Intersection numbers of geodesic arcs

Abstract

For a compact surface S with constant curvature -κ (for some and kappa 0) and genus g ≥ 2, we show that the tails of the distribution of the normalized intersection numbers i(α, β)/l(α)l(β) (where i(α, β) is the intersection number of the closed geodesics α and β and l(·) denotes the geometric length) are estimated by a decreasing exponential function. As a consequence, we find the asymptotic average of the normalized intersection numbers of pairs of closed geodesics on S. In addition, we prove that the size of the sets of geodesic arcs whose T-self-intersection number is not close to κ T² = (2π² (g - 1)) is also estimated by a decreasing exponential function. And, as a corollary of the latter, we obtain a result of Lalley which states that most of the closed geodesics α on S with l(α) ≤ T have roughly κl(α)²/(2π and sup2;(g-1)) self-intersections, when T is large.