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 S = { S g : g ∈ G } \mathcal {S} = \{ S_g : g\in G \} of a right amenable semitopological semigroup G G . We assume that S \mathcal {S} is mean ergodic as a semigroup of linear Markov operators acting on ( C ( K ) , ‖ ⋅ ‖ sup ) (C(K), \| \cdot \|_{\sup }) , where K 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 { χ ( g ) S g : g ∈ G } \{ \chi (g)S_g : g\in G \} , where χ \chi is a semigroup character. Such conditions were obtained before under the additional assumption that S \mathcal {S} is uniquely ergodic.