4 research outputs found
Behavior of the Escape Rate Function in Hyperbolic Dynamical Systems
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.Comment: 21 pages. v2 differs from v1 only by additions to the acknowledgment