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The growth rate of trajectories of a quadratic differential

  • Howard Masur (a1)
  • DOI:
  • Published online: 01 September 2008

Suppose q is a holomorphic quadratic differential on a compact Riemann surface of genus g ≥ 2. Then q defines a metric, flat except at the zeroes. A saddle connection is a geodesic joining two zeroes with no zeroes in its interior. This paper shows the asymptotic growth rate of the number of saddles of length at most T is at most quadratic in T. An application is given to billiards.

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[K-M-S]S. Kerckhoff , H. Masur & J. Smillie . Ergodicity of billiard flows and quadratic differentials. Ann. Math. 124 (1986), 293311.

[R]M. Rees . An alternative approach to the ergodic theory of measured foliations on surfaces. Ergod. Th. & Dynam. Sys. 1 (1981), 461488.

[St]K. Strebel . Quadratic Differentials. Springer-Verlag: New York, 1984.

[Su]D. Sullivan . Disjoint spheres, approximation by imaginary quadratic numbers and the logarithm law for geodesies. Acta Math. 149 (1982), 215238.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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