Intersection numbers of geodesic arcs

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

DOI: https://doi.org/10.15446/recolma.v49n2.60450

Números de Intersección de Arcos Geodésicos

Yoe Alexander Herrera Jaramillo¹

¹ Universidad Autónoma de Bucaramanga, Bucaramanga, Colombia
e-mail: yherrera743@unab.edu.co, yoeherrera@gmail.com

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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.

Key words and phrases. geodesics, geodesic flow, geodesic currents, intersection number, mixing, ergodicity.

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2010 Mathematics Subject Classification. 37d40.

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Resumen

Para una superficie S con curvatura constante −κ (con κ> 0) y género g ≥ 2, mostramos que las colas de la distribución de i(α, β)/l(α)l(β) (donde i(α, β) es el número de intersección de las geodésicas cerradas α y β) se puede estimar con una función exponencial decreciente. Como consecuencia, encontramos el promedio asintótico de los números de intersecciones normalizados de los pares de geodésicas cerradas en S. Además, demostramos que el tamaño de los conjuntos de geodésicas cuyo número de T -auto-intersecciones no es cercano κT ²/(2π²(g − 1)) también decrece exponencialmiente rápido. Y, como corolario de este ultimo, obtenemos un resultado de Lalley que afirma que la mayoría de las geodésicas cerradas α en S con l(α) ≤ T tienen aproximadamente κl(α)²/(2π²(g − 1)) auto-intersecciones, cuando T es grande.

Palabras y frases clave. geodésica, flujo geodésico, corrientes geodésicas, número de intersección, mezcla, ergodicidad.

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References

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(Recibido en mayo de 2015. Aceptado en agosto de 2015)