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The growth rate of trajectories of a quadratic differential

Published online by Cambridge University Press:  19 September 2008

Howard Masur
Howard Masur
Affiliation:
Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois 60680, USA
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Abstract

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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 1 , March 1990 , pp. 151 - 176
Copyright
Copyright © Cambridge University Press 1990

References

REFERENCES

[B]Boshennitzan, M.. A condition for minimal interval exchange maps to be uniquely ergodic. Duke Math. J. 52 (1985), 723–752.Google Scholar
[G]Gutkin, E.. Billiards on almost integrable polyhedral surfaces. Ergodic Th. & Dynam. Sys. 4 (1984), 569–584.CrossRefGoogle Scholar
[K]Katok, A.. The growth rate for the number of singular and periodic orbits for a polygonal billiard. Preprint.CrossRefGoogle Scholar
[K-M-S]Kerckhoff, S., Masur, H. & Smillie, J.. Ergodicity of billiard flows and quadratic differentials. Ann. Math. 124 (1986), 293–311.CrossRefGoogle Scholar
[R]Rees, M.. An alternative approach to the ergodic theory of measured foliations on surfaces. Ergod. Th. & Dynam. Sys. 1 (1981), 461–488.CrossRefGoogle Scholar
[St]Strebel, K.. Quadratic Differentials. Springer-Verlag: New York, 1984.CrossRefGoogle Scholar
[Su]Sullivan, D.. Disjoint spheres, approximation by imaginary quadratic numbers and the logarithm law for geodesies. Acta Math. 149 (1982), 215–238.CrossRefGoogle Scholar