Virtually expanding dynamics

TL;DR

This paper introduces 'virtually expanding' dynamical systems, a broad class including expanding and some partially hyperbolic maps, and demonstrates the quasi-compactness of the Perron-Frobenius operator on associated Sobolev spaces.

Contribution

The paper defines the class of virtually expanding maps and proves the quasi-compactness of the Perron-Frobenius operator for this class.

Findings

Virtually expanding maps include all expanding and some partially hyperbolic volume-expanding maps.

Perron-Frobenius operator is quasi-compact on Sobolev spaces for these maps.

Provides a framework for analyzing statistical properties of a broad class of dynamical systems.

Abstract

We introduce a class of discrete dynamical systems that we call \emph{virtually expanding}. This is an open subset of self-covering maps on a closed manifold which contains all expanding maps and some partially hyperbolic volume-expanding maps. We show that the Perron-Frobenius operator is quasi-compact on a Sobolev space of positive order for such class of dynamical systems.