Ergodic theory for group actions (Lecture 1)

Consider a measure-preserving actions of a group $G$ on a probability space $(X,\mu)$. It is natural to consider ergodic averages of a function over some subsets $F_n$ in the group
$$\begin{equation*} \frac{1}{|F_n|}\sum_{g\in F_n} f(T_g x). \end{equation*}$$
However, for, say, free group there are no unique “natural” way to fix the sequence $F_n$. The theory here is quite different from the usual ergodic theory for amenable groups such as $\mathbb Z$. We will study the case of the free groups, as well as more general settings (Markov, Gromov hyperbolic, and Fuchsian groups).