E-10, BE10 and arithmetical chaos in superstring cosmology

It is shown that the neverending oscillatory behavior of the generic soluti on, near a cosmological singularity, of the massless bosonic sector of supe rstring theory can be described as a billiard motion within a simplex in ni ne-dimensional hyperbolic space. The Coxeter group of reflections of this b illiard is discrete and is the Weyl group of the hyperbolic Kac-Moody algeb ra E-10 (for type II) or BE10 (for type I or heterotic), which are both ari thmetic. These results lead to a proof of the chaotic ("Anosov") nature of the classical cosmological oscillations, and suggest a "chaotic quantum bil liard" scenario of vacuum selection in string theory.