Three-Period Orbits in Billiards on the Surfaces of Constant Curvature

TL;DR

This paper investigates three-period orbits in billiards on surfaces of constant curvature, revealing measure-zero sets on hyperbolic planes and positive measure sets on spheres under certain conditions, using Jacobi fields.

Contribution

It applies Wojtkovski's Jacobi fields approach to analyze 3-period orbits on hyperbolic and spherical surfaces, providing new proofs and extending understanding of their measure properties.

Findings

3-period orbits have zero measure on hyperbolic planes

On spheres, 3-period orbits can have positive measure under certain conditions

New proof of Baryshnikov's theorem for spherical billiards

Abstract

An approach due to Wojtkovski [9], based on the Jacobi fields, is applied to study sets of 3-period orbits in billiards on hyperbolic plane and on two-dimensional sphere. It is found that the set of 3-period orbits in billiards on hyperbolic plane, as in the planar case, has zero measure. For the sphere, a new proof of Baryshnikov's theorem is obtained which states that 3-period orbits can form a set of positive measure provided a natural condition on the orbit length is satisfied.