2013/09/24 by A. V. Romanov · 10 citations
Mathematics · #Advanced Topology and Set Theory #Discrete mathematics #Dynamical system (definition) #Dynamical systems theory #Ergodic theory #Functional analysis #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Metrization theorem #Omega #Physics #Pure mathematics #Semigroup #Topological dynamics #advanced mathematical theories #math.DS #msc:20M20 #msc:37A30
paper · pdf · doi:10.1017/etds.2014.62
published in Ergodic Theory and Dynamical Systems 36(1), 198-214 (Cambridge University Press) · 24 pages
arxiv created 2013/09/24 · openalex publication_date 2014/08/04 · arxiv updated 2015/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
For a continuous semicascade on a metrizable compact set \rmΩ , we consider the weak ∗ convergence of generalized operator ergodic means in End C∗ (\rmΩ) . 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 \rmΩ , 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 \rmΓ⊂ End C∗ (\rmΩ) , and the semigroup G=co \rmΓ . The main results are stated for semicascades with metrizable E and for tame semicascades.