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

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.