Behavior of the Escape Rate Function in Hyperbolic Dynamical Systems

TL;DR

This paper investigates how the escape rate in hyperbolic dynamical systems depends on the size and shape of holes, establishing continuity properties and revealing complex behaviors like devil's staircase structures.

Contribution

It proves the existence and Holder continuity of escape rates for systems with small holes and extends the analysis to general holes in Anosov diffeomorphisms, highlighting the complex nature of escape rate functions.

Findings

Escape rate is Holder continuous for small holes with Young towers.

Escape rate function can form a devil's staircase with jumps.

Bounds on escape rates are derived using pressure on survivor sets.

Abstract

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.