Event № 328
The concept of measurable entropy goes back to Kolmogorov and Sinai who in the late 50ies defined an isomorphism invariant for measure preserving Z-actions. While a similar theory can be developed in an analogous manner for abelian or even amenable groups, the situation gets far more complicated when dealing with groups which are "very" non-commutative, such as free groups. We start the talk with a warm-up about the classical Kolmogorov-Sinai entropy. Using the free group on two generators as an illustrative example, we show how to define cocycle entropy as a new isomorphism invariant for measure preserving actions of quite general countable groups. Further, we draw connections to other notions of entropy and to open problems in the field. We conclude the talk by clarifying pointwise almost sure approximation of cocycle entropy values. To this end, we present a first Shannon-McMillan-Breiman theorem for actions of non-amenable groups. Joint work with Amos Nevo.