  In this paper we prove two asymptotic estimates for pairs of closed
trajectories for open billiards similar to those established by Pollicott and
Sharp for closed geodesics on negatively curved compact surfaces. The first of
these estimates holds for general open billiards in any dimension. The more
intricate second estimate is established for open billiards satisfying the so
called Dolgopyat type estimates. This class of billiards includes all open
billiards in the plane and open billiards in $\R^N, N \geq 3$ satisfying some
additional conditions.
