set $ \mathcal{C} \subseteq X $ that intersects every orbit in a non-empty lacunary[^4] set, i.e., $ \mathcal{C} \cap [x]_{\cber[X]} \ne {\varnothing}$ for all $ x \in X $, and there is a non-empty neighborhood of the identity $ U \subseteq \mathbb{R}^{d} $ such that $ (x + U) \cap \mathcal{C}  = \{x\} $ for all $ x \in \mathcal{C} $. When one wants to specify $ U $ explicitly, $ \mathcal{C} $ is called $ U $-lacunary.

One can think of a cross section as being a discrete version of the ambient equivalence relation. A theorem of A. S. Kechris [@MR1176624] shows that all Borel flows admit cross sections. The notion of a cross section can be further strengthened by requiring cocompactness.

A cross section $ \mathcal{C} $ for a flow $ \mathbb{R}^{d} {\curvearrowright}X $ is *cocompact* if there exists a compact set $ K \subseteq \mathbb{R}^{d} $ such that $ \mathcal{C} + K = X $.

C. Conley proved existence of cocompact cross sections for all Borel flows (see, for instance, [@2015arXiv150400958S Section 2]). Our approach to separate local and global aspects of time change equivalence starts with cocompact cross sections $ \mathcal{C}_{i} \subseteq X_{i} $ for two 