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 < l, 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.

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