An *orbit equivalence* between two flows $ \mathbb{R}^{d} {\curvearrowright}X $ and $ \mathbb{R}^{d} {\curvearrowright}Y $ is a Borel bijection $ \phi : X \to Y $ such that for all $ x, y \in X$ $$x \cber*[X] y \iff \phi(x) \cber*[Y] \phi(y).$$

The notion of orbit equivalence is particularly suited for actions of discrete groups, but it tends to trivialize for certain locally compact groups. For instance, it is known that all non smooth free $ \mathbb{R}^{d} $-flows are orbit equivalent. To remedy this collapse, one often considers various strengthenings of orbit equivalence, usually by imposing “local” restrictions on the orbit equivalence maps.

Given any orbit of a *free* action of $ \mathbb{R}^{d} $, there is a bijective correspondence between points of the orbit and elements of $ \mathbb{R}^{d} $. More precisely, with an equivalence relation $ \cber[X] $ arising from a free flow $ \mathbb{R}^{d} {\curvearrowright}X $ one may associate a *cocycle* map $ \rho : \cber[X] \to \mathbb{R}^{d} $ which is defined by the condition $$x + \rho(x,y) = y \quad \textrm{for all } x \cber*[X] y.$$ The map $ \rho(x, \, \cdot \,) $ establishes a bijection between the orbit of $ x $ and $ \mathbb{R}^{d} 