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For a fixed initial reference measure, we study the dependence of the escaperate on the hole for a smooth or piecewise smooth hyperbolic map. First, weprove the existence and Holder continuity of the escape rate for systems withsmall holes admitting Young towers. Then we consider general holes for Anosovdiffeomorphisms, without size or Markovian restrictions. We prove bounds on theupper and lower escape rates using the notion of pressure on the survivor setand show that a variational principle holds under generic conditions. However,we also show that the escape rate function forms a devil's staircase with jumpsalong sequences of regular holes and present examples to elucidate some of thedifficulties involved in formulating a general theory.

Behavior of the Escape Rate Function in Hyperbolic Dynamical Systems