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.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2010
- DOI:
- 10.48550/arXiv.1006.1145
- arXiv:
- arXiv:1006.1145
- Bibcode:
- 2010arXiv1006.1145M
- Keywords:
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- Mathematics - Dynamical Systems;
- 37B05
- E-Print:
- 8 pages, references added