Intrinsic Ergodicity of Open dynamical systems for the doubling map

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.