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arXiv · 2212.09881

Distribution of Ruelle resonances for real-analytic Anosov diffeomorphisms

Malo Jézéquel
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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 $\mathcal{O}(|\log r|^d)$ 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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Malo Jézéquel. 2022-12-19. Distribution of Ruelle resonances for real-analytic Anosov diffeomorphisms. https://arxiv.org/abs/2212.09881

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