On the uniform equidistribution of long closed horocycles
Andreas Strömbergsson
Source: Duke Math. J. Volume 123, Number 3
(2004), 507-547.
Abstract
It is well known that on any given hyperbolic surface of finite area, a closed horocycle of length ℓ becomes asymptotically equidistributed as ℓ→∞. In this paper we prove that any subsegment of length greater than ℓ1/2 + ε of such a closed horocycle also becomes equidistributed as ℓ→∞. 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 ℓ.
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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1086957715
Digital Object Identifier: doi:10.1215/S0012-7094-04-12334-6
Zentralblatt MATH identifier: 02103765
Mathematical Reviews number (MathSciNet): MR2068968
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