The growth rate of trajectories of a quadratic differential. (English) Zbl 0706.30035

Masur, Howard

Ergodic Theory Dyn. Syst. 10, No. 1, 151-176 (1990).

Given a holomorphic quadratic differential q on a compact Riemann surface of genus \(\geq 2\) which defines a metric, flat except at the zeroes, the author shows the asymptotic growth rate of its singular trajectories (and the number of its parallel families of closed regular trajectories) of length at most T is at most quadratic in T (and does not depend on the genus!). Here a singular trajectory in a geodesic joining two zeroes of q with no zeroes in its interior. Also an application is given to billiards. Particularly, this implies that the geodesic flow on a rational billiard table is uniquely ergodic in almost every direction; cf. S.Kerckhoff, H. Mazur and J. Smillie, Ann. Math., II. Ser. 124, 293-311 (1986; Zbl 0637.58010).

Reviewer: B.N.Apanasov

MSC:

30F30

Differentials on Riemann surfaces

Keywords:

quadratic differential

Citations:

Zbl 0637.58010

References:

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[2]	Strebel, Quadratic Differentials (1984) · doi:10.1007/978-3-662-02414-0
[3]	Rees, Ergod. Th. & Dynam. Sys. 1 pp 461– (1981)
[4]	DOI: 10.1215/S0012-7094-85-05238-X · Zbl 0602.28009 · doi:10.1215/S0012-7094-85-05238-X
[5]	Katok, The growth rate for the number of singular and periodic orbits for a polygonal billiard · Zbl 0631.58020 · doi:10.1007/BF01239021
[6]	Gutkin, Ergodic Th. & Dynam. Sys. 4 pp 569– (1984)
[7]	DOI: 10.2307/1971280 · Zbl 0637.58010 · doi:10.2307/1971280