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Dichotomy law for shrinking target problems in a nonautonomous dynamical system: Cantor series expansion


Authors: Yu Sun and Chun-Yun Cao
Journal: Proc. Amer. Math. Soc. 145 (2017), 2349-2359
MSC (2010): Primary 11K55; Secondary 28A80, 37F35
DOI: https://doi.org/10.1090/proc/13420
Published electronically: January 27, 2017
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Abstract: Let $ Q=\{q_k\}_{k\geq 1}$ be a sequence of positive integers with $ q_k\geq 2$ for every $ k\geq 1$. Then each point $ x\in [0,1]$ is attached with an infinite series expansion

$\displaystyle x=\frac {\varepsilon _1(x)}{q_1}+\frac {\varepsilon _2(x)}{q_1q_2}+\cdots +\frac {\varepsilon _n(x)}{q_1\cdots q_n}+\cdots , $

which is called the Cantor series expansion of $ x$. In this paper, we study the shrinking target problems for the system induced by the Cantor series expansion. More precisely, put $ T_{Q}^{n}(x)=q_1\cdots q_nx-\lfloor q_1\cdots q_nx\rfloor $; the shrinking target problem in such a nonautonomous system can be formulated as considering the size of the set

$\displaystyle \mathbb{E}_y(\varphi ):=\{x\in [0,1]:~ \vert T_{Q}^{n}(x)-y\vert<\varphi (n)$$\displaystyle \text { i. o. }n\},$

where $ y$ is a fixed point in $ [0,1]$ and $ \varphi : \mathbb{N}\to (0,1)$ is a positive function with $ \varphi (n)\to 0$ as $ n\to \infty $. It is proved that both the Lebesgue measure and the Hausdorff measure of $ \mathbb{E}_{y}(\varphi )$ fulfill a dichotomy law according to the divergence or convergence of certain series.

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Additional Information

Yu Sun
Affiliation: Faculty of Science, JiangSu University, Zhenjiang 212013, People’s Republic of China
Email: sunyu@ujs.edu.cn

Chun-Yun Cao
Affiliation: College of Science, Huazhong Agricultural University, Wuhan 430070, People’s Republic of China
Email: caochunyun@mail.hzau.edu.cn

DOI: https://doi.org/10.1090/proc/13420
Keywords: Nonautonomous dynamical system, Cantor series expansion, shrinking target problem, Diophantine approximation, Hausdorff measure
Received by editor(s): April 17, 2016
Received by editor(s) in revised form: July 22, 2016
Published electronically: January 27, 2017
Additional Notes: The second author is the corresponding author
Communicated by: Nimish Shah
Article copyright: © Copyright 2017 American Mathematical Society