We study the family of quadratic maps f_a(x) = 1 - ax^2 on the interval[-1,1] with a between 0 and 2. When small holes are introduced into the system,we prove the existence of an absolutely continuous conditionally invariantmeasure using the method of Markov extensions. The measure has a density whichis bounded away from zero and is analogous to the density for the correspondingclosed system. These results establish the exponential escape rate of Lebesguemeasure from the system, despite the contraction in a neighborhood of thecritical point of the map. We also prove convergence of the conditionallyinvariant measure to the SRB measure for f_a as the size of the hole goes tozero.
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