New criterion of physical measures for partially hyperbolic diffeomorphisms

TL;DR

This paper establishes a new criterion for physical measures in partially hyperbolic diffeomorphisms, showing that Cesaro limits satisfy Pesin's formula and analyzing the structure of physical measures and basins in various settings.

Contribution

It introduces a criterion linking Cesaro limits to Pesin's formula for partially hyperbolic diffeomorphisms and explores the existence and properties of physical measures in different dynamical contexts.

Findings

Cesaro limits satisfy Pesin's formula on a full volume set.

Existence of physical measures with non-vanishing center exponents.

Finiteness and full volume basins of physical measures in certain neighborhoods.

Abstract

We show that for any $C^1$ partially hyperbolic diffeomorphism, there is a full volume subset, such that any Cesaro limit of any point in this subset satisfies the Pesin formula for partial entropy. This result has several important applications. First we show that for any $C^{1+}$ partially hyperbolic diffeomorphism with one dimensional center, there is a full volume subset, such that every point in this set belongs to either the basin of a physical measure with non-vanishing center exponent, or the center exponent of any limit of the sequence $\frac1n\sum_{i=0}^{n-1}\delta_{f^i(x)}$ is vanishing. We also prove that for any diffeomorphism with mostly contracting center, it admits a $C^1$ neighborhood such that every diffeomorphism in a $C^1$ residual subset of this open set admits finitely many physical measure, whose basins have full volume.