Asymptotic escape rates and limiting distributions for multimodal maps

We consider multimodal maps with holes and study the evolution of the open systems with respect to equilibrium states for both geometric and Holder potentials. For small holes, we show that a large class of initial distributions share the same escape rate and converge to a unique absolutely continuous conditionally invariant measure; we also prove a variational principle connecting the escape rate to the pressure on the survivor set, with no conditions on the placement of the hole. Finally, introducing a weak condition on the centre of the hole, we prove scaling limits for the escape rate for holes centred at both periodic and nonperiodic points, as the diameter of the hole goes to zero.