The $k$-transformation on an interval with a hole

Abstract

Let $T_k$ be the expanding map of $[0,1)$ defined by $T_k(x)=kx\text{mod 1}$, where $k\ge 2$ is an integer. Given $0\le a<b\le 1$, let ${\mathcal{W}}_k(a,b)=\{x\in [0,1)|T_k^{n}x\notin (a,b),\text{ for all }n\ge 0\}$ be the maximal $T$-invariant subset of $[0,1)\setminus (a,b)$. We examine the Hausdorff dimension of ${\mathcal{W}}_k(a,b)$ as $a$ and $b$ vary.