## Geometry of metrics and measure concentration in abstract ergodic theory

Tim Austin

New York University

April 30, 2014

Many of the major results of modern ergodic theory can be understood in terms of a sequence of finite metric measure spaces constructed from the marginal distributions of a shift-invariant process. Most simply, the growth rate of their covering numbers gives the entropy of the process, and then one finds that more refined geometric invariants determine other properties of the process.