Measure of maximal entropy for minimal Anosov actions

Abstract

For a minimal Anosov ${\mathbb{R}}^{\kappa }$-action on a closed manifold, we study the measure of maximal entropy constructed by Carrasco and Rodriguez-Hertz in \cite{CarHer} and show that it fits into the theory of Ruelle-Taylor resonances introduced by Guedes Bonthonneau, Guillarmou, Hilgert, and Weich in \cite{GBGHW}. More precisely, we show that the topological entropy corresponds to the first Ruelle-Taylor resonance for the action on a certain bundle of forms and that the measure of maximal entropy can be retrieved as the distributional product of the corresponding resonant and co-resonant states. As a consequence, we prove a Bowen-type formula for the measure of maximal entropy and a counting result on the number of periodic torii.