$-flows, and the latter theorem establishes an analog of Theorem \[thm:extending-cross-section-to-time-change\] in this more general context.

Finally, the last section, contains a remark on the difference between time change equivalence and Lebesgue orbit equivalence, which is defined as an orbit equivalence that preserves the Lebesgue measure between orbits. It illustrates the high complexity of Lebesgue orbit equivalence even in the simplest case of periodic $ \mathbb{R} $-flows.

Rational Grids {#sec:rational-grids}
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This section provides some technical constructions that will be used in Section \[sec:back-forth-constr\]. The main concept here is that of a rational grid which will provide a rigorous justification for why the back-and-forth method in Section \[sec:back-forth-constr\] can be performed in a Borel way.

Let $ \mathbb{R}^{d} {\curvearrowright}X $ be a Borel flow. A *spiral of cross sections* $ (\mathcal{C}_{n}, h_{n}) $, $n \in \mathbb{N} $, is a sequence of cross sections $ \mathcal{C}_{n} $ together with Borel maps $ h_{n} : \mathcal{C}_{n} \to \mathbb{R}^{d} $ such that $ \mathcal{C}_{n+1} = \mathcal{C}_{n} + h_{n} $ for all $ n \in \mathbb{N} $, i.e., $$\mathcal{C}_{n+1} = \{ x + h_{n}(x) : x \in
   \mathcal{C}_{n}\}.$$ With a spiral of cross sections we associate *homomorphism* maps $$\phi_{n,n+1} : \mathcal{C}_{n} \to \mathcal{C}_{n+1}\quad \textrm{given 