# 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} }

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

### Multiple Borel–Cantelli Lemma in dynamics and MultiLog Law for recurrence

- MathematicsJournal 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…

### A strong Borel–Cantelli lemma for recurrence

- MathematicsStudia Mathematica
- 2022

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…

### Dynamical Borel-Cantelli lemmas for gibbs measures

- 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…

### RECURRENCE RATE IN RAPIDLY MIXING DYNAMICAL SYSTEMS

- Mathematics
- 2004

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…

### Logarithm laws for flows on homogeneous spaces

- 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…

### On Kurzweil’s 0-1 law in inhomogeneous Diophantine approximation

- 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…

### A note on Borel–Cantelli lemmas for non-uniformly hyperbolic dynamical systems

- MathematicsErgodic 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…

### Ergodic Theory: with a view towards Number Theory

- 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…

### Diophantine analysis of the expansions of a fixed point under continuum many bases

- 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,…

### Dynamical Borel–Cantelli lemma for recurrence theory

- MathematicsErgodic 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…