On ergodicity of mostly expanding semi-group actions

TL;DR

This paper establishes new sufficient conditions for ergodicity of smooth, finitely generated semi-group actions on closed manifolds, extending previous results and introducing Markov partitions for such actions.

Contribution

It introduces a framework for ergodicity of semi-group actions using Markov partitions and provides new transitivity criteria, advancing understanding beyond prior one-dimensional cases.

Findings

Provided sufficient conditions for ergodicity of semi-group actions.

Introduced Markov partitions for finitely generated semi-group actions.

Established ergodicity for a broad class of C^{1+alpha}-diffeomorphisms.

Abstract

In this work, we address ergodicity of smooth actions of finitely generated semi-groups on an m-dimensional closed manifold M. We provide sufficient conditions for such an action to be ergodic with respect to the Lebesgue measure. Our results improve the main result in [8], where the ergodicity for one dimensional fiber was proved. We will introduce Markov partition for finitely generated semi-group actions and then we establish ergodicity for a large class of finitely generated semi-groups of C^{1+alpha}-diffeomorphisms that admit a Markov partition. Moreover, we present some transitivity criteria for semi-group actions and provide a weaker form of dynamical irreducibility that suffices to ergodicity in our setting.