Abstract

HERRERA JARAMILLO, Yoe Alexander. Intersection numbers of geodesic arcs. Rev.colomb.mat. [online]. 2015, vol.49, n.2, pp.307-319. ISSN 0034-7426.  https://doi.org/10.15446/recolma.v49n2.60450.

Abstract For a compact surface S with constant curvature −κ (for some κ> 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 κT ²/(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/(2π²(g−1)) self-intersections, when T is large.

Keywords : geodesics; geodesic flow; geodesic currents; intersection number; mixing; ergodicity.

        · abstract in Spanish     · text in English     · English ( pdf )