$β$-transformation on an interval with hole

Nikita Agarwal

Published 2017-04-09Version 1

Let $T_{\beta}$ be the expanding map of $[0,1]$ defined by $T_{\beta}(x) = \beta x\ \text{mod 1}$, where $\beta$ is an integer atleast 2. Given $0\leq a<b\leq 1$, let $\mathcal{W}_{\beta}(a,b)=\{x\in [0,1]\ \vert \ T_{\beta}^nx\notin (a,b), \ n\geq 0\}$ be the maximal $T$-invariant subset of $[0,1]\setminus (a,b)$. We characterize the intervals $(a,b)$ for which the Hausdorff dimension of $\mathcal{W}_{\beta}(a,b)$ is positive.