SRB measures for Anosov actions

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 $\href{https://doi.org/10.48550/arXiv.2007.14275}{[GBGHW20]}$ 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.