The vacuum Einstein equations for Bianchi space times (that is space
times that can be foliated into three dimensional space like slices
that are all homogenuous spaces) reduce to a system of ordinary differential equations.
The conjectures of Belinskii, Khalatnikov and Lifshitz predict that
for almost all initial data the solutions of these differential
equation behave like trajectories of a billiard in a Farey triangle in
the hyperbolic plane, that is, a triangle whose three vertices are
ideal points. In joint work with M. Reiterer and E. Trubowitz we show
that, for a set of initial data that has positive measure, this is indeed the case.
We use ideas inspired by scattering theory for approximations of the system.
The fact that billiard in a Farey triangle is chaotic leads us to
small divisor problems similar to those of KAM theory in Hamiltonian dynamics.