# Applied Math Colloquium

### Local escape rates in dynamical systems

### Abstract

If one places a hole of positive measure in an ergodic dynamical system, then almost every point will eventually hit the hole and disappear. The exponential decay rate of the left-over set is the escape rate to the hole. Naturally a smaller hole will have a smaller escape rate. However, if one divides the escape rate by the size (measure) of the hole and takes a limit as the size goes to zero, then one obtains the local escape rate. In this talk we show that if the invariant measure is -mixing (with respect to a generating partition) then the local escape rate is equal to one at every point except at periodic points, where it is given by one minus the extremal index. We apply this result to Young towers, equilibrium states for Axiom A systems, interval maps and conformal maps.