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
$\unicode[STIX]{x1D6FD}\in (1,2)$
.