The mapping class group of a minimal subshift

For a homeomorphism T: X → X of a Cantor set X, the mapping class group M(T ) is the group of isotopy classes of orientation-preserving self-homeomorphisms of the suspension Σ_T X. The group M(T ) can be interpreted as the symmetry group of the system (X, T ) with respect to the flow equivalence relation. We study 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 M(T ) is an extension of 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 M(T ) is virtually abelian. We also show that when (X, T ) is minimal, M(T ) embeds into the Picard group of the crossed product algebra C(X) ⋊_T Z.