ar X iv : m at h / 06 10 21 3 v 1 [ m at h . D S ] 6 O ct 2 00 6 THE DYNAMICAL BOREL - CANTELLI LEMMA AND THE WAITING TIME PROBLEMS
نویسنده
چکیده
We investigate the connection between the dynamical Borel-Cantelli and waiting time results. We prove that if a system has the dynamical BorelCantelli property, then the time needed to enter for the first time in a sequence of small balls scales as the inverse of the measure of the balls. Conversely if we know the waiting time behavior of a system we can prove that certain sequences of decreasing balls satisfies the Borel-Cantelli property. This allows to obtain Borel-Cantelli like results in systems like axiom A and generic interval exchanges.
منابع مشابه
1 3 A pr 2 00 8 THE DYNAMICAL BOREL - CANTELLI LEMMA AND THE WAITING TIME PROBLEMS
We investigate the connection between the dynamical Borel-Cantelli and waiting time results. We prove that if a system has the dynamical BorelCantelli property, then the time needed to enter for the first time in a sequence of small balls scales as the inverse of the measure of the balls. Conversely if we know the waiting time behavior of a system we can prove that certain sequences of decreasi...
متن کاملGeneralizations of Borel-Cantelli Lemma
The Borel-Cantelli Lemma is very important in the probability theory. In this paper, we first describe the general case of the Borel-Cantelli Lemma. The first part of this lemma, assuming convergence and the second part includes divergence and independence assumptions. In the following, we have brought generalizations of the first and second part of this lemma. In most generalizat...
متن کامل2 00 8 Dimension and hitting time in rapidly mixing systems
We prove that if a system has superpolynomial (faster than any power law) decay of correlations then the time τ r (x, x 0) needed for a typical point x to enter for the first time a ball B(x 0 , r) centered in x 0 , with small radius r scales as the local dimension at x 0 , i.e. lim r→0 log τ r (x, x 0) − log r = d µ (x 0). This result is obtained by proving a kind of dynamical Borel-Cantelli l...
متن کاملA Borel-cantelli Lemma for Nonuniformly Expanding Dynamical Systems
Let (An)n=1 be a sequence of sets in a probability space (X,B, μ) such that P∞ n=1 μ(An) =∞. The classical Borel-Cantelli lemma states that if the sets An are independent, then μ({x ∈ X : x ∈ An for infinitely many values of n}) = 1. We present analogous dynamical Borel-Cantelli lemmas for certain sequences of sets (An) inX (including nested balls) for a class of deterministic dynamical systems...
متن کاملar X iv : m at h / 02 05 20 7 v 1 [ m at h . R T ] 1 9 M ay 2 00 2 Can p - adic integrals be computed ?
This article gives an introduction to arithmetic motivic integration in the context of p-adic integrals that arise in representation theory. A special case of the fundamental lemma is interpreted as an identity of Chow motives.
متن کامل