On the Intersection of Dynamical Covering Sets with Fractals

TL;DR

This paper investigates the Hausdorff and packing dimensions of the intersection between dynamical covering sets and fractals in measure-preserving systems with exponential mixing, revealing precise dimension formulas.

Contribution

It establishes exact formulas for the Hausdorff and packing dimensions of intersections between dynamical covering sets and regular fractals, extending understanding of fractal intersections in dynamical systems.

Findings

Hausdorff dimension of intersection equals fractal dimension plus orbit dimension minus measure dimension.

Packing dimension of the intersection is explicitly determined.

Provides estimates for Hausdorff dimension of intersections with analytic sets.

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 $G$ equals $\dim_{\rm H}G+\alpha-s$, where $\alpha=\dim_{\rm H}E(x)$ $\mu$--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 $G$.