On Shrinking Targets and Self-returning Points

Kirsebom, Maxim; Kunde, Philipp; Persson, Tomas

doi:10.2422/2036-2145.202110_004

Cited by 3 publications

(2 citation statements)

References 19 publications

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“…And indeed this was proved in several special cases such as [BF, CWW, HLSW]; see also [KKP,Pe,1] for similar results. Note that a topic closely related to recurrence is the so-called shrinking target problem, which is concerned with determining the speed at which the orbit of a µ-typical point accumulates near a fixed point y ∈ X.…”

Section: Introductionmentioning

confidence: 54%

Dynamical Borel–Cantelli lemma for recurrence under Lipschitz twists

Kleinbock

Zheng

2023

Nonlinearity

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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 zero–one laws for the two problems.

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Section: Introductionmentioning

confidence: 54%

Dynamical Borel–Cantelli lemma for recurrence under Lipschitz twists

Kleinbock

Zheng

2023

Nonlinearity

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“…Hussein et al [11] showed that the measure of E(r n ) obeys a zero-full law for some conformal and expanding systems. Kirsebom et al [12] investigated the measure of E(r n ) for a class of mixing interval maps and some linear maps on tori.…”

Section: Theorem 12 ([2]mentioning

confidence: 99%

Hausdorff dimension of recurrence sets

Hu,

Persson

2024

Nonlinearity

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We consider linear mappings on the 2-dimensional torus, defined by T ( x ) = A x ( mod 1 ) , where A is an invertible 2 × 2 integer matrix, with no eigenvalues on the unit circle. In the case det A = ± 1 , we give a formula for the Hausdorff dimension of the set

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Dynamical Borel-Cantelli lemma for recurrence theory

Hussain¹,

Li²,

Simmons³

et al. 2020

Preprint

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We study the dynamical Borel-Cantelli lemma for recurrence sets in a measure preserving dynamical system (X, µ, T ) with a compatible metric d. We prove that, under some regularity conditions, the µ-measure of the following set R(ψ) = {x ∈ X : d(T n x, x) < ψ(n) for infinitely many n ∈ N} obeys a zero-full law according to the convergence or divergence of a certain series, where ψ : N → R + . Some of the applications of our main theorem include the continued fractions dynamical systems, the beta dynamical systems, and the homogeneous self-similar sets.

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On Shrinking Targets and Self-returning Points

Cited by 3 publications

References 19 publications

Dynamical Borel–Cantelli lemma for recurrence under Lipschitz twists

Dynamical Borel–Cantelli lemma for recurrence under Lipschitz twists

Hausdorff dimension of recurrence sets

Dynamical Borel-Cantelli lemma for recurrence theory

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