Find out how to access previewonly content
Holomorphic Functions and Moduli I
Volume 10 of the series Mathematical Sciences Research Institute Publications pp 215228
Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential
 Howard MasurAffiliated withDepartment of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago
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
$$ \mathop{{T \to \infty }}\limits^{{\lim }} \frac{{N\left( T \right)}}{{{T^{2}}}} = \frac{6}{{{\pi ^{2}}}}. $$
 Title
 Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential
 Book Title
 Holomorphic Functions and Moduli I
 Book Subtitle
 Proceedings of a Workshop held March 13–19, 1986
 Pages
 pp 215228
 Copyright
 1988
 DOI
 10.1007/9781461396024_20
 Print ISBN
 9781461396048
 Online ISBN
 9781461396024
 Series Title
 Mathematical Sciences Research Institute Publications
 Series Volume
 10
 Series ISSN
 09404740
 Publisher
 Springer US
 Copyright Holder
 SpringerVerlag New York Inc.
 Additional Links
 Topics
 Industry Sectors
 eBook Packages
 Editors

 D. Drasin ^{(1)}
 I. Kra ^{(2)}
 C. J. Earle ^{(3)}
 A. Marden ^{(4)}
 F. W. Gehring ^{(5)}
 Editor Affiliations

 1. Department of Mathematics, Purdue University
 2. Department of Mathematics, State University of New York at Stony Brook
 3. Department of Mathematics, Cornell University
 4. Department of Mathematics, University of Minnesota
 5. Department of Mathematics, University of Michigan
 Authors

 Howard Masur ^{(6)}
 Author Affiliations

 6. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, Illinois, 60680, USA
Continue reading...
To view the rest of this content please follow the download PDF link above.