From invariant measures to orbit equivalence, via locally finite groups

TL;DR

This paper offers new proofs connecting invariant measures and orbit equivalence of minimal Cantor space homeomorphisms, utilizing locally finite groups and strengthening Krieger's theorem.

Contribution

It provides novel proofs of orbit equivalence characterization and enhances Krieger's theorem for locally finite group actions.

Findings

New proof of Giordano-Putnam-Skau theorem

Strengthened Krieger's theorem for locally finite groups

Unified approach to orbit equivalence and invariant measures

Abstract

We give a new proof of a theorem of Giordano, Putnam and Skau characterizing orbit equivalence of minimal homeomorphisms of the Cantor space in terms of their sets of invariant Borel probability measures. The proof is based on a strengtehning of a theorem of Krieger concerning minimal actions of certain locally finite groups of homeomorphisms, and we also give a new proof of the Giordano--Putnam--Skau characterization of orbit equivalence for these actions.