\[lem:small-perturbations-are-time-change-equivalent\] above.

\[lem:uniform-rokhlin\] For any free Borel flow $ \mathbb{R}^{d} {\curvearrowright}X $ and any rational grid $ Q \subseteq X $ there exist a Borel cocompressible invariant set $ Z \subseteq X $, a sequence of Borel cross sections $ \mathcal{C}_{n} \subseteq Z $, and an increasing sequence of positive rationals $ (l_{n})_{n=1}^{\infty} $ such that for rectangles $ R_{n}= [-l_{n}, l_{n}]^{d} $ one has:

(i) $ \lim_{n \to \infty} l_{n} = \infty $;

(ii) $ Z = \bigcup_{n} \bigl( \mathcal{C}_{n} + R_{n} \bigr) $;

(iii) $ (c + R_{n}) \cap (c' + R_{n}) = {\varnothing}$ for all distinct $ c, c' \in R_{n} $;

(iv) $ \mathcal{C}_{n} + R_{n} \subseteq \mathcal{C}_{n+1} + R_{n+1}^{{\leftarrow}1} $, where $ R_{n+1}^{{\leftarrow}1} $ is obtained by shrinking the square $ R_{n+1} $ by $ 1 $ in every direction: $$R_{n+1}^{{\leftarrow}1} = [-l_{n+1} + 1, l_{n+1}-1]^{d};$$

(v) \[item:on-the-grid\] $ \mathcal{C}_{n} \subseteq Q $.

For any flow $ \mathbb{R}^{d} {\curvearrowright}X $, we let $ \mathcal{E}(X) $ to denote the Borel space of invariant ergodic probability measures on $ X $. The construction of the time change equivalence given in Section \[sec:back-forth-constr\] would be easier if performed relative to a fixed ergodic measure on $ X $. 