# Meeting Details

Title: The new (and wonderful) notion of entropy for actions of higher rank abelian groups, and its connections to slow entropy and rigidity Center for Dynamics and Geometry Seminars Anatole Katok, Penn State Since ordinary measure-theoretic entropy for a smooth measure preserving action of any countable group other that cyclic or its finite extension vanish, alternative notions of entropy for such actions are of interest. In particular, for a smooth action of Z^k slow entropy based on the scale function n^{1/k} provides a proper normalization and gives a first cut into zero entropy'' and positive entropy'' actions. However, in order to ascribe a numerical value to slow entropy one needs to fix a norm on the acting group and this (unlike fixing a volume element'' ) is somewhat arbitrary. In this talk I will discuss a new and natural notion of average entropy that is equal to the inverse of the volume of the unit ball in the entropy norm. Thus is it positive if and only if all non-identity elements of the action have positive entropy. Average entropy is equal to the infimum of the values of the n^{1/k} slow entropy over all norms on the acting group normalized to the volume element. If it is positive, the infimum is achieved at the entropy norm. A corollary of strong rigidity results that are discussed in Federico Rodriguez Hertz's talk is that for the maximal rank actions (where the rank is at least two and dimension is greater then rank by one), the average entropy is either equal to zero or is bounded from below by a positive number that depends only on the dimension. Conjecturally this bound is uniform in dimension. This is a joint work in progress with Federico Rodriguez Hertz.