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Quantitative recurrence properties and homogeneous self-similar sets


Authors: Yuanyang Chang, Min Wu and Wen Wu
Journal: Proc. Amer. Math. Soc. 147 (2019), 1453-1465
MSC (2010): Primary 28A80, 28D05; Secondary 11K55
DOI: https://doi.org/10.1090/proc/14287
Published electronically: December 31, 2018
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Abstract: Let $ K$ be a homogeneous self-similar set satisfying the strong separation condition. This paper is concerned with the quantitative recurrence properties of the natural map $ T: K\rightarrow K$ induced by the shift. Let $ \mu $ be the natural self-similar measure supported on $ K$. For a positive function $ \varphi $ defined on $ \mathbb{N}$, we show that the $ \mu $-measure of the following set:

$\displaystyle R(\varphi ):=\{x\in K: \vert T^n x-x\vert<\varphi (n)$$\displaystyle \text { for infinitely many } n\in \mathbb{N}\}$    

is null or full according to convergence or divergence of a certain series. Moreover, a similar dichotomy law holds for the general Hausdorff measure, which completes the metric theory of this set.

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Additional Information

Yuanyang Chang
Affiliation: Department of Mathematics, South China University of Technology, Guangzhou, 510640, People’s Republic of China
Email: changyy@scut.edu.cn

Min Wu
Affiliation: Department of Mathematics, South China University of Technology, Guangzhou, 510640, People’s Republic of China
Email: wumin@scut.edu.cn

Wen Wu
Affiliation: Department of Mathematics, South China University of Technology, Guangzhou, 510640, People’s Republic of China
Email: wuwen@scut.edu.cn

DOI: https://doi.org/10.1090/proc/14287
Keywords: Quantitative recurrence, self-similar set, Hausdorff measure
Received by editor(s): January 31, 2018
Received by editor(s) in revised form: April 17, 2018
Published electronically: December 31, 2018
Additional Notes: The third author is the corresponding author.
This work was supported by NSFC (Grant No. 11771153), the Fundamental Research Funds for the Central Universities (No. 2017MS110) and the Characteristic innovation project of colleges and universities in Guangdong (No. 2016KTSCX007).
Communicated by: Nimish Shah
Article copyright: © Copyright 2018 American Mathematical Society