Duke Mathematical Journal

On the uniform equidistribution of long closed horocycles

Andreas Strömbergsson

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It is well known that on any given hyperbolic surface of finite area, a closed horocycle of length becomes asymptotically equidistributed as →∞. In this paper we prove that any subsegment of length greater than 1/2 + ε of such a closed horocycle also becomes equidistributed as →∞. The exponent 1/2 + ε is the best possible and improves upon a recent result by Hejhal [He3]. We give two proofs of the above result; our second proof leads to explicit information on the rate of convergence. We also prove a result on the asymptotic joint equidistribution of a finite number of distinct subsegments having equal length proportional to .

Article information

Duke Math. J., Volume 123, Number 3 (2004), 507-547.

First available in Project Euclid: 11 June 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Secondary: 11F 30F35: Fuchsian groups and automorphic functions [See also 11Fxx, 20H10, 22E40, 32Gxx, 32Nxx]


Strömbergsson, Andreas. On the uniform equidistribution of long closed horocycles. Duke Math. J. 123 (2004), no. 3, 507--547. doi:10.1215/S0012-7094-04-12334-6. https://projecteuclid.org/euclid.dmj/1086957715

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