Fernandez and Demers studied the statistical properties of the Manneville-Pomeau map with the physical measure when a hole is put in the system, overcoming some of the problems caused by subexponential mixing. I’ll discuss the same setup, but with a class of natural equilibrium states. We find conditionally invariant measures and give precise information on the transitions between the fast exponentially mixing, the slow exponentially mixing and the subexponentially mixing phases. This is joint work with Mark Demers.