Ergodic Theory of Nonamenable Group Actions

University Of Texas At Austin, Austin TX

Investigators

Abstract

Classical dynamics studies how systems change in time. Ergodic theory focuses on the statistical behavior of dynamical systems. Applications of ergodic theory are widespread: from traffic modelling to aerospace engineering and population dynamics. It is natural and of practical importance to generalize the role of time in a dynamical system to more complicated groups of symmetries. This generalized notion of dynamics leads to applications in statistical mechanics, number theory, and geometry. However, new tools are needed, especially in the particular case when the group of symmetries is non nonamenable, which means that boundary phenomena are too significant to be safely ignored (unlike intervals in the integers or real numbers). Nonamenable groups naturally arise in many parts of mathematics, such as geometry and number theory. This project is concerned with developing the tools needed to analyze the ergodic theory of nonamenable group actions. There are five specific goals. The first is to continue developing sofic entropy theory. This is a vast generalization of Kolmogorov-Sinai entropy to actions of sofic groups, a class of groups that contains all amenable groups and residually finite groups. One major aim of this research is to determine to what extent Ornstein theory can be extended beyond actions of amenable groups. The second goal is to establish pointwise ergodic theorems for geometrically defined groups (e.g. Lie groups, CAT(0) cubulated groups, relatively hyperbolic groups) using techniques recently discovered by the PI and Amos Nevo. The third goal is to import tools from geometric group theory into the study of measured equivalence relations. The fourth goal is to continue to analyzing the structure of the space of weak equivalence classes of actions of a given group. This space serves as a classifying object for the ways in which the Rokhlin Lemma fails for nonamenable groups. Because the Rokhlin Lemma is of crucial importance in Ornstein theory, this topic is intimately connected with the first goal. The fifth goal is to continue clarifying what the space of stationary actions of a given group "looks like.'' One major focus is on new constructions of stationary actions (via invariant random subgroups or measured equivalence relations). Another is on generic behavior of stationary actions. Yet another is to apply the new techniques to the study of harmonic functions and random walks; topics deeply connected with stationary actions.