Quadratic Irrationals, Closed Geodesics on the Modular Surface and Dynamical Zeta Functions

Abstract

We show that generating functions associated to the sequence of convergents of a quadratic irrational are related in a natural way to the dynam- ical zeta function of a hyperbolic automorphism of the 2-torus. As a corollary, this shows that the L\'evy constant of a quadratic irrational appears naturally as the topological entropy of such maps.