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Hitting time in regular sets and logarithm law for rapidly mixing dynamical systems


Author: Stefano Galatolo
Journal: Proc. Amer. Math. Soc. 138 (2010), 2477-2487
MSC (2010): Primary 37A25, 37C45, 37D40, 37A99
DOI: https://doi.org/10.1090/S0002-9939-10-10275-5
Published electronically: March 4, 2010
MathSciNet review: 2607877
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Abstract: We prove that if a system has superpolynomial (faster than any power law) decay of correlations (with respect to Lipschitz observables), then the time $ \tau (x,S_{r})$ is needed for a typical point $ x$ to enter for the first time a set $ S_{r}=\{x:f(x)\leq r\}$ which is a sublevel of a Lipschitz function $ f$ scales as $ \frac{1}{\mu (S_{r})}$ i.e.,

$\displaystyle \underset{r\rightarrow 0}{\lim }\frac{\log \tau (x,S_{r})}{-\log r}=\underset {r\rightarrow 0}{\lim }\frac{\log \mu (S_{r})}{\log r}. $

This generalizes a previous result obtained for balls. We will also consider relations with the return time distributions, an application to observed systems and to the geodesic flow in negatively curved manifolds.


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Additional Information

Stefano Galatolo
Affiliation: Dipartimento di Matematica Applicata, Universita di Pisa, via Buonarroti 1, Pisa, Italy
Email: s.galatolo@docenti.ing.unipi.it

DOI: https://doi.org/10.1090/S0002-9939-10-10275-5
Keywords: Logarithm law, hitting time, decay of correlations, dimension, return time distribution.
Received by editor(s): June 18, 2009
Received by editor(s) in revised form: October 12, 2009
Published electronically: March 4, 2010
Communicated by: Bryna Kra
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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