Multiplicity of periodic solutions to Birkhoff's billiard ball problem

摘要

THE motion of a perfectly elastic billiard ball upon a convex billiard table is a highly typical system of dynamical systems. Let F be the boundary of a billiard table, strictly convex and C~1 smooth. The ball is assumed to roll on the table. It goesstraight until it hits the boundary Γ where the ball bounces off according to the law that the angle of incidence is equal to the angle of reflection. Its path will be a closed n-sided polygon inscribed in Γ having no coincident sides, if and only if the motion is periodic with the positive number n as its minimal period. This is called n-bounce periodic orbit. If an n-bounce periodic orbit makes k circuits of Γ (see refs. for the exact meaning), it is called Birkhoff's periodic orbit of (n, k)-type.