We construct semi-infinite billiard domains which reverse the direction of most incoming particles. We prove that almost all particles will leave the open billiard domain after a finite number of reflections. Moreover, with high probability the exit velocity is exactly opposite to the entrance velocity, and the particle's exit point is arbitrarily close to its initial position. The method is based on asymptotic analysis of statistics of entrance times to a small interval for irrational circle

more »

... tations. The rescaled entrance times have a limiting distribution in the limit when the length of the interval vanishes. The proof of the main results follows from the study of related limiting distributions and their regularity properties.