Quantitative Recurrence Properties in Some Irregular Sets for Beta Dynamical Systems

Chang, Yuanyang; Liu, Wenna

doi:10.3390/math13111850

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Open AccessFeature PaperArticle

Quantitative Recurrence Properties in Some Irregular Sets for Beta Dynamical Systems

by

Yuanyang Chang

^* and

Wenna Liu

School of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430070, China

^*

Author to whom correspondence should be addressed.

Mathematics 2025, 13(11), 1850; https://doi.org/10.3390/math13111850

Submission received: 13 April 2025 / Revised: 25 May 2025 / Accepted: 31 May 2025 / Published: 2 June 2025

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Abstract

Let

β > 1

be a real number and

T_{β} x = β x (m o d 1)

. This paper is concerned with the quantitative recurrence properties of the system

([0, 1], T_{β})

in some (refined) irregular sets. Specifically, let

α_{1}, α_{2} > 0

and

ψ : N \to (0, 1)

be a positive function; we define the set

E_{α_{1}, α_{2}}^{β} = \{x \in [0, 1) : \underset{n \to \infty}{lim inf} \frac{1}{n} S_{n} (x, β) = α_{1}, \underset{n \to \infty}{lim sup} \frac{1}{n} S_{n} (x, β) = α_{2}\},

and calculate the Hausdorff dimension of the set

E_{α_{1}, α_{2}}^{β} (ψ) : = \{x \in E_{α_{1}, α_{2}}^{β} : | T_{β}^{n} x - x | < ψ (n) i . m . n \in N\},

where

i . m .

stands for infinitely many. Consequently, the Hausdorff dimension of the set

{\hat{E}}^{β} (ψ) = \{x \in [0, 1) : lim_{n \to \infty} \frac{1}{n} S_{n} (x, β) does not exist, | T_{β}^{n} x - x | < ψ (n) i . m . n \in N\}

is also determined.

Keywords: β-expansion; irregular sets; quantitative recurrence; Hausdorff dimension

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MDPI and ACS Style

Chang, Y.; Liu, W. Quantitative Recurrence Properties in Some Irregular Sets for Beta Dynamical Systems. Mathematics 2025, 13, 1850. https://doi.org/10.3390/math13111850

AMA Style

Chang Y, Liu W. Quantitative Recurrence Properties in Some Irregular Sets for Beta Dynamical Systems. Mathematics. 2025; 13(11):1850. https://doi.org/10.3390/math13111850

Chicago/Turabian Style

Chang, Yuanyang, and Wenna Liu. 2025. "Quantitative Recurrence Properties in Some Irregular Sets for Beta Dynamical Systems" Mathematics 13, no. 11: 1850. https://doi.org/10.3390/math13111850

APA Style

Chang, Y., & Liu, W. (2025). Quantitative Recurrence Properties in Some Irregular Sets for Beta Dynamical Systems. Mathematics, 13(11), 1850. https://doi.org/10.3390/math13111850

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