Full Groups and Orbit Equivalence in Cantor Dynamics

Abstract

In this note we consider dynamical systems $(X,G)$ on a Cantor set $X$ satisfying some mild technical conditions. The considered class includes, in particular, minimal and transitive aperiodic systems. We prove that two such systems $(X_1,G_1)$ and $(X_2,G_2)$ are orbit equivalent if and only if their full groups are isomorphic as abstract groups. This result is a topological version of the well-known Dye's theorem established originally for ergodic measure-preserving actions.

8 pages, references added

Cited by in corpus (10)

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• Topological full groups of one-sided shifts of finite type
• Topological Full Groups of Ample Groupoids with Applications to Graph Algebras
• Cantor systems and quasi-isometry of groups
• Finitely presented groups associated with expanding maps
• Rigidity properties of full groups of pseudogroups over the Cantor set
• On the centralizers of minimal aperiodic actions on the Cantor set
• Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces
• Strong Orbit Equivalence in Cantor dynamics and simple locally finite groups
• Full groups of minimal homeomorphisms