JournalslemVol. 58, No. 3/4pp. 249–313

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

  • Manfred Einsiedler

    ETH Zürich, Switzerland
  • Elon Lindenstrauss

    The Hebrew University of Jerusalem, Israel
  • Philippe Michel

    Ecole Polytechnique Fédérale de Lausanne, Switzerland
  • Akshay Venkatesh

    Stanford University, United States
The distribution of closed geodesics on the modular surface, and Duke's theorem cover
Download PDF

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.

Cite this article

Manfred Einsiedler, Elon Lindenstrauss, Philippe Michel, Akshay Venkatesh, The distribution of closed geodesics on the modular surface, and Duke's theorem. Enseign. Math. 58 (2012), no. 3, pp. 249–313

DOI 10.4171/LEM/58-3-2