  We investigate the dependence of the escape rate on the position of a hole
placed in uniformly hyperbolic systems admitting a finite Markov partition. We
derive an exact periodic orbit formula for finite size Markov holes which
differs from other periodic expansions in the literature and can account for
additional distortion to maps with piecewise constant expansion rate. Using
asymptotic expansions in powers of hole size we show that for systems conjugate
to the binary shift, the average escape rate is always larger than the
expectation based on the hole size. Moreover, we show that in the small hole
limit the difference between the two decays like a known constant times the
square of the hole size. Finally, we relate this problem to the random choice
of hole positions and we discuss possible extensions of our results to
non-Markov holes as well as applications to leaky dynamical networks.
