Ergodic Properties of Tame Dynamical Systems

TL;DR

This paper investigates the ergodic decomposition of tame dynamical systems using enveloping semigroups, demonstrating convergence properties of weighted ergodic means and exploring the link between statistical features and minimal sets.

Contribution

It establishes the existence of ergodic decompositions for tame systems via generalized averaging and proves convergence of weighted ergodic means, advancing understanding of their statistical structure.

Findings

Ergodic decomposition exists for tame systems with appropriate averaging.

Weighted ergodic means have pointwise convergent subsequences.

Statistical properties relate to the structure of minimal sets and ergodic measures.

Abstract

We study the problem on the weak-star decomposability of a topological $\mathbb{N}_{0}$-dynamical system $(\Omega,\varphi)$, where $\varphi$ is an endomorphism of a metric compact set $\Omega$, into ergodic components in terms of the associated enveloping semigroups. In the tame case (where the Ellis semigroup $E(\Omega,\varphi)$ consists of $B_{1}$-transformations $\Omega\rightarrow \Omega$), we show that (i) the desired decomposition exists for an appropriate choice of the generalized sequential averaging method; (ii) every sequence of weighted ergodic means for the shift operator $x\rightarrow x\circ\varphi$, $x\in C(\Omega)$, contains a pointwise convergent subsequence. We also discuss the relationship between the statistical properties of $(\Omega,\varphi)$ and the mutual structure of minimal sets and ergodic measures.