ce terme aitL à l’origine désigné, et désigne encore le plus souvent dans la lit- térature, son adjoint. Les propriétés statistiques de l’application T sont liées au spectre de : Si T est lisse, la résolvante de l’opérateur de transfert admet un prolongement méromor-L phe à C 0 et les fonctions de corrélation admettent un développement asympto- tique s’exprimant\{ } de manière explicite à l’aide des pôles et résidus de la résolvante (voir par exemple à cet égard [Rue89] et [BKL02]; le chapitre 1 contient une description plus détaillée de ces principes, notamment la section 1, et la section 5 présente des résultats précis concernant pour des applications k). Les pôles de la résolvante (c’est-à-dire de son prolongement méromorphe) sontC appelées réso- nances de Ruelle-Pollicott, et leur collection spectre de Ruelle-Pollicott. L’existence d’un tel prolongement méromorphe est généralement obtenue par le bi- ais de la construction d’espaces dans lesquels le spectre de l’opérateur de transfert est discret en dehors d’un voisinage arbitrairement petit de l’origine. Dans le cas de flots ou d’applications partiellement hyperboliques, la présence d’une direction neutre complique les choses. La section 5 du chapitre 1 présente des résultats de la littérature à ce sujet. La façon la plus élémentaire de contruire un système qui sort du cadre hyperbolique est peut-être de partir d’une application hyperbolique et d’en construire une extension en cercles : Si T est une application dilatante ou un difféomorphisme d’une variété riemannienne compacte M, et si l’on dispose d’une fonction continue τ : M T := R/Z, alors on peut considérer le système dynamique −→ M T M T (0.2) F : . (x,y× ) −→ (T (×x),y + τ(x)) 7−→ Ce genre de systèmes a été étudié notamment par Dolgopyat [Dol02], qui en a montré le caractère génériquement mélangeant pour des mesures naturelles. Avec cette définition, les modes de Fourier dans la direction y sont préservés : si u ∞(M) et p Z, ∈C ∈ 2iπp 2iπpτ(x) 2iπpy (0.3) (u e ·)(F (x,y)) = e u(T (x))e . ⊗ 4 INTRODUCTION EN FRANÇAIS De ce fait, on se ramène à l’étude d’une famille d’opérateurs de transfert ∞(M) ∞(M) (0.4) ξ,τ : C −→ C L u eiξτ u T 7−→ ◦ associés à une application dilatante ou hyperbolique, et indexés par un paramètre fréquentiel ξ R. Comme première étape dans l’exploration du spectre de Ruelle, on prend pour∈ τ une fonction aléatoire et on s’intéresse aux traces plates des itérés de l’opérateur de transfert : Si ξ,τ n’est en général pas un opérateur à trace, il admet une trace plate (voir chapitreL 1 section 5.1), définie comme l’intégrale de son noyau de Schwartz le long de la diagonale. Pour les opérateurs à trace, cette définition coïncide avec la notion habituelle de trace. La trace plate d’un itéré de l’opérateur de transfert s’exprime comme une somme sur les orbites périodiques : n iξτx ♭ n e (0.5) Tr ( ξ,τ )= n , L n det(1 dTx ) x,TX(x)=x | − | n où τx est la somme de Birkhoff n n 1 (0.6) τ = τ(x)+ + τ(T − (x)). x ··· Notons lx la période (minimale) d’un point périodique x, et 1 − 2 lx (0.7) An := n 2 1. n det(1 dTx ) ≫ x,TX(x)=x | − | Remarquons qu’An est exponentiellement grand : n Pr( 2Ju)+o(n) (0.8) An = e− 2 − où Ju = log det(dT Eu ) est le jacobien instable, et Pr( 2Ju) < 0 sa pression topologique (voir (3.B.24)).| − Dans l’hypothèse où les phases comprenant les sommes de Birkhoff, qui inter- viennent dans l’expression de la trace plate (0.5), se comporteraient comme des variables aléatoires indépendantes, uniformes sur S1, un rapide calcul montre que ♭ n les traces plates Tr ( ξ,τ ), renormalisées par le facteur An convergent vers des vari- ables aléatoires gaussiennesL complexes de variance 1. Dans [FW17], Faure et Weich établissent une forme normale globale, dans un modèle ouvert très proche, appelé IFS (iterated function scheme) : ils expriment les itérés n de l’opérateur de transfert comme une somme d’opérateurs de rang Lξ,τ 1 indexés par les points périodiques, dans la limite où n et ξ tendent vers l’infini, avec n plus grand qu’un temps d’Ehrenfest c log ξ. Ils font usage de cette forme nor- male pour obtenir une borne supérieure sur le rayon spectral de ξ,τ pour ξ large, ce qui fournit un trou spectral asymptotique. Ils rencontrent lesL mêmes phases, faisant intervenir de sommes de Birkhoff, que dans la formule des traces (0.5), et obtiendraient une borne optimale si les phases se comportaient comme des variables aléatoires uniformes indépendantes. Cependant, il n’est pas très raisonnable de considérer des fonctions aléatoires τ pour lesquelles les valeurs prises en différents points périodiques d’une même période sont indépendantes, car cela mènerait à une grande irrégularité (si par exemple τ est un champ gaussien, cette hypothèse implique que τ est partout localement non RÉSULTATS 5 borné). Il n’y a alors plus lieu de parler de traces plates. De plus, l’ensemble captif, qui apparaît de manière centrale dans le papier [FW17] est sensible aux perturbations 1, il est donc naturel de souhaiter prendre pour fonction toit τ une peturbation, petiteC en norme 1, d’une fonction τ donnée. C 0 Résultats Les résultats contenus dans cette thèse sont les deux théorèmes suivants: Applications dilatantes du cercle, chapitre 2. Ce théorème est démontré dans le chapitre 2,qui est une reproduction de [Gos20a]. Soit k N. Soit E une application dilatante k du cercle, de degré l, préservant l’orientation.∈ Soit C M = max E′. k Theorem 0.1. Soit τ0 (T). Soit (cp)p N une famille variables aléatoires ∈ C 2 2∈ ν gaussiennes complexes vérifiant E[ c ]= O(p− − ) pour un certain ν > 0 (afin de | p| s’assurer que δτ est presque sûrement continu). Soit c p = cp et − 2iπpx (0.9) δτ(x)= cpe . p Z X∈ Si 2 C (0.10) ǫ> 0, C > 0, p Z∗, E c , ∃ ∃ ∀ ∈ | p| ≥ p2k+2+ǫ h i alors on a la convergence en loi ♭ n (0.11) A Tr C(0, 1) n Lξ,τ0+δτ −→ N pour n et ξ tendant tous deux vers l’infini, en suivant la relation log ξ (0.12) 0 <c< 1, n,ξ, n c 1 ǫ . ∃ ∀ ≤ log l +(k + 2 + 2 ) log M On verra que si l’on inverse l’inégalité dans la condition (0.10), c’est-à-dire, s’il existe α> 0 et C > 0 tels que 2 C (0.13) p Z∗, E c , ∀ ∈ | p| ≤ p2k+2+α h i alors la fonction δτ est presque sûrement de classe Ck. Prendre k plus grand permet donc d’énoncer le théorème pour des champs plus réguliers, la contrepartie étant qu’il est alors valable pour des temps plus courts.
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