Sofic entropy, after Lewis Bowen, David Kerr and Hanfeng Li

TL;DR

This paper reviews the development of sofic entropy, a generalization of classical entropy for dynamical systems, highlighting key contributions by Bowen, Kerr, and Li in extending entropy concepts to non-amenable groups.

Contribution

It summarizes the foundational work and recent advances in sofic entropy, including Bowen's introduction and Kerr and Li's topological and variational principles.

Findings

Sofic entropy extends classical entropy to non-amenable groups.

Bowen's work enabled entropy analysis for sofic group actions.

Kerr and Li established a topological version with a variational principle.

Abstract

The entropy in dynamical systems was introduced by A. Kolmogorov. Initially dedicated to iterations of one finite measure preserving transformation, the notion was gradually generalized so as to encompass amenable group actions and topological actions. L. Bowen (2008) succeeded in breaking the non-amenable frontier by introducing the sofic entropy. This invariant provides the same services as the classical entropy for the measured actions of the sofic groups (a class which contains the residually finite groups). In 2010, D. Kerr et H. Li established a topological version together with a variational principle.