International Conference Contemporary mathematics in honor of the 80th birthday of Vladimir Arnold, Moscow, December 18-23, 2017
For a continuous semicascade on a metrizable compact set Ω, we consider the weak* convergence of generalized operator ergodic means in End C*(Ω). We discuss conditions on the dynamical system 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 Ω, 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 E, the Kohler operator semigroup Г, and the semigroup G that is the weak* closure of the convex hull of Г in End C*(Ω). The main results are stated for semicascades with metrizable E and for tame semicascades.
A relationship is considered between ergodic properties of a discrete dynamical system on a compact metric space Ω and characteristics of companion algebro-topological objects, namely, the Ellis enveloping semigroup E, the Kohler enveloping operator semigroup Γ, and the semigroup G being the closure of the convex hull of Γ in the weak-star topology on the operator space End C*(Ω). The main results are formulated for ordinary (having metrizable semigroup E) semicascades and for tame dynamical systems determined by the condition card E ≤ c. A classification of compact semicascades in terms of topological properties of the semigroups specified above is given.