Ergodic Properties of Tame Dynamical Systems

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.