Journal of Modern Dynamics (JMD)

Billiards in nearly isosceles triangles

Pages: 159 - 231, Issue 2, April 2009      doi:10.3934/jmd.2009.3.159

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W. Patrick Hooper - Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208-2730, United States (email)
Richard Evan Schwartz - Department of Mathematics, Brown University, Providence, RI 02912, 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.
Mathematics Subject Classification:  37E15.

Received: July 2008;      Available Online: May 2009.