Dynamical Borel–Cantelli lemma for recurrence under Lipschitz twists

  title={Dynamical Borel–Cantelli lemma for recurrence under Lipschitz twists},
  author={Dmitry Kleinbock and Jiajie Zheng},
  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… 

Recurrence rates for shifts of finite type

Let Σ A be a topologically mixing shift of finite type, let σ : Σ A → Σ A be the usual left-shift, and let µ be the Gibbs measure for a H¨older continuous potential that is not cohomologous to a

Recurrent set on some Bedford-McMullen carpets

. In this paper, we study the Hausdorff dimension of the quantitative recurrent set of the canonical endomorphism on the Bedford–McMullen carpets whose Hausdorff dimension and box dimension are equal.

Quantitative recurrence properties for piecewise expanding map on $ [0,1]^d $

Let $ T\colon[0,1]^d\to [0,1]^d $ be a piecewise expanding map with an absolutely continuous invariant measure $ \mu $. Let $ (H_n) $ be a sequence of hyperrectangles or hyperboloids centered at the

Twisted recurrence for dynamical systems with exponential decay of correlations

We study the set of points returning infinitely often to a sequence of targets dependent on the starting points. With an assumption of decay of correlations for L 1 against bounded variations, we



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

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

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

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

Logarithm laws for flows on homogeneous spaces

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

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

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

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

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

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