Distribution of Ruelle resonances for real-analytic Anosov diffeomorphisms

Jézéquel, Malo

doi:10.5802/ahl.208

Distribution of Ruelle resonances for real-analytic Anosov diffeomorphisms
[Distribution des résonances de Ruelle pour les difféomorphismes d’Anosov analytiques]

Jézéquel, Malo ¹

¹CNRS, Univ. Brest, UMR 6205, Laboratoire de Mathématiques de Bretagne Atlantique (France)

Annales Henri Lebesgue, Tome 7 (2024), pp. 673-726

Abstract (VO)
Résumé

We prove an upper bound for the number of Ruelle resonances for Koopman operators associated to real-analytic Anosov diffeomorphisms: in dimension $d$ , the number of resonances larger than $r$ is a $𝒪 (| log r |^{d})$ when $r$ goes to $0$ . For each connected component of the space of real-analytic Anosov diffeomorphisms on a real-analytic manifold, we prove a dichotomy: either the exponent $d$ in our bound is never optimal, or it is optimal on a dense subset. Using examples constructed by Bandtlow, Just and Slipantschuk, we see that we are always in the latter situation for connected components of the space of real-analytic Anosov diffeomorphisms on the $2$ -dimensional torus.

Nous prouvons une borne supérieure sur le nombre de résonances de Ruelle pour les opérateurs de Koopman associés aux difféomorphismes d’Anosov analytiques : en dimension $d$ , le nombre de résonances plus grandes que $r$ est un $𝒪 (| log r |^{d})$ lorsque $r$ tend vers $0$ . Dans chaque composante connexe de l’espace des difféomorphismes d’Anosov analytique réels sur une variété analytique réelle, nous établissons une dichotomie : soit l’exposant $f$ dans notre borne n’est jamais optimal, soit il l’est sur un ensemble dense. En utilisant des exemples construits par Bandtlow, Just et Slipantschuk, nous montrons que nous sommes toujours dans le deuxième cas pour les composantes connexes de l’espace des difféomorphismes d’Anosov analytiques réels sur le tore de dimension $2$ .

Reçu le : 2023-02-03
Révisé le : 2024-04-17
Accepté le : 2024-05-07
Publié le : 2024-09-05

MR Zbl

DOI : 10.5802/ahl.208

Classification : 37C30, 37D20
Keywords: Anosov diffeomorphism, Ruelle resonances, Koopman operator, real-analytic

Affiliations des auteurs :

Jézéquel, Malo ¹

¹ CNRS, Univ. Brest, UMR 6205, Laboratoire de Mathématiques de Bretagne Atlantique (France)

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Jézéquel, Malo. Distribution of Ruelle resonances for real-analytic Anosov diffeomorphisms. Annales Henri Lebesgue, Tome 7 (2024), pp. 673-726. doi: 10.5802/ahl.208

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