Billiards in Nearly Isosceles Triangles

TL;DR

This paper proves that small perturbations of isosceles triangles have periodic billiard paths, revealing complex behaviors and self-similarity phenomena in near-Veech triangles through Fourier series analysis.

Contribution

It introduces a novel Fourier series approach to analyze billiards in nearly isosceles triangles and uncovers unexpected complexity near Veech triangles.

Findings

Small perturbations of isosceles triangles have periodic billiard paths

Billiard dynamics near Veech triangles are highly complex

Self-similarity phenomena are observed in irrational triangular billiards

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.