| Résume | We present an overview of the theory of orbit equivalence relations of Borel flows, (i.e. free Borel actions of the Euclidean space). While more familiar in the framework of countable group actions, orbit equivalence is an important tool in understanding the structure of $\mathbb{R}^n$ actions just as well. We will survey a number of related results including:
- the classification of Borel flows up to Lebesgue Orbit Equivalence
(which can be viewed as the analog of Dougherty--Jackson--Kechris classification of hyperfinite equivalence relations);
- connections of this classification to Rudolph's theorem about regular cross sections;
- Topological Orbit Equivalence, including the Miller--Rosendal theorem on time-change equivalence. |
