  In this paper we obtain the asymptotic behavior of solutions of the
Klein-Gordon equation on Lorentzian manifolds $(X^\circ,g)$ which are de
Sitter-like at infinity. Such manifolds are Lorentzian analogues of the
so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces.
Under global assumptions on the (null)bicharacteristic flow, namely that the
boundary of the compactification X is a union of two disjoint manifolds, Y+ and
Y-, and each bicharacteristic converges to one of these two manifolds as the
parameter along the bicharacteristic goes to plus infinity, and to the other
manifold as the parameter goes to minus infinity, we also define the scattering
operator, and show that it is a Fourier integral operator associated to the
bicharacteristic flow from Y+ to Y-.
