The distribution of closed geodesics on the modular surface, and Duke's theorem

TL;DR

This paper provides an ergodic theoretic proof of Duke's theorem on the equidistribution of closed geodesics on the modular surface, replacing congruence conditions with entropy-based methods.

Contribution

It introduces a conceptual entropy-based approach to prove equidistribution, removing the need for congruence assumptions used in previous proofs.

Findings

Ergodic theoretic proof of Duke's theorem

Entropy methods can replace congruence conditions

Enhanced understanding of geodesic distribution on modular surfaces

Abstract

We give an ergodic theoretic proof of a theorem of Duke about equidistribution of closed geodesics on the modular surface. The proof is closely related to the work of Yu. Linnik and B. Skubenko, who in particular proved this equidistribution under an additional congruence assumption on the discriminant. We give a more conceptual treatment using entropy theory, and show how to use positivity of the discriminant as a substitute for Linnik's congruence condition.