Topological version of the Wiener-Wintner theorem
characterizes $\lambda$ on complex unit circle for which averages
$\frac{1}{N} \sum \limits_{n=0}^{N-1} \lambda^n f(T^n x)$ are uniformly
convergent for every continuous function $f \in C(X)$, 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.