Galois Theory for Subshifts of Finite Type and Representations of Automorphism Groups

Abstract

The purpose of this paper is to constructively develop a Galois theory on irreducible shifts of finite type (SFTs) and to analyze the automorphism groups of SFTs using this framework. Let $X$ and $Y$ be irreducible SFTs. We demonstrate that a factor map $Y \to X$ satisfying algebraic conditions exhibits Galois-theoretic behavior. Specifically, we prove that a Galois correspondence, analogous to those known in the Galois theory of fields and covering spaces in topology, holds for irreducible SFTs. Furthermore, we introduce the absolute Galois group and its cohomology as conjugacy invariants for irreducible SFTs. Finally, we construct a representation of the automorphism group of an irreducible SFT into this cohomology to extract information from the automorphism group, whose structure is not yet fully understood.