flows $ \mathbb{R}^{d} {\curvearrowright}X_{i} $, $ i = 1,2 $, and a *given* orbit equivalence $ \phi : \mathcal{C}_{1} \to \mathcal{C}_{2} $. The question is then whether $ \phi $ can be extended to a *time change equivalence* $ \phi : X_{1} \to X_{2} $. Since $ \phi $ is given on points from $ \mathcal{C}_{1} $, its global behavior is uniquely defined, and one can concentrate on the local aspect of the problem. This approach is already implicit in the aforementioned work of Feldman [@MR1113569]. Before stating our results, we need one more definition.

A Borel flow $ \mathbb{R}^{d} {\curvearrowright}X $ is said to be *compressible* if it admits no invariant Borel probability measures. An invariant Borel subset $ Z \subseteq X $ is *compressible* if the restriction of the flow onto $ Z $ is compressible. An invariant set $ Z \subseteq X $ is *cocompressible* if its complement is compressible.

The definition of compressibility given above is concise, but not very useful. The term “compressible” is explained by an important characterization due to M. G. Nadkarni [@MR1081705] (see also [@MR1425877 Theorem 4.3.1]) of the direct analog of this notion for countable equivalence relations.

Our first result in 