Entropy rigidity of $u$-Gibbs measures

Abstract

We obtain new entropy rigidity results for $u$-Gibbs measures by showing that whenever a $u$-Gibbs measure of a partially hyperbolic diffeomorphism admits an unstable Margulis family, the unstable Jacobian data of the system must to be constant. We apply our result to center isometries and flow type diffeomorphisms showing that if a measure of maximal entropy is also $u$-Gibbs then Jacobian periodic data along the unstable bundle are constant. In the case of smooth jointly integrable partially hyperbolic diffeomorphisms of $\mathbb{T}^3$, assuming that there exists some $u$-Gibbs measure which is also a measure of maximal unstable entropy, we obtain smooth conjugacy along the center-unstable foliation and uniqueness of $u$-Gibbs measures in this case.