Symbolic extensions of amenable group actions and the comparison property

Abstract

Symbolic Extension Entropy Theorem (SEET) describes the possibility of a lossless digitalization of a dynamical system by extending it to a subshift. It gives an estimate on the entropy of symbolic extensions (and the necessary number of symbols). Unlike in the measure-theoretic case, where Kolmogorov--Sinai entropy is the estimate, in the topological setup the task reaches beyond the classical theory of entropy. Tools from an extended theory of entropy structures are needed. The main goal of this paper is to prove the SEET for actions of countable amenable groups: Let a countable amenable group $G$ act by homeomorphisms on a compact metric space $X$ and let $\mathcal M_G(X)$ denote the simplex of $G$-invariant probability measures on $X$. A function $E $ on $\mathcal M_G(X)$ equals the extension entropy function $h^π$ of a symbolic extension $π:(Y,G)\to (X,G)$, where $h^π(μ)=\sup\{h_ν(Y,G): ν\inπ^{-1}(μ)\}$ ($μ\in\mathcal M_G(X)$), if and only if $E $ is an affine superenvelope of the entropy structure of $(X,G)$. The statement is preceded by presentation of the concepts of an entropy structure and superenvelopes, adapted from $\mathbb Z$-actions. In full generality we prove a slightly weaker version of SEET, in which symbolic extensions are replaced by quasi-symbolic extensions, i.e., extensions in form of a joining of a subshift with a zero-entropy tiling system. The notion of a tiling system is a subject of earlier works and in this paper we review and complement the theory developed there. The full version of the SEET is proved for groups which are either residually finite or enjoy the comparison property. In order to describe the range of our theorem, we devote a large portion of the paper to the comparison property. Our main result in this aspect shows that all subexponential groups have the comparison property (and thus satisfy the SEET).