Ergodic Properties of Discrete Dynamical Systems and Enveloping Semigroups

For a continuous semicascade on a metrizable compact set $\Omega$, we consider the weak$^{*}$ convergence of generalized operator ergodic means in ${\rm End}\, \, C^{*} (\Omega)$. We discuss conditions on the dynamical system under which (a) every ergodic net contains a convergent subsequence; (b) all ergodic nets converge; (c) all ergodic sequences converge. We study the relationships between the convergence of ergodic means and the properties of transitivity of the proximality relation on $\Omega$, minimality of supports of ergodic measures, and uniqueness of minimal sets in the closure of trajectories of a semicascade. These problems are solved in terms of three algebraic-topological objects associated with the dynamical system: the Ellis enveloping semigroup, the K\"{o}hler operator semigroup $\Gamma$, and the semigroup $G$ that is the weak$^{*}$ closure of the convex hull of $\Gamma$ in ${\rm End}\, C^{*} (\Omega)$. The main results are stated for ordinary semicascades (whose Ellis semigroup is metrizable) and tame semicascades. For a dynamics, being ordinary is equivalent to being "nonchaotic" in an appropriate sense. We present a classification of compact dynamical systems in terms of topological properties of the above-mentioned semigroups.