Spectrum of the Ruelle operator and exponential decay of correlations for open billiard flows (Q2747079)

The paper deals with the billiard flow in the exterior of several convex disjoint obstacles on the plane satisfying an additions visibility condition (called ``no eclipse''). Using a modification of Dolgopyat's techniques, the author gets spectral estimates for the Ruelle operator related to a Markov family for the nonwandering (trapped) trajectories of the flow. As a consequence, the author gets exponential decay of correlations for the billiard flow restricted to the nonwandering set. In addition, he derives the existence of a meromorphic extension of the dynamical zeta function of the billiard flow to a half-plane Re\((s)<h_T-\varepsilon\), where \(h_T\) 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 \(\lambda>0\).