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 [Ergod. Th. & Dynam. Sys.40(9) (2020) 2482–2514] proved 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. Consequently, the dimension
$\dim _{H} K_{\beta }(t)$ is not jointly continuous in
$\beta $ and t. Our strategy to find the critical value
$\tau (\beta )$ depends on certain substitutions of Farey words and a renormalization scheme from dynamical systems.