We consider escape from chaotic maps through a subset of phase space, the
hole. Escape rates are known to be locally constant functions of the hole
position and size. In spite of this, for the doubling map we can extend the
current best result for small holes, a linear dependence on hole size h, to
include a smooth h^2 ln h term and explicit fractal terms to h^2 and higher
orders, confirmed by numerical simulations. For more general hole locations the
asymptotic form depends on a dynamical Diophantine condition using periodic
orbits ordered by stability.Comment: This version has a new section investigating different hole
locations. Now 9 pages, 3 figure
