For $\unicode[STIX]{x1D6FD}\in (1,2]$ the $\unicode[STIX]{x1D6FD}$-transformation $T_{\unicode[STIX]{x1D6FD}}:[0,1)\rightarrow [0,1)$ is defined by $T_{\unicode[STIX]{x1D6FD}}(x)=\unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$. For $t\in [0,1)$ let $K_{\unicode[STIX]{x1D6FD}}(t)$ be the survivor set of $T_{\unicode[STIX]{x1D6FD}}$ with hole $(0,t)$ given by
$$\begin{eqnarray}K_{\unicode[STIX]{x1D6FD}}(t):=\{x\in [0,1):T_{\unicode[STIX]{x1D6FD}}^{n}(x)\not \in (0,t)\text{ for all }n\geq 0\}.\end{eqnarray}$$ In this paper we characterize the bifurcation set
$E_{\unicode[STIX]{x1D6FD}}$ of all parameters
$t\in [0,1)$ for which the set-valued function
$t\mapsto K_{\unicode[STIX]{x1D6FD}}(t)$ is not locally constant. We show that
$E_{\unicode[STIX]{x1D6FD}}$ is a Lebesgue null set of full Hausdorff dimension for all
$\unicode[STIX]{x1D6FD}\in (1,2)$. We prove that for Lebesgue almost every
$\unicode[STIX]{x1D6FD}\in (1,2)$ the bifurcation set
$E_{\unicode[STIX]{x1D6FD}}$ contains infinitely many isolated points and infinitely many accumulation points arbitrarily close to zero. On the other hand, we show that the set of
$\unicode[STIX]{x1D6FD}\in (1,2)$ for which
$E_{\unicode[STIX]{x1D6FD}}$ 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
$\unicode[STIX]{x1D6FD}\in (1,2)$ a lower and an upper bound for the value
$\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$ such that the Hausdorff dimension of
$K_{\unicode[STIX]{x1D6FD}}(t)$ is positive if and only if
$t<\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$. We show that
$\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}\leq 1-(1/\unicode[STIX]{x1D6FD})$ for all
$\unicode[STIX]{x1D6FD}\in (1,2)$.