Topological Wiener–Wintner theorems for amenable operator semigroups

Abstract

Inspired by topological Wiener-Wintner theorems we study the mean ergodicity of amenable semigroups of Markov operators on C(K) and show the connection to the convergence of strong and weak ergodic nets. The results are then used to characterize mean ergodicity of Koopman semigroups corresponding to skew product actions on compact group extensions.

A note on uniform convergence of Wiener-Wintner ergodic averages

“…Note that the first part of the theorem is a generalization of a result by Lenz (see [17, Corollary 1]), who considered only uniquely ergodic actions of discrete LCA-groups. The second part of the theorem is very similiar but still different to results by Lenz (see [17,Theorem 2]) and Schreiber (see [26,Corollary 1.13]). We want to point out that in the latter works the Følnersequence in question is not assumed to be tempered.…”

“…Furthermore the Wiener-Wintner ergodic theorem has been transferred to actions of LCAgroups or more general amenable groups on probability spaces (see for example Zorin-Kranich [28] as well as Schreiber [26] and Bartoszek, Śpiewak [3] for the topological case). In the case of abelian groups the corresponding averages look as follows 1 m G (F n ) Fn ξ(g)f (gx)dm G (g), (1.3) where (F n ) is a (tempered strong) Følner-sequence in G, m G is the Haar-measure on G, ξ is a character of G and gx describes the action of an element g ∈ G on x ∈ X.…”

Compact operator semigroups applied to dynamical systems

“…In this section we study different notions of mean ergodicity in topological dynamics (see [Sch14]). We note that for a topological dynamical system (K; S) the mapping…”

“…Inspired by the approach of R. Nagel to mean ergodic semigroups (see [Nag73] and the supplement of Chapter 8 of [EFHN15]) we use techniques developed by A. Romanov in [Rom11] as well as M. Schreiber in [Sch13a] and [Sch13b] to discuss mean ergodicity of operator semigroups on locally convex spaces (see also [Ebe49], [Sat78], Section 2.1.2 in [Kre85], [GK14]; see [ABR12] for mean ergodicity of one-parameter semigroups).…”

“…In this section we study different notions of mean ergodicity in topological dynamics (see [Sch14]). We note that for a topological dynamical system (K; S) the mapping S −→ T S , s → T s is an epimorphism of semitopological semigroups, if we reverse the order of multiplication in S and equip T S with the strong operator topology (see Theorem 4.17 in [EFHN15]).…”