Title: Markov extensions for some dynamical systems with holes.

Abstract: Analysis of dynamical systems with holes generally centers on conditionally invariant measures for the systems and their associated escape rates. I will present a general construction of families of absolutely continuous conditionally invariant measures under quite general conditions from which follow some considerations on what defines a natural measure for the system. I will briefly review Young's Tower Map (Markov extension) which is a useful tool in the study of the statistical properties of dynamical systems and explain how it can be applied to systems with holes. I will then state some recent results for expanding maps of the interval and the logistic family using this method.