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Abstract

For a homeomorphism $T \colon X \to X$ of a Cantor set $X$, the mapping class group $\mathcal {M}(T)$ is the group of isotopy classes of orientation-preserving self-homeomorphisms of the suspension $\Sigma _{T}X$. The group $\mathcal {M}(T)$ can be interpreted as the symmetry group of the system $(X,T)$ with respect to the flow equivalence relation. We study $\mathcal {M}(T)$, focusing on the case when $(X,T)$ is a minimal subshift. We show that when $(X,T)$ is a subshift associated to a substitution, the group $\mathcal {M}(T)$ is an extension of $\mathbb {Z}$ by a finite group; for a large class of substitutions including Pisot type, this finite group is a quotient of the automorphism group of $(X,T)$. When $(X,T)$ is a minimal subshift of linear complexity satisfying a no-infinitesimals condition, we show that $\mathcal {M}(T)$ is virtually abelian. We also show that when $(X,T)$ is minimal, $\mathcal {M}(T)$ embeds into the Picard group of the crossed product algebra $C(X) \rtimes _{T} \mathbb {Z}$.