Entropic stability beyond partial hyperbolicity

TL;DR

This paper studies specific small deformations of Anosov diffeomorphisms that preserve high entropy dynamics despite breaking topological conjugacy, revealing new insights into entropy stability beyond partial hyperbolicity.

Contribution

It introduces a class of deformations that maintain high entropy dynamics and establishes partial conjugacy and expansiveness properties, extending understanding of entropy stability beyond partial hyperbolicity.

Findings

High entropy measures are preserved under certain deformations.

A partial conjugacy exists between deformed and original systems.

Includes nonpartially hyperbolic, robustly transitive diffeomorphisms.

Abstract

We analyze a class of deformations of Anosov diffeomorphisms: these $C^0$-small, but $C^1$-macroscopic deformations break the topological conjugacy class but leave the high entropy dynamics unchanged. More precisely, there is a partial conjugacy between the deformation and the original Anosov system that identifies all invariant probability measures with entropy close to the maximum. We also establish expansiveness around those measures. This class of deformations contains many of the known nonhyperbolic robustly transitive diffeomorphisms. In particular, we show that it includes a class of nonpartially hyperbolic, robustly transitive diffeomorphisms described by Bonatti and Viana.