The Mapping Class Group of a Minimal Subshift

TL;DR

This paper investigates the structure of the mapping class group of minimal subshifts, revealing its extensions, abelian properties, and embeddings into algebraic structures, thereby advancing understanding of symmetries in symbolic dynamical systems.

Contribution

It characterizes the mapping class group for minimal subshifts, especially those from substitutions, and establishes its algebraic properties and connections to crossed product algebras.

Findings

For substitution subshifts, the group is an extension of Z by a finite group.

For certain minimal subshifts, the group is virtually abelian.

The group embeds into the Picard group of the associated crossed product algebra.

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 $C(X) \rtimes_{T} \mathbb{Z}$.