Spectrum of the Ruelle operator and exponential decay of correlations for open billiard flows

The paper deals with the billiard flow in the exterior of several strictly convex disjoint domains in the plane with smooth boundaries satisfying an a dditional (visibility) condition. Using a modification of the technique of Dolgopyat, we get spectral estimates for the Ruelle operator related to a M arkov family for the nonwandering (trapping) set of the flow similar to tho se of Dolgopyat in the case of transitive Anosov flows on compact manifolds with smooth jointly nonintegrable horocycle foliations. As a consequence, we get exponential decay of correlation for Holder continuous potentials on the nonwandering set. Combining the spectral estimate for the Ruelle opera tor with an argument of Pollicott and Sharp, we also derive the existence o f a meromorphic continuation of the dynamical zeta function of the billiard flow to a half-plane Re (s) < h(T) - epsilon, where hT is the topological entropy of the billiard flow, and an asymptotic formula with an error term for the number pi(lambda) of closed orbits of least period lambda > 0.