Periodic orbit sum rules for billiards: accelerating cycle expansions

We show that the periodic orbit sums for two-dimensional billiards satisfy an infinity of exact sum rules. We demonstrate their utility by using the f low conservation sum rule to accelerate the convergence of cycle expansions for the overlapping three-disc billiard. The effectiveness of the approach is studied by applying the method on averages, known explicitly by other s um rules. The method is then applied to the Lyapunov exponent.