Modular curves, C^* algebras, and chaotic cosmology

Abstract

We make some brief remarks on the relation of the mixmaster universe model of chaotic cosmology to the geometry of modular curves and to noncommutative geometry. We show that the full dynamics of the mixmaster universe is equivalent to the geodesic flow on the modular curve X_(Γ0(2)). We then consider a special class of solutions, with bounded number of cycles in each Kasner era, and describe their dynamical properties (invariant density, Lyapunov exponent, topological pressure). We relate these properties to the noncommutative geometry of a moduli space of such solutions, which is given by a Cuntz–Krieger C^∗-algebra.