, $ n \in \mathbb{N} $, and rationals $ l_{i,n} \in \mathbb{Q} $ such that for the squares $ R_{i,n} = [-l_{i,n}, l_{i,n}]^{d} $ , $ i=1,2 $, one has

1.  \[item:on-rational-grid\] cross sections $ \mathcal{C}_{i,n} \subseteq Q_{i} $ are on the rational grids;

2.  \[item:exhaust-orbits\] $ \tilde{Z}_{i} = \bigcup_{n} \bigl( \mathcal{C}_{i,n} + R_{i,n}\bigr) $;

3.  boxes $ c + R_{i,n} $ are pairwise disjoint;

4.  \[item:nested\] $ \mathcal{C}_{i,n} + R_{i,n} \subseteq \mathcal{C}_{i,n+1} +
        R_{i,n+1}^{{\leftarrow}1} $;

5.  \[item:sizes-increase\] sequences $ (l_{i,n})_{n \in \mathbb{N}} $ are increasing and unbounded.

Since cross sections $ \mathcal{D}_{i} $ are cocompact, we may omit, if necessary, finitely many cross sections $ \mathcal{C}_{i,n} $ and assume without loss of generality that $ l_{i,n} $ are so large that $ \mathcal{D}_{i} \cap (c + R_{i,n}) \ne {\varnothing}$ for all $ c \in \mathcal{C}_{i,n} $. We shall further decrease sets $ \tilde{Z}_{i} $ by throwing away invariant compressible sets, so for notational convenience we assume that $ \tilde{Z}_{i} = X_{i} $.

For each of $ X_{i} $ we pick an ergodic decomposition $ x \mapsto \mu_{x} $ as in Lemma \[lem:ergodic-decomposition\]. We also let $ \fber[n][i] $ to denote finite Borel equivalence relations on $ \mathcal{D}_{i} \cap \bigl(\mathcal{C}_{i,n} 