The $\beta$-transformation with a hole
Year: 2015
Type: article
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 $T_\beta(x)=\beta x$ mod 1. Let $\mathcal{J}_\beta(a,b) := \{ x \in (0,1) : T_\beta^n(x) \notin (a,b) \text{ for all } n \geq 0 \}$. An integer $n$ is bad for $(a,b)$ if every periodic point of period $n$ 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:\begin{align*} D_0(\beta) &= \{ (a,b) \in [0,1)^2 : \mathcal{J}(a,b) \neq \emptyset \}, \\D_1(\beta) &= \{ (a,b) \in [0,1)^2 : \mathcal{J}(a,b) \text{ is uncountable} \}, \\D_2(\beta) &= \{ (a,b) \in [0,1)^2 : B_\beta(a,b) \text{ is finite} \}.\end{align*}
Author: Lyndsey Clark
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W1804733242