Quasi-invariant measures, escape rates and the effect of the hole

Let T be a piecewise expanding interval map and T H be an abstract perturbation of T into an interval map with a hole. Given a number l, 0 < l < 1, we compute an upper-bound on the size of a hole needed for the existence of an absolutely continuous conditionally invariant measure (accim) with escape rate not greater than-In(1-l). The two main ingredients of our approach are Ulam's method and an abstract perturbation result of Keller and Liverani