Topological Wiener-Wintner theorems for amenable operator semigroups

TL;DR

This paper investigates the mean ergodicity of amenable semigroups of Markov operators on continuous functions, establishing connections to ergodic net convergence and characterizing ergodicity in skew product actions on compact groups.

Contribution

It extends topological Wiener-Wintner theorems to amenable operator semigroups, linking ergodic properties with convergence of ergodic nets and Koopman semigroup behavior.

Findings

Characterization of mean ergodicity for amenable semigroups of Markov operators.

Connection established between ergodic net convergence and mean ergodicity.

Application to skew product actions on compact group extensions.

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.