Smooth orbit equivalence of multidimensional Borel flows

TL;DR

This paper proves that all free non-tame Borel $ ^d$-flows are smoothly equivalent for dimensions $d geq 2$, establishing a universal classification in this setting.

Contribution

It demonstrates that in dimensions $d geq 2$, all free non-tame Borel $ ^d$-flows are smoothly equivalent, resolving a question posed by Miller and Rosendal.

Findings

All free non-tame Borel $ ^d$-flows are smoothly equivalent for $d geq 2$

Answers a longstanding open question in the field

Provides a classification result for Borel flows in higher dimensions

Abstract

Free Borel $\mathbb{R}^{d}$-flows are smoothly equivalent if there is a Borel bijection between the phase spaces that maps orbits onto orbits and is a $C^{\infty}$-smooth orientation preserving diffeomorphism between orbits. We show that all free non-tame Borel $\mathbb{R}^{d}$-flows are smoothly equivalent in every dimension $d \ge 2$. This answers a question of B. Miller and C. Rosendal.