On the intersection of dynamical covering sets with fractals

• DOI: 10.1007/s00209-021-02924-2
• Publication Date: 2022-01-18T00:03:57Z
• Authors: Zhang-nan Hu; Bing Li; Yimin Xiao
• Keywords: Dynamical system (definition)

Let $(X,\mathscr{B}, ��,T,d)$ be a measure-preserving dynamical system with exponentially mixing property, and let $��$ 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 $G$ equals $\dim_{\rm H}G+��-s$, where $��=\dim_{\rm H}E(x)$ $��$--a.e. Moreover, we obtain the packing dimension of $E(x)\cap G$ and an estimate for $\dim_{\rm H}(E(x)\cap G)$ for any analytic set $G$.<br/>27pages<br/>