On the uniform equidistribution of long closed horocycles

Andreas Strömbergsson
2004 Duke mathematical journal  
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
more » ... vergence. We also prove a result on the asymptotic joint equidistribution of a finite number of distinct subsegments having equal length proportional to . The proof in [EM] is for closed horocycles, that is, α = 0, β = 1, but the proof can be adapted to work for any fixed α < β.
doi:10.1215/s0012-7094-04-12334-6 fatcat:bb22ao65hbfdhkwuhz7ysun4f4