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Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces
Let be a compact, negatively curved surface. From the (finite)
set of all closed geodesics on of length , choose one, say
, at random and let be the number of its
self-intersections. It is known that there is a positive constant
depending on the metric such that in
probability as . The main results of this paper concern
the size of typical fluctuations of about . It
is proved that if the metric has constant curvature -1 then typical
fluctuations are of order , in particular,
converges weakly to a nondegenerate probability distribution. In contrast, it
is also proved that if the metric has variable negative curvature then
fluctuations of are of order , in particular, converges weakly to a Gaussian
distribution. 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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