Quantitative recurrence properties and homogeneous self-similar sets

Yuanyang Chang; Min Wu; Wen Wu

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arXiv · 1801.10334

Quantitative recurrence properties and homogeneous self-similar sets

Yuanyang Chang·Min Wu·Wen Wu

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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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Yuanyang Chang, Min Wu, Wen Wu. 2018-01-31. Quantitative recurrence properties and homogeneous self-similar sets. https://arxiv.org/abs/1801.10334

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