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

Given β>1 , let T_β be the β -transformation on the unit circle $[0,1)$ , defined by T_β (x)=β x- β x . For each t∈ [0,1) , let K_β (t) be the survivor set consisting of all x∈ [0,1) whose orbit ⁿ_β (x): n≥ 0\ never enters the interval $[0,t)$ . Kalle et al [ Ergod. Th. & Dynam. Sys. 40 (9) (2020), 2482–2514] considered the case β ∈ (1,2] . They studied the set-valued bifurcation set E_β :=∈ [0,1): K_β (t')≠ K_β (t)~ for all t'>t\ and proved that the Hausdorff dimension function t↦ H K_β (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 β ∈ (1,2] , the critical value τ (β ):=min >0: η _β (t)=0\ . The purpose of the present article is to extend these results to all β>1 . In addition to calculating τ (β ) , we show that: (i) the function τ : β ↦ τ (β ) is left-continuous on (1,∞ ) with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) τ has no downward jumps; and (iii) there exists an open set O⊂ (1,∞ )