Let $X$ be a finite set, $(X^{\mathbb Z}, T)$ is the shift, i.e. the Tikhonov topological space of all biinfinite words in $X$, $T$ shifts the words one letter to the right. A subshift is a closed subset of $X^{\mathbb{Z}}$ stable under $T$.
Is there a recent survey about the problem of equivalence of subshifts?



The paper Open Problems in Symbolic Dynamics by Mike Boyle discusses the conjugacy problem for shifts of finite type and sofic shifts. More details can be found in books such as An Introduction to Symbolic Dynamics and Coding. 

