Shrinking target problems for beta-dynamical system

Shen, Luming; Wang, BaoWei

doi:10.1007/s11425-012-4478-8

Cited by 41 publications

(22 citation statements)

References 20 publications

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“…However, this should not cause any trouble and we may consider that the following result, established by Shen and Wang [15], extends (1.4) to β-expansions.…”

Section: Introduction and Resultsmentioning

confidence: 83%

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Uniform Diophantine approximation related to -ary and -expansions

Bugeaud¹,

Liao²

2014

Ergod. Th. Dynam. Sys.

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Let b ≥ 2 be an integer andv a real number. Among other results, we compute the Hausdorff dimension of the set of real numbers ξ with the property that, for every sufficiently large integer N , there exists an integer n such that 1 ≤ n ≤ N and the distance between b n ξ and its nearest integer is at most equal to b −v N . We further solve the same question when replacing b n ξ by T n β ξ , where T β denotes the classical β-transformation.

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“…However, this should not cause any trouble and we may consider that the following result, established by Shen and Wang [15], extends (1.4) to β-expansions.…”

Section: Introduction and Resultsmentioning

confidence: 83%

“…PROPOSITION 3.9 [15,. Lemma 2.7] For any w ∈ n β N viewed as an element of n β , we have β −(n+N ) ≤ |I n (w)| ≤ β −n .…”

mentioning

confidence: 99%

Uniform Diophantine approximation related to -ary and -expansions

Bugeaud¹,

Liao²

2014

Ergod. Th. Dynam. Sys.

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“…This is initiated by Hill & Velani who studied the dimension of W y (φ) for expanding rational maps acting on its Julia set [12,13]. For infinite iterated function systems or some concrete systems without finite Markov partitions, one is referred to the works [5,15,22,24,20,27,29] etc.…”

Section: Chao Ma Baowei Wang and Jun Wumentioning

confidence: 99%

Diophantine approximation of the orbits in topological dynamical systems

Ma¹,

Wang²,

Wu³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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We would like to present a general principle for the shrinking target problem in a topological dynamical system. More precisely, let (X, d) be a compact metric space and T : X → X a continuous transformation on X. For any integer valued sequence {an} and y ∈ X, define Ey({an}) = δ>0 x ∈ X : T n x ∈ Ba n (y, δ), for infinitely often n ∈ N , the set of points whose orbit can well approximate a given point infinitely often, where Bn(x, r) denotes the Bowen-ball. It is shown that htop(Ey({an}), T) = 1 1 + a htop(X, T), with a = lim inf n→∞ an n , if the system (X, T) has the specification property. Here htop denotes the topological entropy. An example is also given to indicate that the specification property required in the above result cannot be weakened even to almost specification.

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“…Philipp [28] showed that the Lebesgue measure or Parry measure of the set W y (T β , Ψ) is zero or full according to the convergence or divergence of the series n 1 Ψ(n). The Hausdorff dimension of W y (T β , Ψ) was given by Shen and Wang [32] (see also Bugeaud and Wang [9]). As stated above, in this paper, we focus on the Hausdorff measure of W y (T β , Ψ).…”

Section: Introductionmentioning

confidence: 99%

A Dichotomy Law for the Diophantine Properties in ‐dynamical Systems

2016

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Let β > 1 be a real number and define the β-transformation on [0, 1] by T β : x → βx mod 1. Further, definefor infinitely many n} andfor infinitely many n}, where Ψ : N → R >0 is a positive function such that Ψ(n) → 0 as n → ∞. In this paper, we show that each of the above sets obeys a Jarník-type dichotomy, that is, the generalised Hausdorff measure is either zero or full depending upon the convergence or divergence of a certain series. This work completes the metrical theory of these sets.

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Shrinking target problems for beta-dynamical system

Cited by 41 publications

References 20 publications

Uniform Diophantine approximation related to -ary and -expansions

Uniform Diophantine approximation related to -ary and -expansions

Diophantine approximation of the orbits in topological dynamical systems

A Dichotomy Law for the Diophantine Properties in ‐dynamical Systems

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