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.
Categories: math.DS