Smooth Orbit Equivalence (SOE) is an orbit equivalence relation between free ℝᵈ-flows that acts by diffeomorphisms between orbits. This idea originated in ergodic theory of ℝ-flows under the name of time-change equivalence, where it is closely connected with the concept of Kakutani equivalence of induced transformations. When viewed from the ergodic theoretical viewpoint, SOE has a rich structure in dimension one, but, as discovered by Rudolph, all ergodic measure-preserving ℝᵈ-flows, d > 1, are SOE.
Miller and Rosendal initiated the study of this concept from the point of view of descriptive set theory, where phase spaces of flows aren't endowed with any measures. This significantly enlarges the class of potential orbit equivalences, and they proved that all nontrivial free Borel ℝ-flows are SOE. They posed a question of whether the same remains to be true in dimension d>1. In this talk, we answer their question in the affirmative, and show that all nontrivial Borel ℝᵈ-flows are SOE.
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