Anosov maps with small holes
by
Serge Troubetzkoy
University of Alabama at Birmingham
Coauthors: R. Markarian, N. Chernov

Open billiards and other open Hamiltonian systems have become very popular in physics under the name of chaotic scattering theory in the past ten years. They have been studied numerically and heuristically. Together with Chernov and Markarian we study a model problem: Anosov diffeomorphisms on surfaces with small holes. The points that are mapped into the holes disappear and never return. We proved the existence of a conditionally invariant measure \mu₊. 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.

Date received: February 24, 1998