The $\beta$-transformation with a hole

Abstract

This paper extends those of Glendinning and Sidorov [3] and of Hare and Sidorov [6] from the case of the doubling map to the more general $\beta$-transformation. Let $\beta \in (1,2)$ and consider the $\beta$-transformation $T_{\beta }(x)=\beta x(\mathrm{mod}1)$. Let ${\mathcal{J}}_{\beta }(a,b):=\{x\in (0,1):T_{\beta }^n(x)\notin (a,b)\text{ for all }n\ge 0\}$. An integer $n$ is bad for $(a,b)$ if every $n$-cycle for $T_{\beta }$ intersects $(a,b)$. Denote the set of all bad $n$ for $(a,b)$ by $B_{\beta }(a,b)$. In this paper we completely describe the following sets: \[ D_0(β) = \{ (a,b) \in [0,1)^2 : \mathcal{J}_β(a,b) \neq \emptyset \}, \] \[ D_1(β) = \{ (a,b) \in [0,1)^2 : \mathcal{J}_β(a,b) \text{ is uncountable} \}, \] \[ D_2(β) = \{ (a,b) \in [0,1)^2 : B_β(a,b) \text{ is finite} \}. \]

20 pages, 8 figures. Updated version has added examples and a small mistake in Lemma 3.4 has been fixed

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