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

TL;DR

This paper determines the critical value (eta) for the -transformation with a hole at 0, revealing its properties and behavior across the parameter range, using Farey words and renormalization techniques.

Contribution

It provides an explicit formula and detailed analysis of the critical value (eta) for all eta in (1,2], answering a question posed by Kalle et al. (2020).

Findings

(eta) is left continuous with countably many discontinuities.

(eta) has no downward jumps, with (1+)=0 and (2)=1/2.

(eta) is real-analytic, convex, and strictly decreasing on certain open subsets of (1,2].

Abstract

Given $\beta\in(1,2]$, let $T_\beta$ be the $\beta$-transformation on the unit circle $[0,1)$ such that $T_\beta(x)=\beta x\pmod 1$. For each $t\in[0,1)$ let $K_\beta(t)$ be the survivor set consisting of all $x\in[0,1)$ whose orbit $\{T^n_\beta(x): n\ge 0\}$ never hits the open interval $(0,t)$. Kalle et al. proved in [Ergodic Theory Dynam. Systems, 40 (9): 2482--2514, 2020] that the Hausdorff dimension function $t\mapsto\dim_H K_\beta(t)$ is a non-increasing Devil's staircase. So there exists a critical value $\tau(\beta)$ such that $\dim_H K_\beta(t)>0$ if and only if $t<\tau(\beta)$. In this paper we determine the critical value $\tau(\beta)$ for all $\beta\in(1,2]$, answering a question of Kalle et al. (2020). For example, we find that for the Komornik-Loreti constant $\beta\approx 1.78723$ we have $\tau(\beta)=(2-\beta)/(\beta-1)$. Furthermore, we show that (i) the function $\tau: \beta\mapsto\tau(\beta)$ is left continuous on $(1,2]$ with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) $\tau$ has no downward jumps, with $\tau(1+)=0$ and $\tau(2)=1/2$; and (iii) there exists an open set $O\subset(1,2]$, whose complement $(1,2]\setminus O$ has zero Hausdorff dimension, such that $\tau$ is real-analytic, convex and strictly decreasing on each connected component of $O$. Our strategy to find the critical value $\tau(\beta)$ depends on certain substitutions of Farey words and a renormalization scheme from dynamical systems.