  Given $\beta\in(1,2]$, let $T_\beta$ be the $\beta$-transformation on the
unit 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 $t\mapsto
K_\beta(t)$, Kalle et al. [Ergodic Theory Dynam. Systems, 40 (9): 2482--2514,
2020] conjectured that \[ \dim_H\big(\mathscr{E}_\beta\cap[t,1]\big)=\dim_H
K_\beta(t) \qquad \forall\,t\in(0,1). \] The main purpose of this article is to
prove this conjecture. We do so by investigating dynamical properties of the
symbolic equivalent of the survivor set $K_\beta(t)$, in particular its entropy
and topological transitivity. In addition, we compare $\mathscr{E}_\beta$ with
the 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.), and
show that, for Lebesgue-almost every $\beta\in(1,2]$, the difference
$\mathscr{E}_\beta\backslash\mathscr{B}_\beta$ has positive Hausdorff
dimension, but for every $k\in\{0,1,2,\dots\}\cup\{\aleph_0\}$, there are
infinitely many values of $\beta$ such that the cardinality of
$\mathscr{E}_\beta\backslash\mathscr{B}_\beta$ is exactly $k$. For a countable
but dense subset of $\beta$'s, we also determine the intervals of constancy of
the function $t\mapsto \dim_H K_\beta(t)$.
  Some connections with other topics in dynamics, such as kneading invariants
of Lorenz maps and the doubling map with an arbitrary hole, are also discussed.
