Contribution to the ergodic theory of robustly transitive maps

TL;DR

This paper investigates the ergodic properties of robustly transitive local diffeomorphisms on compact manifolds, establishing generic existence of dense pre-orbits and ergodic measures with strong statistical properties.

Contribution

It proves the existence of a residual subset of maps with dense pre-orbits and shows that generically, these maps admit uncountably many ergodic expanding measures with full support.

Findings

Residual subset of maps with dense pre-orbits

Existence of uncountably many ergodic measures with exponential decay of correlations

Results apply to a significant class of robustly transitive maps

Abstract

In this article we intend to contribute in the understanding of the ergodic properties of the set RT of robustly transitive local diffeomorphisms on a compact manifold M without boundary. We prove that there exists a C^1 residual subset R_0 of RT such that any f in R_0 has a residual subset of M with dense pre-orbits. Moreover, C^1 generically in the space of local diffeomorphisms with no splitting and all points with dense pre-orbit, there are uncountably many ergodic expanding invariant measures with full support and exhibiting exponential decay of correlations. In particular, these results hold for an important class of robustly transitive maps considered in [Lizana-Pujals'12].