Entropy rigidity of $u$-Gibbs measures

TL;DR

This paper establishes new entropy rigidity results for $u$-Gibbs measures in partially hyperbolic systems, showing conditions under which the unstable Jacobian data must be constant, leading to implications for smooth conjugacy and measure uniqueness.

Contribution

It proves that $u$-Gibbs measures with unstable Margulis families have constant unstable Jacobian data, and applies this to derive smooth conjugacy and measure uniqueness results.

Findings

Unstable Jacobian data must be constant for certain $u$-Gibbs measures.

Measures of maximal entropy that are $u$-Gibbs have constant Jacobian periodic data.

Smooth conjugacy along center-unstable foliation in specific partially hyperbolic systems.

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.