Ergodic Properties of Tame Dynamical Systems

Abstract

The problem of the *-weak decomposability into ergodic components of a topological ℕ₀-dynamical system (Ω, φ), where φ is a continuous endomorphism of a compact metric space Ω, is considered in terms of the associated enveloping semigroups. It is shown that, in the tame case (where the Ellis semigroup E(Ω, φ) consists of endomorphisms of Ω of the first Baire class), such a decomposition exists for an appropriately chosen generalized sequential averaging method. A relationship between the statistical properties of (Ω, φ) and the mutual structure of minimal sets and ergodic measures is discussed.