---
abstract: 'This paper addresses the notion of time change equivalence for Borel $ \mathbb{R}^{d} $-flows. We show that all free $ \mathbb{R}^{d} $-flows are time change equivalent up to a compressible set. An appropriate version of this result for non-free flows is also given.'
address: |
    Department of Mathematics, Statistics, and Computer Science\
    University of Illinois at Chicago\
    322 Science and Engineering Offices (M/C 249)\
    851 S Morgan Street\
    Chicago, IL 60607–7045\
    United States
author:
- Konstantin Slutsky
bibliography:
- 'refs.bib'
title: On time change equivalence of Borel flows
---

Introduction {#sec:introduction}
============

A Borel flow $ {\mathfrak{F}}$ is a Borel measurable action of $ \mathbb{R}^{d} $ on a standard Borel space. The action will be denoted additively: $ x + \vec{r} $ is the action of $ \vec{r} \in \mathbb{R}^{d} $ upon the point $ x \in X $. With any flow $ {\mathfrak{F}}$ on $ X $ we associate an equivalence relation $ \cber[X][{\mathfrak{F}}] $ given by $ x \cber*[X][{\mathfrak{F}}] y $ whenever there is $ \vec{r} \in \mathbb{R}^{d} $ such that $ x + \vec{r} = y $. An equivalence class of $ x \in X $ will be denoted by $ [ x ]_{\cber[X][{\mathfrak{F}}]} $. 