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

الأصل · EN

Let Υ be a compact, negatively curved surface. From the (finite) set of all closed geodesics on Υ of length ≤ 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)/L² → κ in probability as L→ ∞. The main results of this paper 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, (N (γₗ)-κL²)/L converges weakly to a nondegenerate probability 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, (N (γₗ)-κL²)/L³/² 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.