Subsystem entropies of shifts of finite type and sofic shifts on countable amenable groups

Abstract

Abstract In this work, we study the entropies of subsystems of shifts of finite type (SFTs) and sofic shifts on countable amenable groups. We prove that for any countable amenable group G, if X is a G-SFT with positive topological entropy $h(X)> 0$ , then the entropies of the SFT subsystems of X are dense in the interval $[0, h(X)]$ . In fact, we prove a ‘relative’ version of the same result: if X is a G-SFT and $Y \subset X$ is a subshift such that $h(Y) < h(X)$ , then the entropies of the SFTs Z for which $Y \subset Z \subset X$ are dense in $[h(Y), h(X)]$ . We also establish analogous results for sofic G-shifts.