What is known about first return times to Markov partitions for Anosov diffeomorphisms?

Question

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 $T$.

Answer 1

My best recommendation is that you have a look at Barreira's book Dimension and Recurrence in Hyperbolic Dynamics, also for references. It seems to contain the most complete account of the relation between quantitative recurrence (along the approach starting with Boshernitzan, as well as with Ornstein and Weiss although in a different setting) and hyperbolic dynamics. Reading your question above, I regret that I cannot understand why you say that what Barreira and Saussol proved is not what you ask. Perhaps then you should be clearer.

In any event it is really difficult to expect that the product structure of recurrence, as described in your comment, could have generalizations to dynamics with a weaker hyperbolicity, the reason being that all seems to get together from various areas to be able to establish such a relation (this includes decay of correlations, difficulty of estimating returns times when passing from balls to rectangles of the Markov partition, even finite Markov partitions, etc).