The classical problem of two fixed centers is integrable and has no chaos at all. However the relativistic problem is highly chaotic. In the case of particles with energy smaller than the escape energy most orbits fall into a black hole; the black holes act as attractors, although the system is conservative. Of special interest is the study of the asymptotic curves of the unstable periodic orbits, and of the homoclinic and heteroclinic orbits. We conclude that the system of two black holes is different in several ways from a generic nonlinear dynamical system without singularities.