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Working paper

Ergodic Properties of Discrete Dynamical Systems and Enveloping Semigroups

arxiv.org. math. Cornell University, 2013. No. 1309.6283.
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: (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 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, 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 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.