W. Patrick Hooper - Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208-2730, United States (email)
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 well understood.
Keywords: triangular billiards, periodic orbits, Veech triangles, isoceles triangles,
trigonometric series, unfoldings.
Received: July 2008; Published: May 2009.