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Given $\beta\in(1,2]$, let $T_\beta$ be the $\beta$-transformation on theunit 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 enters the interval $[0,t)$. Letting$\mathscr{E}_\beta$ denote the bifurcation set of the set-valued map $t\mapstoK_\beta(t)$, Kalle et al. [Ergodic Theory Dynam. Systems, 40 (9): 2482--2514,2020] conjectured that \[ \dim_H\big(\mathscr{E}_\beta\cap[t,1]\big)=\dim_HK_\beta(t) \qquad \forall\,t\in(0,1). \] The main purpose of this article is toprove this conjecture. We do so by investigating dynamical properties of thesymbolic equivalent of the survivor set $K_\beta(t)$, in particular its entropyand topological transitivity. In addition, we compare $\mathscr{E}_\beta$ withthe bifurcation set $\mathscr{B}_\beta$ of the map $t\mapsto \dim_H K_\beta(t)$(which is a decreasing devil's staircase by a theorem of Kalle et al.), andshow that, for Lebesgue-almost every $\beta\in(1,2]$, the difference$\mathscr{E}_\beta\backslash\mathscr{B}_\beta$ has positive Hausdorffdimension, but for every $k\in\{0,1,2,\dots\}\cup\{\aleph_0\}$, there areinfinitely many values of $\beta$ such that the cardinality of$\mathscr{E}_\beta\backslash\mathscr{B}_\beta$ is exactly $k$. For a countablebut dense subset of $\beta$'s, we also determine the intervals of constancy ofthe function $t\mapsto \dim_H K_\beta(t)$.  Some connections with other topics in dynamics, such as kneading invariantsof Lorenz maps and the doubling map with an arbitrary hole, are also discussed.

Entropy plateaus, transitivity and bifurcation sets for the $\beta$-transformation with a hole at $0$