American Journal of Mathematics

The paper deals with the billiard flow in the exterior of several strictly convex disjoint domains in the plane with smooth boundaries satisfying an additional (visibility) condition. Using a modification of the technique of Dolgopyat, we get spectral estimates for the Ruelle operator related to a Markov family for the nonwandering (trapping) set of the flow similar to those of Dolgopyat in the case of transitive Anosov flows on compact manifolds with smooth jointly non-integrable horocycle foliations. As a consequence, we get exponential decay of correlation for Hölder continuous potentials on the nonwandering set. Combining the spectral estimate for the Ruelle operator with an argument of Pollicott and Sharp, we also derive the existence of a meromorphic continuation of the dynamical zeta function of the billiard flow to a half-plane Re(s) < h_T - ε, where h_T is the topological entropy of the billiard flow, and an asymptotic formula with an error term for the number π(λ) of closed orbits of least period λ > 0.