Ergodic Properties of Tame Dynamical Systems

Abstract

We study the problem on the weak-star decomposability of a topological N 0 -dynamical system (Ω, ϕ), where ϕ is an endomorphism of a metric compact set Ω, into ergodic components in terms of the associated enveloping semigroups. In the tame case (where the Ellis semigroup E(Ω, ϕ) consists of B 1 -transformations Ω → Ω), 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 → x • ϕ, x ∈ C(Ω), contains a pointwise convergent subsequence. We also discuss the relationship between the statistical properties of (Ω, ϕ) and the mutual structure of minimal sets and ergodic measures.2010 Mathematics Subject Classification: Primary 37A30, 47A35; Secondary 20M20