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On the intersection of dynamical covering sets with fractals

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Abstract

Let \((X,\mathscr {B}, \mu ,T,d)\) be a measure-preserving dynamical system with exponentially mixing property, and let \(\mu \) be an Ahlfors s-regular probability measure. The dynamical covering problem concerns the set E(x) of points which are covered by the orbits of \(x\in X\) infinitely many times. We prove that the Hausdorff dimension of the intersection of E(x) and any regular fractal G with \(\dim _\mathrm{H}G>s-\alpha \) equals \(\dim _\mathrm{H}G+\alpha -s\), where \(\alpha =\dim _\mathrm{H}E(x)\) \(\mu \)–a.e. Moreover, we obtain the packing dimension of \(E(x)\cap G\) and an estimate for \(\dim _\mathrm{H}(E(x)\cap G)\) for any analytic set G.

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Authors and Affiliations

  1. School of Mathematics, South China University of Technology, Guangzhou, 510641, China

    Zhang-nan Hu & Bing Li

  2. Department of Statistics and Probability, Michigan State University, East Lansing, MI, 48824, USA

    Yimin Xiao

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  1. Zhang-nan Hu
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  2. Bing Li
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  3. Yimin Xiao
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Correspondence to Bing Li.

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This research of Bing Li and Zhangnan Hu was supported by NSFC 11671151 and Guangdong Natural Science Foundation 2018B0303110005. The research of Yimin Xiao is partially supported by the NSF Grant DMS-1855185.

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Hu, Zn., Li, B. & Xiao, Y. On the intersection of dynamical covering sets with fractals. Math. Z. 301, 485–513 (2022). https://doi.org/10.1007/s00209-021-02924-2

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