Three-Period Orbits in Billiards on the Surfaces of Constant Curvature
Dynamical Systems
2011-11-01 v2
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.
Keywords
Cite
@article{arxiv.1108.0987,
title = {Three-Period Orbits in Billiards on the Surfaces of Constant Curvature},
author = {Victoria Blumen and Ki Yeun Kim and Joe Nance and Vadim Zharnitsky},
journal= {arXiv preprint arXiv:1108.0987},
year = {2011}
}
Comments
10 pages, 3 figures