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

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.