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arXiv · 1412.6384

The $β$-transformation with a hole

Lyndsey Clark
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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 $β$-transformation. Let $β\in (1,2)$ and consider the $β$-transformation $T_β(x)=βx \pmod 1$. Let $\mathcal{J}_β (a,b) := \{ x \in (0,1) : T_β^n(x) \notin (a,b) \text{ for all } n \geq 0 \}$. An integer $n$ is bad for $(a,b)$ if every $n$-cycle for $T_β$ intersects $(a,b)$. Denote the set of all bad $n$ for $(a,b)$ by $B_β(a,b)$. In this paper we completely describe the following sets: \[ D_0(β) = \{ (a,b) \in [0,1)^2 : \mathcal{J}_β(a,b) \neq \emptyset \}, \] \[ D_1(β) = \{ (a,b) \in [0,1)^2 : \mathcal{J}_β(a,b) \text{ is uncountable} \}, \] \[ D_2(β) = \{ (a,b) \in [0,1)^2 : B_β(a,b) \text{ is finite} \}. \]

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Lyndsey Clark. 2015-09-18. The $β$-transformation with a hole. https://doi.org/10.3934/dcds.2016.36.1249

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