Dye's theorem in the almost continuous category

TL;DR

This paper establishes an almost continuous version of Dye's theorem, showing that non-atomic measure-preserving homeomorphisms of Polish spaces are orbit equivalent via a map that is continuous on a full measure subset, extending recent results.

Contribution

It introduces an almost continuous orbit equivalence framework for measure-preserving homeomorphisms, including cases with infinite invariant measures, broadening the scope of Dye's theorem.

Findings

Proves almost continuous orbit equivalence for non-atomic measures

Extends results to infinite invariant measures

Provides a framework for orbit equivalence with full measure subsets

Abstract

We prove an almost continuous version of Dye's theorem: any two non-atomic probability measure preserving homeomorphisms of Polish spaces are almost continuously orbit equivalent. More precisely they are orbit equivalent by a map which is defined and continuous on a Polish subset of full measure with an inverse satisfying the same conditions. This result includes all of the recent results on almost continuous orbit equivalence. We also deal with the case of infinite invariant measures.