Critical values for the beta-transformation with a hole at 0

Abstract: Given $\beta \in (1,2]$, let $T$ be the $\beta$-transformation on the unit circle $[0,1)$. For $t \in [0,1)$ let $K(t)$ be the survivor set consisting of all $x$ whose orbit under $T$ never hits the open interval $(0,t)$. Kalle et al. [ETDS, 2020] proved that the Hausdorff dimension function $\dim K(t)$ is a non-increasing Devil's staircase in $t$. So there exists a critical value such that $\dim K(t)$ is vanishing when $t$ is passing through this critical value. In this paper we will describe this critical value and analyze its interesting properties. Our strategy to find the critical value depends on certain substitutions of Farey words and a renormalization scheme from dynamical systems. This is joint work with Pieter Allaart.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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