Ergodic properties of discrete dynamical systems and enveloping semigroups

For a continuous semicascade on a metrizable compact set ${\rm\Omega}$, we consider the weak$^{\ast }$ convergence of generalized operator ergodic means in $\text{End}\,C^{\ast }({\rm\Omega})$. We discuss conditions under which: every ergodic net contains a convergent sequence; all ergodic nets converge; all ergodic sequences converge. We study the relationships between the convergence of ergodic means and the properties of transitivity of the proximality relation on ${\rm\Omega}$, 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 associated algebraic-topological objects: the Ellis semigroup $E$, the Köhler operator semigroup ${\rm\Gamma}\subset \text{End}\,C^{\ast }({\rm\Omega})$, and the semigroup $G=\overline{\text{co}}\,{\rm\Gamma}$. The main results are stated for semicascades with metrizable $E$ and for tame semicascades.