Ruelle spectrum of linear pseudo-Anosov maps

TL;DR

This paper fully characterizes the Ruelle resonances of linear pseudo-Anosov maps, linking spectral data to cohomological actions, and applies this to invariant distributions and solving the cohomological equation for the flow.

Contribution

It provides a complete classification of Ruelle resonances for linear pseudo-Anosov maps based on their cohomological action, advancing understanding of their spectral properties.

Findings

Complete classification of Ruelle resonances for these maps

Description of invariant distributions under the flow

Solution to the cohomological equation for the flow

Abstract

The Ruelle resonances of a dynamical system are spectral data describing the precise asymptotics of correlations. We classify them completely for a class of chaotic two-dimensional maps, the linear pseudo-Anosov maps, in terms of the action of the map on cohomology. As applications, we obtain a full description of the distributions which are invariant under the linear flow in the stable direction of such a linear pseudo-Anosov map, and we solve the cohomological equation for this flow.