On dichotomy law for beta-dynamical system in parameter space
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Mathematische Zeitschrift
On dichotomy law for beta-dynamical system in parameter space Fan Lü1 · Jun Wu2 Received: 22 August 2018 / Accepted: 4 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Let ϕ : N → (0, 1] be a positive function and Tβ be the beta-transformation for any β > 1. We prove that the set E(0, ϕ) = {β > 1 : |Tβn 1 − 0| < ϕ(n) for infinitely many n ∈ N} is of zero or full Lebesgue measure in (1, +∞) according to ϕ(n) < +∞ or not. As an application, we determine the exact Lebesgue measure of the set E(0, {ln }) = {β > 1 : |Tβn 1 − 0| < β −ln for infinitely many n ∈ N}, where {ln }n≥1 is a sequence of non-negative real numbers. Keywords Beta-dynamical system · Diophantine approximation · Shrinking target problem · Paley–Zygmund inequality · Lebesgue measure Mathematics Subject Classification Primary 11K55; Secondary 28A80
1 Introduction Given a real number β > 1, the beta-transformation Tβ : [0, 1] → [0, 1] is defined by Tβ (x) = βx − βx
for all x ∈ [0, 1],
where · denotes the integral part of a real number. In 1957, Rényi [13] introduced this map as a model for expanding real numbers in non-integer bases. Parry [10] proved that the
This work was supported by NSFC Nos. 11225101, 11271114, 11601358, 11831007.
B
Fan Lü [email protected] Jun Wu [email protected]
1
Department of Mathematics, Sichuan Normal University, Chengdu 610066, People’s Republic of China
2
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China
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F. Lü, J. Wu
transformation Tβ has an invariant ergodic measure νβ , which is equivalent to the Lebesgue measure L on [0, 1]. This measure, called Parry measure, is the unique measure of maximal entropy [6]. For any β > 1, since Tβ is ergodic with respect to the Parry measure νβ , Birkhoff’s ergodic theorem yields the following hitting property, namely, for L-almost all x ∈ [0, 1], lim inf |Tβn x − 0| = 0. n→∞
(1.1)
It is a natural question to ask about the speed of convergence in (1.1). This leads to the study of the metric properties of the set Dβ (0, ϕ) = {x ∈ [0, 1] : |Tβn x − 0| < ϕ(n) for infinitely many n ∈ N}, where ϕ : N → (0, 1] is a positive function, in the sense of measure and in the sense of Hausdorff dimension. In 1967, Philipp [12] proved that 0, if ϕ(n) < +∞; L(Dβ (0, ϕ)) = 1, if ϕ(n) = +∞. When ϕ(n) < +∞, Shen and Wang [15] studied the Hausdorff dimension of the set Dβ (0, ϕ), and found that dim H Dβ (0, ϕ) =
− logβ ϕ(n) 1 with α = lim inf , n→∞ 1+α n
where dim H denotes the Hausdorff dimension. This kind of study in a measure-preserving dynamical system is called the dynamical Borel–Cantelli Lemma [2] or the shrinking target problem in dynamical systems [5]. Comparing with the shrinking target problem, there is a modified shrinking target problem, which concerns the distribution properties of the orbit of some given point in a family of dynamical systems, instead of considering the properties in one given system. For example, let Rα
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