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We study two classes of dynamical systems with holes: expanding maps of theinterval and Collet-Eckmann maps with singularities. In both cases, we provethat there is a natural absolutely continuous conditionally invariant measure$\mu$ (a.c.c.i.m.) with the physical property that strictly positive H\"{o}ldercontinuous functions converge to the density of $\mu$ under the renormalizeddynamics of the system. In addition, we construct an invariant measure $\nu$,supported on the Cantor set of points that never escape from the system, thatis ergodic and enjoys exponential decay of correlations for H\"{o}lderobservables. We show that $\nu$ satisfies an equilibrium principle whichimplies that the escape rate formula, familiar to the thermodynamic formalism,holds outside the usual setting. In particular, it holds for Collet-Eckmannmaps with holes, which are not uniformly hyperbolic and do not admit a finiteMarkov partition.  We use a general framework of Young towers with holes and first prove resultsabout the \accim and the invariant measure on the tower. Then we show how totransfer results to the original dynamical system. This approach can beexpected to generalize to other dynamical systems than the two above classes.

Existence and convergence properties of physical measures for certain dynamical systems with holes