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

TL;DR

This paper investigates the distribution of entropies among subsystems of shifts of finite type and sofic shifts on countable amenable groups, showing that these entropies are densely distributed within certain intervals.

Contribution

It proves that for countable amenable groups, the entropies of SFT subsystems are dense in the interval from zero to the system's entropy, extending to sofic shifts and relative cases.

Findings

Entropies of SFT subsystems are dense in [0, h(X)] for positive entropy shifts.

The result extends to sofic shifts, not just SFTs.

For subshifts Y with lower entropy, SFTs between Y and X have entropies dense in [h(Y), h(X)].

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.