Ergodic Properties of Tame Dynamical Systems

The problem of the *-weak decomposability into ergodic components of a topological ℕ 0 -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.