equivalence between flows $ \mathbb{R}^{d} {\curvearrowright}X $ and $ \mathbb{R}^{d} {\curvearrowright}Y $ is global, as it requires different orbits to be mapped to different orbits. To be more specific, if a partial construction of $ \phi $ yields $ X \ni x \mapsto \phi(x) \in Y $ for some $ x \in X $, then no $ x' \in X $ such that $ \neg \bigl( x' \cber*[X] x \bigr) $ can be mapped into $ \bigl[ \phi(x) \bigr]_{\cber[Y]} $. In this sense, before defining $ \phi(x') $, any $ x' $ needs to know something about points $ x \in X $ from other orbits. On the other hand, the property for an orbit equivalence $ \phi : X \to Y $ to be a time change equivalence is purely local as it can be checked by looking at each orbit individually.

It is oftentimes beneficial (if nothing else for pedagogical reasons) to decouple whenever possible local and global aspects of a problem at hand. For various constructions of orbit equivalence this is achieved via the notion of a cross section.

Let $ \mathbb{R}^{d} {\curvearrowright}X $ be a Borel flow. A *cross section* for the flow is a Borel 