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15 April 2014 Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces
Steven P. Lalley
Duke Math. J. 163(6): 1191-1261 (15 April 2014). DOI: 10.1215/00127094-2649425

Abstract

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.

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Steven P. Lalley. "Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces." Duke Math. J. 163 (6) 1191 - 1261, 15 April 2014. https://doi.org/10.1215/00127094-2649425

Information

Published: 15 April 2014
First available in Project Euclid: 11 April 2014

zbMATH: 1328.37033
MathSciNet: MR3192528
Digital Object Identifier: 10.1215/00127094-2649425

Subjects:
Primary: 57M05
Secondary: 37D40, 53C22

Rights: Copyright © 2014 Duke University Press

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Vol.163 • No. 6 • 15 April 2014
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