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

Let $T_{k}$ be the expanding map of $[0,1)$ defined by $T_{k}(x) = k x\ \text{mod 1}$, where $k\geq 2$ is an integer. Given $0\leq a<b\leq 1$, let $\mathcal{W}_{k}(a,b)=\{x\in [0,1)\ \vert \ T_{k}^nx\notin (a,b), \text{ for all } n\geq 0\}$ 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.

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