  We give sufficient conditions for intervals $(a,b)$ such that the associated
open dynamical system for the doubling map is intrinsically ergodic. We also
show that the set of parameters $(a,b) \in (\frac{1}{4}, \frac{1}{2}) \times
(\frac{1}{2},\frac{3}{4})$ such that the attractor $(\Lambda_{(a,b)},
f_{(a,b)})$ is intrinsically ergodic has full Lebesgue measure and we construct
a set of points where intrinsic ergodicity does not hold.
