Distribution of Ruelle resonances for real-analytic Anosov diffeomorphisms

Jézéquel, Malo

doi:10.5802/ahl.208

Jézéquel, Malo

Distribution of Ruelle resonances for real-analytic Anosov diffeomorphisms

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

Metadata

DOI10.5802/ahl.208

Keywords Anosov diffeomorphism, Ruelle resonances, Koopman operator, real-analytic

Abstract

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.

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