The $\beta$-transformation with a hole

Lyndsey  Clark

doi:10.3934/dcds.2016.36.1249

Article Contents

2016, Volume 36, Issue 3: 1249-1269. Doi: 10.3934/dcds.2016.36.1249

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The $\beta$-transformation with a hole

Lyndsey Clark^1,

1.
School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL

Received: December 31, 2014

Revised: April 30, 2015

Published: May 2017

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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 $\beta$-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*}

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Mathematics Subject Classification: Primary: 28D05; Secondary: 37B10, 68R15.

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