Minerva Mini-Course

Let $G$ be a locally compact group acting by measure-preserving transformations on a probability space $(X,\mu)$. To every probability measure on $G$ there is an associated averaging operator on $L^p(X,\mu)$. Ergodic theorems describe the pointwise and norm limits of sequences of such operators. In joint work with Amos Nevo, we develop a new general approach based on reducing the problem to the amenable case. From this we obtain ergodic theorems for sector and spherical averages when $G$ is a rank $1$ Lie group or a countable Gromov hyperbolic group.