Markov Extensions for Dynamical Systems with Holes: An Application to Expanding Maps of the Interval

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We introduce the Markov extension, represented schematically as a tower, to the study of dynamical systems with holes. For tower maps with small holes, we prove the existence of conditionally invariant probability measures which are absolutely continuous with respect to Lebesgue measure (abbreviated a.c.c.i.m.). We develop restrictions on the Lebesgue measure of the holes and simple conditions on the dynamics of the tower which ensure existence and uniqueness in a class of Holder continuous densities. We then use these results to study the existence and properties of a.c.c.i.m. for expanding maps of the interval with holes. We obtain the convergence of the a.c.c.i.m. to the SRB measure of the corresponding closed system as the measure of the hole shrinks to zero.