Billiards in Nearly Isosceles Triangles
Abstract
We prove that any sufficiently small perturbation of an isosceles triangle has a periodic billiard path. Our proof involves the analysis of certain infinite families of Fourier series that arise in connection with triangular billiards, and reveals some self-similarity phenomena in irrational triangular billiards. Our analysis illustrates the surprising fact that billiards on a triangle near a Veech triangle is extremely complicated even though Billiards on a Veech triangle is very well understood.
Keywords
Cite
@article{arxiv.0807.3498,
title = {Billiards in Nearly Isosceles Triangles},
author = {W. Patrick Hooper and Richard Evan Schwartz},
journal= {arXiv preprint arXiv:0807.3498},
year = {2013}
}
Comments
Errors have been corrected in Section 9 from the prior and published versions of this paper. In particular, the formulas associated to homology classes of curves corresponding to stable periodic billiard paths in obtuse Veech triangles were corrected. See Remark 9.1 of the paper for more information. The main results and the results from other sections are unaffected. 82 pages, 43 figures