Wiener-Wintner theorem for semigroups of Markov operators

Topological version of the Wiener-Wintner theorem characterizes parameters z from the complex unit circle for which averages \frac{1}{N} \sum_{n=0}^{N-1} z^n f(T^n x) are uniformly convergent for every continuous function f, where (X,T) is a topological dynamical system. I will present extensions of this result to amenable semigroups of Markov operators on C(X). This is joint work with W. Bartoszek.