For a fixed initial reference measure, we study the dependence of the escape rate on the hole for a smooth or piecewise smooth hyperbolic map. First, we prove the existence and Holder continuity of the escape rate for systems with small holes admitting Young towers. Then we consider general holes for Anosov diffeomorphisms, without size or Markovian restrictions. We prove bounds on the upper and lower escape rates using the notion of pressure on the survivor set and show that a variational principle holds under generic conditions. However, we also show that the escape rate function forms a devil's staircase with jumps along sequences of regular holes and present examples to elucidate some of the difficulties involved in formulating a general theory.
Demers, Mark and Wright, Paul, "Behavior of the escape rate function in hyperbolic dynamical systems" (2012). Math & Computer Science Faculty Publications. 44.
Mark Demers and Paul Wright. "Behavior of the escape rate function in hyperbolic dynamical systems." Nonlinearity 25 (2012), 2133-2150.