$. While concrete identification depends on the choice of $ x $, any translation invariant structure on $ \mathbb{R}^{d} $ can be transferred onto orbits of such an action unambiguously. Time change equivalence between free flows is defined as an orbit equivalence that preserves the topology on every orbit.

Let $ \mathbb{R}^{d} {\curvearrowright}X $ and $ \mathbb{R}^{d} {\curvearrowright}Y $ be free Borel flows. An orbit equivalence $ \phi : X \to Y $ is said to be a *time change equivalence* if for any $ x \in X $ the map $ \xi(x, \,\cdot\,) : \mathbb{R}^{d} \to \mathbb{R}^{d} $ specified by $$\phi(x + \vec{r}) = \phi(x) + \xi(x, \vec{r})$$ is a homeomorphism[^1]-4pt.

The concept of time change equivalence has been studied quite extensively in ergodic theory, where the set up differs from the one of Borel dynamics in the following aspects. In ergodic theory phase spaces are assumed to be endowed with probability measures, which flows are required to preserve (or to “quasi-preserve”, i.e., to preserve the null sets). Moreover, all conditions of interest may hold only up to a set of measure zero as opposed to holding everywhere. In these regards ergodic theory is less restrictive than Borel dynamics. 