Consider an Anosov diffeomorphism $T: M \rightarrow M$ and a corresponding Markov partition $\mathcal{R}$ of $M$. For $x \in M$, let $\mathcal{R}(x)$ denote the element of $\mathcal{R}$ containing $x$ and let $t_\infty^{(\mathcal{R})}(x)$ be the first return to $\mathcal{R}(x)$, i.e.

$t_\infty^{(\mathcal{R})}(x) := \inf \{n: \exists m \mbox{ s.t. } \left( 0 < m < n \land T^mx \notin \mathcal{R}(x) \right) \land T^nx \in \mathcal{R}(x)\}$.

What, if anything, is already known about this quantity? (I am not interested in similar functions [unless perhaps only the first exit requirement is dropped], but only this one; it is of particular interest from the point of view of statistical physics.) Any theorems, references, etc. would be helpful. In particular, I would be interested to know if there are results demonstrating some sort of local product structure w/r/t the expanding and contracting directions of $T$.