Ergodic Properties of Tame Dynamical Systems

The problem of the $*$-weak decomposability into ergodic components of a topological $\mathbb N_0$-dynamical system $(\Omega,\varphi)$, where $\varphi$ is a continuous endomorphism of a compact metric space $\Omega$, is considered in terms of the associated enveloping semigroups. It is shown that, in the tame case (where the Ellis semigroup $E(\Omega,\varphi)$ consists of endomorphisms of $\Omega$ of the first Baire class), such a decomposition exists for an appropriately chosen generalized sequential averaging method. A relationship between the statistical properties of $(\Omega,\varphi)$ and the mutual structure of minimal sets and ergodic measures is discussed.