Sharp polynomial bounds on the number of Pollicott-Ruelle resonances

TL;DR

This paper establishes a precise polynomial upper bound on the number of Pollicott-Ruelle resonances for Anosov contact flows, advancing the understanding of their spectral properties and correlation decay.

Contribution

It introduces a novel approach using exponentially weighted spaces to improve bounds on Pollicott-Ruelle resonances, surpassing previous results by Faure and Sj"ostrand.

Findings

Established a sharp polynomial bound on the number of resonances

Replaced complex scaling with exponentially weighted spaces in the analysis

Improved upon previous bounds by Faure-Sj"ostrand

Abstract

We give a sharp polynomial bound on the number of Pollicott-Ruelle resonances. These resonances, which are complex numbers in the lower half-plane, appear in expansions of correlations for Anosov contact flows. The bounds follow the tradition of upper bounds on the number of scattering resonances and improve a recent bound of Faure-Sj\"ostrand. The complex scaling method used in scattering theory is replaced by an approach using exponentially weighted spaces introduced by Helffer-Sj\"ostrand in scattering theory and by Faure-Sj\"ostrand in the theory of Anosov flows.