The $\beta $-transformation with a hole at $0$: the general case

Abstract Given $\beta>1$ , let $T_\beta $ be the $\beta $ -transformation on the unit circle $[0,1)$ , defined by $T_\beta (x)=\beta x-\lfloor \beta x\rfloor $ . 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 enters the interval $[0,t)$ . Kalle et al [ Ergod. Th. & Dynam. Sys. 40 (9) (2020), 2482–2514] considered the case $\beta \in (1,2]$ . They studied the set-valued bifurcation set $\mathscr {E}_\beta :=\{t\in [0,1): K_\beta (t')\ne K_\beta (t)~\text { for all } t'>t\}$ and proved that the Hausdorff dimension function $t\mapsto \dim _H K_\beta (t)$ is a non-increasing Devil’s staircase. In a previous paper [ Ergod. Th. & Dynam. Sys. 43 (6) (2023), 1785–1828], we determined, for all $\beta \in (1,2]$ , the critical value $\tau (\beta ):=\min \{t>0: \eta _\beta (t)=0\}$ . The purpose of the present article is to extend these results to all $\beta>1$ . In addition to calculating $\tau (\beta )$ , we show that: (i) the function $\tau : \beta \mapsto \tau (\beta )$ is left-continuous on $(1,\infty )$ with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) $\tau $ has no downward jumps; and (iii) there exists an open set $O\subset (1,\infty )$