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

TL;DR

This paper reveals a connection between quadratic irrationals, closed geodesics on the modular surface, and dynamical zeta functions, linking number theory and dynamical systems through generating functions and entropy.

Contribution

It establishes a natural relation between generating functions of quadratic irrational convergents and the dynamical zeta function of hyperbolic torus automorphisms, highlighting the Lévý constant as topological entropy.

Findings

Generating functions relate to dynamical zeta functions

Lévý constant appears as topological entropy

Connection between number theory and dynamical systems

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.