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 - ln(1 - l). The two main ingredients of our approach are Ulam's method and an abstract perturbation result of Keller and Liverani.