Invariant measures for Anosov maps with small holes

We study Anosov diffeomorphisms on surfaces with small holes. The points th at are mapped into the holes disappear and never return. In our previous pa per we proved the existence of a conditionally invariant measure mu(+). Her e we show that the iterations of any initially smooth measure, after renorm alization, converge to mu(+). We construct the related invariant measure on the repeller and prove that it is ergodic and K-mixing. We prove the escap e rate formula, relating the escape rate to the positive Lyapunov exponent and the entropy.