Additivity properties of sofic entropy and measures on model spaces

TL;DR

This paper investigates the additivity properties of sofic entropy for Cartesian products of systems, introducing a new entropy notion to establish bounds and conditions for when sofic entropy is additive.

Contribution

It introduces a new entropy concept based on model spaces, providing bounds and conditions for the additivity of sofic entropy in product systems.

Findings

Established a lower bound for sofic entropy of products

Proved the lower bound is optimal under certain conditions

Derived sufficient conditions for strict additivity of sofic entropy

Abstract

Sofic entropy is an invariant for probability-preserving actions of sofic groups. It was introduced a few years ago by Lewis Bowen, and shown to extend the classical Kolmogorov-Sinai entropy from the setting of amenable groups. Some parts of Kolmogorov-Sinai entropy theory generalize to sofic entropy, but in other respects this new invariant behaves less regularly. This paper explores conditions under which sofic entropy is additive for Cartesian products of systems. It is always subadditive, but the reverse inequality can fail. We define a new entropy-notion in terms of probability distributions on the spaces of good models of an action. Using this, we prove a general lower bound for the sofic entropy of a Cartesian product in terms of separate quantities for the two factor systems involved. We also prove that this lower bound is optimal in a certain sense, and use it to derive some sufficient conditions for the strict additivity of sofic entropy itself. Various other properties of this new entropy notion are also developed.