# Subshifts (of finite type) and the homoclinic relation

Let $$A$$ be a finite alphabet (just some finite set), $$G$$ a group and $$X\subseteq A^G$$ some closed subshift, i.e. a closed subset of $$A^G$$ invariant with respect to the shift action of $$G$$. For $$x,y\in X$$ denote by $$\Delta(x,y)$$ the set where their supports differ, i.e. $$\Delta(x,y):=\{g\in G\colon x(g)\neq y(g)\}$$. Finally, let $$\sim$$ denote the homoclinic equivalence relation on $$X$$. That is, $$x\sim y$$ if $$\Delta(x,y)$$ is finite.

When $$X$$ is the full shift, $$x\sim y\in X$$ and $$|\Delta(x,y)|=n$$, then we can find a sequence $$x_0=x,x_1,\ldots,x_n=y\in X$$ such that $$|\Delta(x_i,x_{i+1})|=1$$, for every $$i. That is, we can find a path from $$x$$ to $$y$$ where on each step we change the configuration just on one element.

This is unlikely to work on more general subshifts. I wonder if there is some (reasonable) class of subshifts (probably necessarily of finite type) such that there is a constant $$K$$ so that for every $$x\sim y\in X$$ we can find a sequence $$x_0=x,x_1,\ldots,x_n=y\in X$$ such that $$|\Delta(x_i,x_{i+1})|\leq K$$, for every $$i. I was thinking about irreducible subshifts of finite type, without success though.

If $$X$$ is a group shift (meaning that $$A$$ is a finite group and $$X$$ happens to be a subgroup of $$A^G$$) and the subgroup $$\Delta_X \subset X$$ of homoclinic points is generated by translates of a finite set $$S \subset X$$ of configurations in the sense that $$\Delta_X = \langle \{ g \cdot x : g \in G, x \in S \} \rangle, \tag{1}$$ then $$X$$ has the desired property. This is because $$x \sim y$$ is equivalent to $$x^{-1} y \in \Delta_X$$, which is a product of translates of elements of $$S$$. If $$G$$ is polycyclic, then every group shift $$X \subset A^G$$ is necessarily of finite type and has a finite set $$S$$ with (1).
• Thanks! If I under correctly and I want to find such a subshift, then for groups $G$ and $A$ I choose some finite set $S$ of finitely supported elements of $A^G$. I let $\Delta_S$ to be the group $\langle\{g\cdot x: g\in G, x\in S\}\rangle$ and let $X$ to be the closure of $\Delta_S$. $X$ is then a closed subshift whose $\sim$-classes are determined by $\Delta_S$. Is there any (algorithmic) way to see if $X$ is of finite type for general groups $A$ and $G$? If $G$ is a free group? Jan 3, 2020 at 10:33