Weak Pseudo-Physical Measures and Pesin's Entropy Formula for Anosov C1 diffeomorphisms

TL;DR

This paper studies C1 Anosov diffeomorphisms on compact manifolds, introducing weak pseudo-physical measures, proving their ergodic existence, and characterizing measures satisfying Pesin's Entropy Formula as convex combinations of these measures.

Contribution

It introduces weak pseudo-physical measures for C1 Anosov diffeomorphisms, establishing their ergodic existence and linking Pesin's Entropy Formula to these measures.

Findings

Ergodic weak pseudo-physical measures exist.

Measures satisfying Pesin's Entropy Formula form a convex hull of these measures.

Provides a physical-like characterization of Pesin's Entropy Formula in C1 hyperbolic systems.

Abstract

We consider C1 Anosov diffeomorphisms on a compact Riemannian manifold. We define the weak pseudo-physical measures, which include the physical measures when these latter exist. We prove that ergodic weak pseudo-physical measures do exist, and that the set of invariant probability measures that satisfy Pesin's Entropy Formula is the weak*-closed convex hull of the ergodic weak pseudo-physical measures. In brief, we give in the C1-scenario of uniform hyperbolicity, a characterization of Pesin's Entropy Formula in terms of physical-like properties.