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On the uniform equidistribution of long closed horocycles. (English) Zbl 1060.37023

Given a hyperbolic surface of finite area with cusps, a horocycle is a level set of the Busemann function associated to a ray go in into the cusp. When we are inside the cusp, the horocycle is a simple closed curve that becomes arbitrarily short when we go to the corresponding end of the manifold. If we move in the other direction, the horocycles become arbitrarily long and are not simple curves anymore. In fact, they become equidistributed when the length goes to infinity. Namely, if \(l\) denotes the length or the horocycle, and \(A\) any regular subset of the hyperbolic surface, the ratio between the length of the intersection of the horocycle with \(A\) and the total length \(l\) converges to the ratio between the area of \(A\) and the area of the surface.
The main theorem of this paper shows that the equidistribution holds for subsegments of the horocycle provided that the length is larger than \(l^{1/2+\varepsilon}\). The power \(1/2+\varepsilon\) is the best possible: segments of length \(l^{1/2}\) can be not equidistributed. This improves previous versions of this result: for instance, D. A. Hejhal [Asian J. Math. 4, 839–854 (2000; Zbl 1014.11038)] proved this for some power depending on the group, and a proof when the length is precisely equal to \(l^{1/2+\varepsilon}\) for some fixed exponent \(1/2+\varepsilon\) can be deduced from the techniques of N. A. Shah [Proc. Indian Acad. Sci., Math. Sci. 106, 105–125 (1996; Zbl 0864.22004)] and A. Eskin and C. McMullen [Duke Math. J. 71, 181–209 (1993; Zbl 0798.11025)].
If one replaces sets by tests functions, this theorem is read in terms of functions and two proofs are given, one using ergodic theory and the other one spectral theory. The former allows one to obtain asymptotic equidistribution on the unit tangent bundle, the latter one gives explicit information of the rate of convergence.
There is also a result on the asymptotic joint equidistribution of a finite number of distinct subsegments having length proportional to the length of the horocycle.

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
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