Computing Conditionally Invariant Measures and Escape Rates

We consider expanding discontinuous maps with holes and the associated Markov partitions. These partitions are characterized by the orbits of the turning points and the discontinuity points of the maps. For these maps we study the process of escape of points from the interval, that is characterized by a conditionally invariant measure. We construct this measure which naturally generates an unique invariant probability measure. A direct consequence of these results is to compute explicitly the escape rate, with connection to the transfer operator. We also introduce a weighted kneading theory which allows a rigorous computation of the escape rate