Regular cross sections of Borel flows

TL;DR

This paper demonstrates that all free Borel flows can be simplified to cross sections with only two distances between points and characterizes non-smooth flows by their invariant measures for Lebesgue orbit equivalence.

Contribution

It introduces a method to obtain cross sections with two distances and characterizes non-smooth flows via invariant measures for Lebesgue orbit equivalence.

Findings

Existence of cross sections with two distances for all free Borel flows

Non-smooth flows are Lebesgue orbit equivalent iff they share the same number of invariant ergodic measures

Provides a classification criterion for non-smooth flows based on invariant measures

Abstract

Any free Borel flow is shown to admit a cross section with only two possible distances between adjacent points. Non smooth flows are proved to be Lebesgue orbit equivalent if and only if they admit the same number of invariant ergodic probability measures.