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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Quantitative recurrence properties and homogeneous self-similar sets
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by Yuanyang Chang, Min Wu and Wen Wu PDF
Proc. Amer. Math. Soc. 147 (2019), 1453-1465 Request permission

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: \begin{equation*} R(\varphi ):=\{x\in K: |T^n x-x|<\varphi (n) \text { for infinitely many } n\in \mathbb {N}\} \end{equation*} 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
  • MR Author ID: 1223628
  • Email: changyy@scut.edu.cn
  • Min Wu
  • Affiliation: Department of Mathematics, South China University of Technology, Guangzhou, 510640, People’s Republic of China
  • MR Author ID: 214816
  • 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
  • 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
  • © Copyright 2018 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 1453-1465
  • MSC (2010): Primary 28A80, 28D05; Secondary 11K55
  • DOI: https://doi.org/10.1090/proc/14287
  • MathSciNet review: 3910412