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