Given a subshift $(X,\sigma )$, the mapping class group $\mathcal{M}(\sigma )$ is the group of self-flow equivalences of $(X,\sigma )$, up to isotopy. For a minimal shift, there is an embedding $\text{Aut}(X)/⟨\sigma ⟩\text{xhookrightarrow}\mathcal{M}(\sigma )$, where $\text{Aut}(X)$ is the group of automorphisms.

If $(X,\sigma )$ is conjugate to a primitive substitutive shift, then $\mathcal{M}(\sigma )$ is a finite extension of $\mathbb{Z}$, and under mild conditions, this finite group is precisely $\text{Aut}(X)/⟨\sigma ⟩$.

We discuss more the general case when $(X,\sigma )$ is a minimal subshift of linear complexity, subject to a technical condition, and give some examples.

This is joint work with Scott Schmieding.