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We give sufficient conditions for intervals $(a,b)$ such that the associatedopen dynamical system for the doubling map is intrinsically ergodic. We alsoshow 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 constructa set of points where intrinsic ergodicity does not hold.

Intrinsic Ergodicity of Open dynamical systems for the doubling map