Upper bound on the density of Ruelle resonances for Anosov flows

TL;DR

This paper establishes an upper bound on the density of Ruelle resonances for Anosov flows using a semiclassical approach, demonstrating the discreteness of the spectrum in anisotropic Sobolev spaces.

Contribution

It introduces a semiclassical method to bound the density of Ruelle resonances for Anosov flows, advancing understanding of their spectral properties.

Findings

Spectrum of Anosov vector fields is discrete in anisotropic Sobolev spaces.

Provides an explicit upper bound for the density of Ruelle resonances.

Shows the spectrum's eigenvalues cluster near the real axis for large real parts.

Abstract

Using a semiclassical approach we show that the spectrum of a smooth Anosov vector field V on a compact manifold is discrete (in suitable anisotropic Sobolev spaces) and then we provide an upper bound for the density of eigenvalues of the operator (-i)V, called Ruelle resonances, close to the real axis and for large real parts.