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 End C* (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 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 Kohler operator semigroup Gamma subset of End C*(Omega), and the semigroup G = (co) over bar Gamma. The main results are stated for semicascades with metrizable E and for tame semicascades.