Quantitative recurrence properties and homogeneous self-similar sets

Chang, Yuanyang; Wu, Min; Wu, Wen

doi:10.1090/proc/14287

Quantitative recurrence properties and homogeneous self-similar sets
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by Yuanyang Chang, Min Wu and Wen Wu

Proc. Amer. Math. Soc. 147 (2019), 1453-1465

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: \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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