# On the entropies of subshifts of finite type on countable amenable groups

## TL;DR

This paper explores the entropies of subshifts of finite type on countable amenable groups, introducing group charts to embed subshifts and characterizing entropy sets across various groups, with implications for computability and group actions.

## Contribution

It introduces the concept of group charts for embedding subshifts, derives an entropy addition formula, and characterizes entropy sets for SFTs on certain groups, advancing understanding of entropy in symbolic dynamics.

## Key findings

- Entropy of H-SFTs is contained in G-SFTs entropy set within an additive constant.
- On groups with decidable word problem and a translation-like Z^2 action, entropy set equals non-negative upper semi-computable reals.
- Complete characterization of SFT entropies in several group classes achieved.

## Abstract

Let $G,H$ be two countable amenable groups. We introduce the notion of group charts, which gives us a tool to embed an arbitrary $H$-subshift into a $G$-subshift. Using an entropy addition formula derived from this formalism we prove that whenever $H$ is finitely presented and admits a subshift of finite type (SFT) on which $H$ acts freely, then the set of real numbers attained as topological entropies of $H$-SFTs is contained in the set of topological entropies of $G$-SFTs modulo an arbitrarily small additive constant for any finitely generated group $G$ which admits a translation-like action of $H$. In particular, we show that the set of topological entropies of $G$-SFTs on any such group which has decidable word problem and admits a translation-like action of $\mathbb{Z}^2$ coincides with the set of non-negative upper semi-computable real numbers. We use this result to give a complete characterization of the entropies of SFTs in several classes of groups. Corrigendum: An error has been found in the proof of Theorem 4.7. We have added a corrigendum appendix which explains the error, discusses possible solutions and details which results from Section 5 still hold (the only result that is no longer proven is Corollary 5.12). We also provide an update on the state of the art concerning the questions asked in Section 6.