Invariant measures for Anosov maps with small holes

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 [6] we proved the existence of a conditionally invariant measure ¯+ . Here we show that the iterations of any initially smooth measure, after renormalization, converge to ¯+ . 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. AMS classification numbers: 58F12, 58F15, 58F11 Keywords: Repellers, chaotic scattering theory, escape rates. 1 Introduction A pictorial model of a chaotic dynamical system with holes (also known as open dynamical systems) was proposed by Pianigiani and Yorke [14]. Imagine a Sinai billiard table (with dispersing boundary) in which the dynamics of the ball is strongly chaotic. Let one or more holes be cut in the table, so that the ball can fall through...