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.
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