A topological dynamical system with two different positive sofic entropies

TL;DR

This paper constructs a mixing subshift of finite type that demonstrates the dependence of sofic entropy on the choice of sofic approximation, revealing new complexity in dynamical systems with sofic groups.

Contribution

It provides the first explicit example of a mixing subshift with two different positive sofic entropies, highlighting the non-uniqueness of sofic entropy.

Findings

Demonstrates dependence of sofic entropy on approximation choice

Provides explicit example of a mixing subshift with two positive entropies

Connects statistical physics models to dynamical systems theory

Abstract

A sofic approximation to a countable group is a sequence of partial actions on finite sets that asymptotically approximates the action of the group on itself by left-translations. A group is sofic if it admits a sofic approximation. Sofic entropy theory is a generalization of classical entropy theory in dynamics to actions by sofic groups. However, the sofic entropy of an action may depend on a choice of sofic approximation. All previously known examples showing this dependence rely on degenerate behavior. This paper exhibits an explicit example of a mixing subshift of finite type with two different positive sofic entropies. The example is inspired by statistical physics literature on 2-colorings of random hyper-graphs.