Duke Mathematical Journal

Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces

Steven P. Lalley

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Let ϒ be a compact, negatively curved surface. From the (finite) set of all closed geodesics on ϒ of length at most L, choose one, say, γL, at random, and let N(γL) be the number of its self-intersections. It is known that there is a positive constant κ depending on the metric such that N(γL)/L2κ in probability as L. The main results of this article concern the size of typical fluctuations of N(γL) about κL2. It is proved that if the metric has constant curvature 1, then typical fluctuations are of order L; in particular, as L the random variables (N(γL)κL2)/L converge in distribution. In contrast, it is also proved that if the metric has variable negative curvature, then fluctuations of N(γL) are of order L3/2; in particular, the random variables (N(γL)κL2)/L3/2 converge in distribution to a Gaussian distribution with positive variance. Similar results are proved for generic geodesics, that is, geodesics whose initial tangent vectors are chosen randomly according to normalized Liouville measure.

Article information

Duke Math. J., Volume 163, Number 6 (2014), 1191-1261.

First available in Project Euclid: 11 April 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M05: Fundamental group, presentations, free differential calculus
Secondary: 53C22: Geodesics [See also 58E10] 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)


Lalley, Steven P. Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces. Duke Math. J. 163 (2014), no. 6, 1191--1261. doi:10.1215/00127094-2649425. https://projecteuclid.org/euclid.dmj/1397223299

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