On the Ergodicity of Rotation Extensions of Hyperbolic Endomorphisms

TL;DR

This paper investigates the ergodic properties of certain partially hyperbolic endomorphisms, especially skew products over Anosov endomorphisms, establishing conditions for ergodicity and topological transitivity.

Contribution

It proves ergodicity and accessibility are generic in this setting and extends ergodic stability results to rotation extensions with one-dimensional center.

Findings

Accessibility is open and dense among these endomorphisms.

Conservative accessible systems are topologically transitive.

Ergodicity is established for skew products with circle fibers.

Abstract

We study the ergodicity of partially hyperbolic endomorphisms, focusing on skew products where the base dynamics are governed by Anosov endomorphisms. For this family, we establish ergodicity and prove that accessibility holds for an open and dense subset. By analyzing the topological implications of accessibility, we demonstrate that conservative accessible partially hyperbolic endomorphisms are topologically transitive. Leveraging accessibility, we further show ergodicity for skew products with $\mathbb{S}^1$-fibers. Finally, although out the context of rotation extensions, we prove ergodic stability results for partially hyperbolic endomorphisms with $\dim(E^c) = 1.$