A winning approach to the intersections of twisted non-recurrent sets with fractals

Abstract

In this paper, we prove that if $S\subseteq {\mathbb{R}}^d$ is hyperplane absolute winning on a closed hyperplane diffuse set $L\subseteq {\mathbb{R}}^d$, then ${\dim }_{H}S\cap K={\dim }_{H}K$ for any irreducible self-conformal set $K\subseteq L$ without assuming any separation condition on $K$. The result is then applied to obtain the Hausdorff dimension of intersections between irreducible self-conformal sets and twisted non-recurrent sets $\mathrm{N}(T,\mathcal{G})$ defined as $$\mathrm{N}(T,\mathcal{G}):=\left\{\mathbf{x}\in [0,1{]}^d:\underset{n\to \infty }{\mathrm{liminf}}\Vert T^n(\mathbf{x})-g_n(\mathbf{x})\Vert >0\right\},$$ where $T:[0,1{]}^d\to [0,1{]}^d$ belongs to a broad class of product maps, $\mathcal{G}:=\{g_n{\}}_{n\in \mathbb{N}}$ is a sequence of self-maps on $[0,1{]}^d$ with uniform Lipschitz constant and $\Vert \cdot \Vert$ denotes the maximal norm in ${\mathbb{R}}^d$. When $T$ is the $\beta$-transformation on $[0,1]$, it provides a positive answer to a question raised informally by Broderick, Bugeaud, Fishman, Kleinbock and Weiss (Math. Res. Lett., 2010). For the case $T$ is a $d\times d$ diagonal matrix transformations, our results provide a partial answer asked in a paper of Li, Liao, Velani and Zorin (Adv. Math., 2023). A natural generalization to non-autonomous setting is also obtained.

arXiv comment30 pages