A note on a Wiener-Wintner theorem for mean ergodic Markov amenable semigroups

Abstract:

We prove a Wiener-Wintner ergodic type theorem for a Markov representation $\mathcal {S} = \{ S_g : g\in G \}$ of a right amenable semitopological semigroup $G$. We assume that $\mathcal {S}$ is mean ergodic as a semigroup of linear Markov operators acting on $(C(K), \| \cdot \|_{\sup })$, where $K$ is a fixed Hausdorff, compact space. The main result of the paper is necessary and sufficient conditions for mean ergodicity of a distorted semigroup $\{ \chi (g)S_g : g\in G \}$, where $\chi$ is a semigroup character. Such conditions were obtained before under the additional assumption that $\mathcal {S}$ is uniquely ergodic.