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

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 $\{T_{\beta }^n(x):n\ge 0\}$ never hits the interval $[0,t)$. Kalle et al.~[{\em Ergodic Theory Dynam. Systems} {\bf 40} (2020), no.~9, 2482–2514] considered the case $\beta \in (1,2]$. They studied the set-valued bifurcation set ${\mathcal{E}}_{\beta }:=\{t\in [0,1):K_{\beta }(t^{\mathrm{\prime }})\mathrm{\ne }{K}_{\beta }(t)\mathrm{\forall }{t}^{\mathrm{\prime }}>t\}$ 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 [{\em Ergodic Theory Dynam. Systems} {\bf 43} (2023), no.~6, 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 $(1,\mathrm{\infty })$ with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) $\tau$ has no downward jumps; and (iii) there exists an open set $O\subset (1,\mathrm{\infty })$, whose complement $(1,\mathrm{\infty })\mathrm{\backslash }O$ has zero Hausdorff dimension, such that $\tau$ is real-analytic, strictly convex and strictly decreasing on each connected component of $O$. We also prove several topological properties of the bifurcation set ${\mathcal{E}}_{\beta }$. The key to extending the results from $\beta \in (1,2]$ to all $\beta >1$ is an appropriate generalization of the Farey words that are used to parametrize the connected components of the set $O$. Some of the original proofs from the above-mentioned papers are simplified.