Open dynamics with a moving hole: inhomogeneous IFS and exponential separation condition

Abstract

Let $\mathcal F=\{f_i(x)=r_i x+t_i\}_{i=0}^{b-1}$ be a self-similar iterated function system (IFS) on $\mathbb R$ with $b\ge 3$, and let $K_{\mathcal F}$ be the associated self-similar set. Given $m\in\mathbb N$, for any sequence $ω=ω^0ω^1\ldots$ with each $ω^i\in \{0,1,\ldots,b-1\}^m$ a block of length $m$, let \[ K_{\mathcal F}^ω:=π_{\mathcal F}\left(\{(d_i)\in \{0,1,\ldots,b-1\}^{\mathbb N}: d_{k+1}d_{k+2}\ldots d_{k+m}\neω^k\quad\forall k\ge 0\}\right), \] where $π_{\mathcal F}$ is the natural coding map from $\{0,1,\ldots,b-1\}^{\mathbb N}$ to $K_{\mathcal F}$. The set $K_{\mathcal F}^ω$ is called the survivor set of an open dynamical system with a moving hole related to $ω$. When $\mathcal F$ is a homogeneous IFS satisfying the open set condition (OSC), the dimensions of $K_{\mathcal F}^ω$ were determined by the first author, Sun and Wang (Adv. Math., 2026). In this paper we generalize these results to inhomogeneous IFSs. When $\mathcal F$ satisfies the OSC, we give explicit formulae for the dimensions of $K_{\mathcal F}^ω$. In particular, we show that the packing dimension and upper box-counting dimension of $K_{\mathcal F}^ω$ always coincide, while the Hausdorff dimension and lower box-counting dimension of $K_{\mathcal F}^ω$ are not necessarily equal, which is different from the homogeneous case. Furthermore, when $\mathcal F$ satisfies the exponential separation condition and the similarity dimension $s_{\mathcal F}\le 1$, we are able to determine the Hausdorff dimension of $K_{\mathcal F}^ω$. Finally, we apply our results to badly approximable numbers and non-recurrence points in the setting of inhomogeneous IFS satisfying the OSC.