SRB measures for Anosov actions

TL;DR

This paper investigates special invariant measures for Anosov $ ext{R}^$ actions, demonstrating their properties akin to classical SRB measures and establishing conditions for their uniqueness and full support.

Contribution

It introduces and analyzes invariant measures for Anosov actions using Ruelle-Taylor resonances, extending SRB measure properties to higher-rank actions.

Findings

Measures have smooth disintegrations along unstable foliations

They possess positive Lebesgue measure basins of attraction

Uniqueness and full support when the action is transitive in the positive Weyl chamber

Abstract

Given a general Anosov $\mathbb{R}^\kappa$ action on a closed manifold, we study properties of certain invariant measures that have recently been introduced in \cite{BGHW20} using the theory of Ruelle-Taylor resonances. We show that these measures share many properties of Sinai-Ruelle-Bowen measures for general Anosov flows such as smooth disintegrations along the unstable foliation, positive Lebesgue measure basins of attraction and a Bowen formula in terms of periodic orbits. Finally we show that if the action in the positive Weyl chamber is transitive, the measure is unique and has full support.