Abstract
Suppose X is the torus the complex plane C divided by the group of translations generated by z → z + 1 and 2 → z + i and q = dz 2 is the unique up to scalar multiple quadratic differential on X. The trajectories of q are straight lines and a trajectory is closed if and only if its slope is a rational p/q. Its length is (p 2 + q 2)1/2. Parallel closed trajectories fill up X. The number N(T) of parallel families of length ≤ T is then the number of lattice points (p, q) with p, q relatively prime inside a circle of radius T. It is classical that
This work was supported in part by NSF Grant DMS-8601977.
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Dedicated to the memory of my father, Ernest F. Masur.
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© 1988 Springer-Verlag New York Inc.
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Masur, H. (1988). Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential. In: Drasin, D., Kra, I., Earle, C.J., Marden, A., Gehring, F.W. (eds) Holomorphic Functions and Moduli I. Mathematical Sciences Research Institute Publications, vol 10. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9602-4_20
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DOI: https://doi.org/10.1007/978-1-4613-9602-4_20
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