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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 $T$.

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Hi Steve, maybe you already know about: AN INTRODUCTION TO QUANTITATIVE POINCARE RECURRENCE IN DYNAMICAL SYSTEMS - Benoit Saussol Reviews in Mathematical Physics Vol. 21, No. 8 (2009) 949–979. It is not "difeo paper", but since you have markov partitions perhaps you can find a way to translate the results. – Leandro Feb 23 '10 at 5:41
@Leandro: I realize now that I had grabbed that paper, but off the arxiv. Still need to read it. – Steve Huntsman Feb 23 '10 at 15:59
From the article mentioned above I found the paper "Product Structure of Poincaré Recurrence" by Barreira and Saussol, which states in the abstract that "for locally maximal hyperbolic sets the recurrence rate possesses a certain local product structure, which closely imitates the product structure provided by the families of local stable and unstable manifolds, as well as the almost product structure of hyperbolic measures." – Steve Huntsman Feb 23 '10 at 16:49
Looking at theorem 10 in this paper, I find the points of closest approach to what I'd like, but still not really what I'm after. I've looked at all the conceivably relevant papers that cite this in Google Scholar. So if there is anything out there it's probably very new, unpublished, etc. – Steve Huntsman Feb 23 '10 at 21:54

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).

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Thanks for your answer. A bit after I asked this question some physical considerations led me to be primarily concerned with variants of the $L^2$ mixing time and topological entropy as timescales of interest--I still don't know how to define the timescale I really want, but these are close for systems that can't be written as products. – Steve Huntsman Dec 24 '15 at 0:23

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