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 $\textrm{Aut}(X)/\langle \sigma \rangle \xhookrightarrow{} \mathcal{M}(\sigma)$, where $\textrm{Aut}(X)$ 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 $\textrm{Aut}(X)/\langle \sigma \rangle$.

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.