On shrinking targets and self-returning points

Kirsebom, Maxim; Kunde, Philipp; Persson, Tomas

doi:10.48550/arxiv.2003.01361

Cited by 7 publications

(13 citation statements)

References 11 publications

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“…) is bounded away from 0 for large k, it is possible to find a K such that (35) holds for all large k. 10.2. Proof of Theorem 3.4 Case (2).…”

Section: Proof Of Theorem 34mentioning

confidence: 99%

Dichotomy results for eventually always hitting time statistics and almost sure growth of extremes

Holland¹,

Kirsebom²,

Kunde³

et al. 2021

Preprint

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Suppose (f, X , µ) is a measure preserving dynamical system and φ : X → Ê a measurable function. Consider the maximum process Mn := max{X1, . . . , Xn}, where Xi = φ • f i−1 is a time series of observations on the system. Suppose that (un) is a non-decreasing sequence of real numbers, such that µ(X1 > un) → 0. For certain dynamical systems, we obtain a zero-one measure dichotomy for µ(Mn ≤ un i.o.) depending on the sequence un. Specific examples are piecewise expanding interval maps including the Gauß map. For the broader class of non-uniformly hyperbolic dynamical systems, we make significant improvements on existing literature for characterising the sequences un. Our results on the permitted sequences un are commensurate with the optimal sequences (and series criteria) obtained by Klass (1985) for i.i.d. processes. Moreover, we also develop new series criteria on the permitted sequences in the case where the i.i.d. theory breaks down. Our analysis has strong connections to specific problems in eventual always hitting time statistics and extreme value theory.

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“…) is bounded away from 0 for large k, it is possible to find a K such that (35) holds for all large k. 10.2. Proof of Theorem 3.4 Case (2).…”

Section: Proof Of Theorem 34mentioning

confidence: 99%

Dichotomy results for eventually always hitting time statistics and almost sure growth of extremes

Holland¹,

Kirsebom²,

Kunde³

et al. 2021

Preprint

Self Cite

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“…Theorem 1.1 was improved by who showed that the exponent α can be replaced by the lower local dimension of a measure at x. For piecewise C 2 expanding maps with the ergodic measure equivalent to Lebsgue measure, Kirsebom-Kunde-Persson [6] improved the speed in (1.1) from n to n(log n) θ with θ < 1/2.…”

Section: Introductionmentioning

confidence: 99%

“…As far as a general error function ψ is concerned, hardly anything is known. The only known results for µ-measure of R(ψ) are recently proven by Chang-Wu-Wu [4], Baker-Farmer [1], and Kirsebom-Kunde-Persson [6]. Chang-Wu-Wu [4] considered homogeneous self similar set satisfying the strong separation condition.…”

Section: Introductionmentioning

confidence: 99%

“…Baker-Farmer [1] generalised Chang-Wu-Wu's result to the finite conformal iterated function systems with open set condition. On the other hand, Kirsebom-Kunde-Persson [6] presented the recurrence and shrinking target theory when T is an integer matrix action with some condition about the eigenvalues. However, all of these results are not applicable to some well known dynamical systems, for example, β-dynamical systems or the dynamical systems of continued fractions.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Recent improvements of Boshernitzan's result for particular classes of dynamical systems can be found in papers by Pawelec [12]; Chang, Wu and Wu [5]; Baker and Farmer [1]; Hussain, Li, Simmons and Wang [7]; and by Kirsebom, Kunde and Persson [9].…”

mentioning

confidence: 99%