On mostly expanding diffeomorphisms

TL;DR

This paper investigates mostly expanding partially hyperbolic diffeomorphisms, showing their stability, existence of physical measures, and statistical properties, with examples including Mañé's derived-from-Anosov diffeomorphism.

Contribution

It establishes that the class of mostly expanding diffeomorphisms is open in the $C^r$ topology and guarantees the existence of physical measures with detailed statistical behavior.

Findings

The class is $C^r$-open among partially hyperbolic diffeomorphisms.

Existence of physical measures is guaranteed under the mostly expanding condition.

Includes examples like Mañé's derived-from-Anosov diffeomorphism.

Abstract

In this work we study the class of mostly expanding partially hyperbolic diffeomorphisms. We prove that such class is $C^r$-open, $r>1$, among the partially hyperbolic diffeomorphisms (in the narrow sense) and we prove that the mostly expanding condition guarantee the existence of physical measures and provide more information about the statistics of the system. Ma\~n\'e's classical derived-from-Anosov diffeomorphism on $\mathbb{T}^3$ belongs to this set.