Full Groups and Orbit Equivalence in Cantor Dynamics

Details Full Groups and Orbit Equivalence in Cantor Dynamics Full Groups and Orbit Equivalence in Cantor Dynamics

2010-06-06

arxiv.org/abs/1006.1145v2

Mathematics

Dynamical Systems

8 pages, references added

Scientific paper

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.

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