Symbolic Extensions of Amenable Group Actions and the Comparison Property

In topological dynamics, the Symbolic Extension Entropy Theorem (SEET) (Boyle and Downarowicz, 2004) describes the possibility of a lossless digitalization of a dynamical system by extending it to a subshift on finitely many symbols. The theorem gives a precise estimate on the entropy of such a symbolic extension (and hence on the necessary number of symbols). Unlike in the measure-theoretic case, where Kolmogorov–Sinai entropy serves as an estimate in an analogous problem, in the topological setup the task reaches beyond the classical theory of measure-theoretic and topological entropy. Necessary are tools from an extended theory of entropy, the theory of entropy structures developed in Downarowicz (2005). The main goal of this paper is to prove the analog of the SEET for actions of (discrete infinite) countable amenable groups: Let a countable amenable group G G act by homeomorphisms on a compact metric space X X and let M G ( X ) \mathcal {M}_{G}(X) denote the simplex of all G G -invariant Borel probability measures on X X . A function E A {E}_{\mathsf {A}} on M G ( X ) \mathcal {M}_{G}(X) equals the extension entropy function h π h^\pi of a symbolic extension π : ( Y , G ) → ( X , G ) \pi :(Y,G)\to (X,G) , where h π ( μ ) = sup { h ν ( Y , G ) : ν ∈ π − 1 ( μ ) } h^\pi (\mu )=\sup \{h_\nu (Y,G): \nu \in \pi ^{-1}(\mu )\} ( μ ∈ M G ( X ) \mu \in \mathcal {M}_{G}(X) ), if and only if E A {E}_{\mathsf {A}} is a finite affine superenvelope of the entropy structure of ( X , G ) (X,G) . Of course, the statement is preceded by the presentation of the concepts of an entropy structure and its superenvelopes, adapted from the case of Z \mathbb {Z} -actions. In full generality we are able to 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 s