The $\beta$-transformation with a hole

TL;DR

This paper generalizes previous results on the doubling map to the broader class of $eta$-transformations, characterizing sets related to orbits avoiding a hole and the finiteness of bad cycles.

Contribution

It provides a complete description of the sets where the orbit set is nonempty, uncountable, and where bad cycles are finite for $eta$-transformations.

Findings

Characterization of the set $D_0(eta)$ where orbits avoid the hole.

Description of the set $D_1(eta)$ where the orbit set is uncountable.

Determination of the set $D_2(eta)$ where bad cycles are finite.

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 $T_{\beta}(x)=\beta x \pmod 1$. 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 $n$-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: \[ D_0(\beta) = \{ (a,b) \in [0,1)^2 : \mathcal{J}_{\beta}(a,b) \neq \emptyset \}, \] \[ D_1(\beta) = \{ (a,b) \in [0,1)^2 : \mathcal{J}_{\beta}(a,b) \text{ is uncountable} \}, \] \[ D_2(\beta) = \{ (a,b) \in [0,1)^2 : B_\beta(a,b) \text{ is finite} \}. \]