Invariant measures for Anosov maps with small holes

Abstract

We study Anosov diffeomorphisms on surfaces with small holes. The points that are mapped into the holes disappear and never return. In our previous paper we proved the existence of a conditionally invariant measure $\mu_+$. Here we show that the iterations of any initially smooth measure, after renormalization, converge to $\mu_+$. We construct the related invariant measure on the repeller and prove that it is ergodic and K-mixing. We prove the escape rate formula, relating the escape rate to the positive Lyapunov exponent and the entropy.