New directions in entropy theory
In recent years, the classical theory of entropy for a dynamical system has been revolutionized by the ground-breaking work of several researchers. Two definitions were proposed and developed for actions of general groups : sofic entropy (initiated by L. Bowen) and Rokhlin entropy (initiated by B. Seward). We will start with a very brief account of the latter, and then describe our own recently developed approach to entropy theory for free probability-measure-preserving actions of all countable groups. We will then formulate our main result, namely that Rokhlin entropy satisfies a Shannon-McMillan-Breiman pointwise convergence theorem. We will demonstrate the geometric significance of this convergence theorem in the case of actions of free non-Abelian groups
Based on joint work with F. Pogorzelski (Leipzig University).