  For $\beta\in(1,2]$ the $\beta$-transformation $T_\beta: [0,1) \to [0,1)$ is
defined by $T_\beta ( x) = \beta x \pmod 1$. For $t\in[0, 1)$ let $K_\beta(t)$
be the survivor set of $T_\beta$ with hole $(0,t)$ given by
\[K_\beta(t):=\{x\in[0, 1): T_\beta^n(x)\not \in (0, t) \textrm{ for all }n\ge
0\}.\] In this paper we characterise the bifurcation set $E_\beta$ of all
parameters $t\in[0,1)$ for which the set valued function $t\mapsto K_\beta(t)$
is not locally constant. We show that $E_\beta$ is a Lebesgue null set of full
Hausdorff dimension for all $\beta\in(1,2)$. We prove that for Lebesgue almost
every $\beta\in(1,2)$ the bifurcation set $E_\beta$ contains both infinitely
many isolated and accumulation points arbitrarily close to zero. On the other
hand, we show that the set of $\beta\in(1,2)$ for which $E_\beta$ contains no
isolated points has zero Hausdorff dimension. These results contrast with the
situation for $E_2$, the bifurcation set of the doubling map. Finally, we give
for each $\beta \in (1,2)$ a lower and upper bound for the value $\tau_\beta$,
such that the Hausdorff dimension of $K_\beta(t)$ is positive if and only if
$t< \tau_\beta$. We show that $\tau_\beta \le 1-\frac1{\beta}$ for all $\beta
\in (1,2)$.
