On Virtual Conjugacy of Generalized Odometers

Abstract

The paper is focused on the study of continuous orbit equivalence for generalized odometers (profinite actions). We show that two generalized odometers are continuously orbit equivalent if and only if the acting groups have finite index subgroups (having the same index) whose actions are piecewise conjugate. This result extends M.~Boyle's flip-conjugacy theorem originally established for $\mathbb{Z}$-actions. As a corollary we obtain a dynamical classification of the restricted isomorphism between generalized Bunce-Deddens $C^{\ast }$-algebras. We also show that the full group associated with a generalized odometer is amenable if and only if the acting group is amenable.

Proofs simplified, references added

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Cited by in corpus (8)

• Classifying matchbox manifolds
• Homology of odometers
• On tameness of almost automorphic dynamical systems for general groups
• Arboreal Cantor actions
• Prime spectrum and dynamics for nilpotent Cantor actions
• Amorphic complexity of group actions with applications to quasicrystals
• On Intermediate $C^{\ast }$-subalgebras of $C^{\ast }$-simple Group Actions
• The prime spectrum of solenoidal manifolds