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

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

2011-11-08

arxiv.org/abs/1111.2060v1

Mathematics

Dynamical Systems

Scientific paper

Let $S$ be a compact surface with constant negative curvature -1. From among all closed geodesics on $\Upsilon$ of length $\leq T$, choose one at random and let $N_{T}$ be the number of its self-intersections. We prove that for a certain constant $\kappa =\kappa_{\Upsilon}>0$ the random variable $(N_{T}-\kappa T^{2})/T$ has a limit distribution as $T \rightarrow \infty$. We conjecture that for surfaces of \emph{variable} negative curvature the order of magnitude of typical variations is $T^{3/2}$, rather than $T$. We also prove analogous results for generic geodesics, that is geodesics whose initial tangent vectors are chosen randomly according to normalized Liouville measure.

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