Entropy of automorphisms of II_1-factors arising from the dynamical systems theory

TL;DR

This paper establishes a relationship between the entropy of automorphisms of II_1-factors derived from ergodic group actions and the classical measure-theoretic entropy, demonstrating their equality under certain conditions and constructing examples with prescribed entropies.

Contribution

It proves that the Connes-Stormer entropy of automorphisms extending measure-preserving transformations equals the Kolmogorov-Sinai entropy when they commute with the group action, and constructs automorphisms with arbitrary entropy values.

Findings

H(alpha_T)=h(T) when T commutes with G

Existence of automorphisms with prescribed entropy values

Extension of measure-preserving automorphisms to II_1-factors

Abstract

Let a countable amenable group G acts freely and ergodically on a Lebesgue space (X,mu), preserving the measure mu. If T is an automorphism of the equivalence relation defined by G then T can be extended to an automorphism alpha_T of the II_1-factor M=L^\infty(X,\mu)\rtimes G. We prove that if T commutes with the action of G then H(alpha_T)=h(T), where H(alpha_T) is the Connes- Stormer entropy of alpha_T, and h(T) is the Kolmogorov-Sinai entropy of T. We prove also that for given s and t, 0\le s\le t\le\infty, there exists a T such that h(T)=s and H(alpha_T)=t.