Ergodic theorems beyond amenable groups

Minerva Distinguished Visitor Lectures at Princeton University: Benjamini-Schramm Convergence in Groups and Dynamics
Topic:Ergodic theorems beyond amenable groups
Speaker:Lewis Bowen
Affiliation:University of Texas, Austin and Princeton University; Minerva Distinguished Visitor
Date:Friday, December 4
Time/Room:1:30pm - 2:30pm/Fine 110, Princeton University

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.