# Dynamical Borel–Cantelli lemma for recurrence under Lipschitz twists

@article{Kleinbock2022DynamicalBL,
title={Dynamical Borel–Cantelli lemma for recurrence under Lipschitz twists},
author={Dmitry Kleinbock and Jiajie Zheng},
journal={Nonlinearity},
year={2022},
volume={36},
pages={1434 - 1460}
}
• Published 24 May 2022
• Mathematics
• Nonlinearity
In the study of some dynamical systems the limsup set of a sequence of measurable sets is often of interest. The shrinking targets and recurrence are two of the most commonly studied problems that concern limsup sets. However, the zero–one laws for the shrinking targets and recurrence are usually treated separately and proved differently. In this paper, we introduce a generalized definition that can specialize into the shrinking targets and recurrence; our approach gives a unified proof of the…
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## References

SHOWING 1-10 OF 32 REFERENCES

• Mathematics
Journal of Modern Dynamics
• 2022
A classical Borel–Cantelli Lemma gives conditions for deciding whether an infinite number of rare events will happen almost surely. In this article, we propose an extension of Borel–Cantelli Lemma to
ABSTRACT. Consider a mixing dynamical systems ([0, 1], T, μ), for instance a piecewise expanding interval map with a Gibbs measure μ. Given a non-summable sequence (mk) of non-negative numbers, one
• Mathematics
• 1999
LetT: X→X be a deterministic dynamical system preserving a probability measure μ. A dynamical Borel-Cantelli lemma asserts that for certain sequences of subsetsAn⊃ X and μ-almost every pointx∈X the
For measure preserving dynamical systems on metric spaces we study the time needed by a typical orbit to return back close to its starting point. We prove that when the decay of correlation is
• Mathematics
• 1999
Abstract.In this paper we generalize and sharpen D. Sullivan’s logarithm law for geodesics by specifying conditions on a sequence of subsets {At | t∈ℕ} of a homogeneous space G/Γ (G a semisimple Lie
• Mathematics
• 2015
We give a sufficient and necessary condition such that for almost all s ∈ R kn� −sk < (n) for infinitely many n ∈ N, whereis fixed and (n) is a positive, non-increasing sequence. This improves upon
• Mathematics
Ergodic Theory and Dynamical Systems
• 2012
Abstract Let (Bi) be a sequence of measurable sets in a probability space (X,ℬ,μ) such that ∑ ∞n=1μ(Bi)=∞. The classical Borel–Cantelli lemma states that if the sets Bi are independent, then
• Mathematics
• 2010
Motivation.- Ergodicity, Recurrence and Mixing.- Continued Fractions.- Invariant Measures for Continuous Maps.- Conditional Measures and Algebras.- Factors and Joinings.- Furstenberg's Proof of
• Mathematics
• 2021
In this paper, we study the Diophantine properties of the orbits of a fixed point in its expansions under continuum many bases. More precisely, let Tβ be the beta-transformation with base β > 1,
• Mathematics
Ergodic Theory and Dynamical Systems
• 2021
Abstract We study the dynamical Borel–Cantelli lemma for recurrence sets in a measure-preserving dynamical system $(X, \mu , T)$ with a compatible metric d. We prove that under some regularity