Distribution of Ruelle resonances for real-analytic Anosov diffeomorphisms

TL;DR

This paper establishes an upper bound on the number of Ruelle resonances for real-analytic Anosov diffeomorphisms, revealing a dimension-dependent logarithmic growth and a dichotomy regarding the optimality of this bound.

Contribution

It provides a new upper bound on Ruelle resonances for real-analytic Anosov diffeomorphisms and analyzes the optimality of this bound across different components.

Findings

Number of resonances > r is O(|log r|^d) as r → 0.

Dichotomy in the optimality of the bound across connected components.

In 2D torus cases, the bound is always optimal on a dense subset.

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.