## Invariant measures for Anosov maps with small holes.(English)Zbl 0963.37030

Authors’ 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 [ibid. 18, 1049-1073 (1998; Zbl 0982.37011)] 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.

### MSC:

 37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010) 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37A25 Ergodicity, mixing, rates of mixing 37A35 Entropy and other invariants, isomorphism, classification in ergodic theory

Zbl 0982.37011
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