Fuchsian groups and polygonal billiards
| dc.contributor.advisor | Veech, William A. | en_US |
| dc.creator | Ward, Clayton Collin | en_US |
| dc.date.accessioned | 2009-06-04T00:23:45Z | en_US |
| dc.date.available | 2009-06-04T00:23:45Z | en_US |
| dc.date.issued | 1996 | en_US |
| dc.description.abstract | Let P be a simple, closed polygon in the plane, all interior angles of which are rational multiples of $\pi$. We consider the possible paths of a point, rebounding in the interior of P with constant speed and elastic reflections. Such a dynamical system is known as "billiards in P". By means of a well-known construction, "billiard" trajectories in such a polygon P are identified with geodesic paths on a closed Riemann surface $X\sp{P}$, where the Riemannian metric is one of zero curvature with isolated singularities, and is given by a holomorphic one-form $\omega$ on the surface. To this holomorphic one-form one can canonically associate a discrete subgroup $\Gamma$ of $PSL(2,\IR$). If $\Gamma$ happens to be a lattice (has cofinite volume), then it is known that all geodesic paths in the zero-curvature metric given by $\omega$ must either be closed or uniformly distributed in the surface $X\sp{P}$. As a corollary, all billiard paths in the original polygon P must either be finite or uniformly distributed in P. A new class of examples of polygons P, whose associated group $\Gamma$ is, in fact, a lattice have been discovered. At the same time, we have discovered the first examples of triangles P, as above, for which the associated groups $\Gamma$ are not lattices (i.e. have infinite covolume). Finally, it is shown how to derive, in an explicit way, algebraic equations which specify the Riemann surface $X\sp{P}$ and one-form $\omega$, which before were only described geometrically. | en_US |
| dc.format.extent | 51 p. | en_US |
| dc.format.mimetype | application/pdf | en_US |
| dc.identifier.callno | THESIS MATH. 1996 WARD | en_US |
| dc.identifier.citation | Ward, Clayton Collin. "Fuchsian groups and polygonal billiards." (1996) Diss., Rice University. <a href="https://hdl.handle.net/1911/16978">https://hdl.handle.net/1911/16978</a>. | en_US |
| dc.identifier.uri | https://hdl.handle.net/1911/16978 | en_US |
| dc.language.iso | eng | en_US |
| dc.rights | Copyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder. | en_US |
| dc.subject | Mathematics | en_US |
| dc.title | Fuchsian groups and polygonal billiards | en_US |
| dc.type | Thesis | en_US |
| dc.type.material | Text | en_US |
| thesis.degree.department | Mathematics | en_US |
| thesis.degree.discipline | Natural Sciences | en_US |
| thesis.degree.grantor | Rice University | en_US |
| thesis.degree.level | Doctoral | en_US |
| thesis.degree.name | Doctor of Philosophy | en_US |
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