given by $ \xi(x + \vec{r}) = x $ for all $ x \in \mathcal{C} $ and $ \vec{r} \in B(\delta) $. The required time change equivalence $ \psi : X \to X $ is defined by the formula: $$\psi(x) =
    \begin{cases}
      x & \textrm{if } x \not \in \mathcal{C} + B(\delta),\\
      \xi(x) + f\bigl(\rho\bigl(\xi(x), \phi \circ \xi(x)\bigr), \rho\bigl(\xi(x),x\bigr)\bigr) &
      \textrm{otherwise}.\\
    \end{cases}$$ The somewhat cryptic definition of $ \psi(x) $ is really simple: within a ball $ c + B(\delta) $, $ c \in \mathcal{C} $, we apply the diffeomorphism $ f(\vec{r}, \, \cdot \,) $, $ \vec{r} = \rho(c, \phi(c)) $, ensuring that $ \psi(c) = \phi(c) $. By assumption $ \mathcal{C} $ is $ B(\delta) $-lacunary, and therefore $ \psi $ is injective, and hence is a time change equivalence.

One of the primary tools to construct orbit equivalences is Rokhlin’s Lemma. The following provides a concrete form that we are going to use. The statement essentially coincides with that of Theorem 6.3 in [@2015arXiv150400958S] with addition of item , which asserts that cross section $ \mathcal{C}_{n} $ can be taken to be on the given grid $ Q $. This modification is straightforward in view of Lemma 