The $\beta$-transformation with a hole at 0

TL;DR

This paper studies the bifurcation set of parameters in the $eta$-transformation with a hole at zero, revealing its measure, dimension, and structural properties for $eta$ in (1,2), and contrasting it with the doubling map case.

Contribution

It characterizes the bifurcation set $E_eta$, establishes its measure and Hausdorff dimension properties, and compares these with the classical doubling map case.

Findings

$E_eta$ is a Lebesgue null set with full Hausdorff dimension for all $eta ext{ in }(1,2)$.

For almost every $eta$, $E_eta$ has both isolated and accumulation points near zero.

The Hausdorff dimension of $K_eta(t)$ is positive if and only if $t< au_eta$, with bounds on $ au_eta$.

Abstract

For $\beta\in(1,2]$ the $\beta$-transformation $T_\beta: [0,1) \to [0,1)$ is defined by $T_\beta ( x) = \beta x \pmod 1$. For $t\in[0, 1)$ let $K_\beta(t)$ be the survivor set of $T_\beta$ with hole $(0,t)$ given by \[K_\beta(t):=\{x\in[0, 1): T_\beta^n(x)\not \in (0, t) \textrm{ for all }n\ge 0\}.\] In this paper we characterise the bifurcation set $E_\beta$ of all parameters $t\in[0,1)$ for which the set valued function $t\mapsto K_\beta(t)$ is not locally constant. We show that $E_\beta$ is a Lebesgue null set of full Hausdorff dimension for all $\beta\in(1,2)$. We prove that for Lebesgue almost every $\beta\in(1,2)$ the bifurcation set $E_\beta$ contains both infinitely many isolated and accumulation points arbitrarily close to zero. On the other hand, we show that the set of $\beta\in(1,2)$ for which $E_\beta$ contains no isolated points has zero Hausdorff dimension. These results contrast with the situation for $E_2$, the bifurcation set of the doubling map. Finally, we give for each $\beta \in (1,2)$ a lower and upper bound for the value $\tau_\beta$, such that the Hausdorff dimension of $K_\beta(t)$ is positive if and only if $t< \tau_\beta$. We show that $\tau_\beta \le 1-\frac1{\beta}$ for all $\beta \in (1,2)$.