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 $\{T^n_\beta(x): n\ge 0\}$ 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 $\mathscr{E}_\beta:=\{t\in[0,1): K_\beta(t')\ne K_\beta(t)~\forall t'>t\}$ 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 [{\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 $(1,\infty)$ with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) $\tau$ has no downward jumps; and (iii) there exists an open set $O\subset(1,\infty)$, whose complement $(1,\infty)\backslash O$ 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 $\mathscr{E}_\beta$. 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.