Smooth orbit equivalence of multidimensional Borel flows

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.