Duke Mathematical Journal

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

Steven P. Lalley

Abstract

Let $\Upsilon$ be a compact, negatively curved surface. From the (finite) set of all closed geodesics on $\Upsilon$ of length at most $L$, choose one, say, $\gamma_{L}$, at random, and let $N(\gamma_{L})$ be the number of its self-intersections. It is known that there is a positive constant $\kappa$ depending on the metric such that $N(\gamma_{L})/L^{2}\rightarrow\kappa$ in probability as $L\rightarrow\infty$. The main results of this article concern the size of typical fluctuations of $N(\gamma_{L})$ about $\kappa L^{2}$. It is proved that if the metric has constant curvature $-1$, then typical fluctuations are of order $L$; in particular, as $L\rightarrow\infty$ the random variables $(N(\gamma_{L})-\kappa L^{2})/L$ converge in distribution. In contrast, it is also proved that if the metric has variable negative curvature, then fluctuations of $N(\gamma_{L})$ are of order $L^{3/2}$; in particular, the random variables $(N(\gamma_{L})-\kappa L^{2})/L^{3/2}$ 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

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

Dates
First available in Project Euclid: 11 April 2014

https://projecteuclid.org/euclid.dmj/1397223299

Digital Object Identifier
doi:10.1215/00127094-2649425

Mathematical Reviews number (MathSciNet)
MR3192528

Zentralblatt MATH identifier
1328.37033

Citation

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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