Periodic billiard trajectories and Morse theory on loop spaces

2014/03/08 by Kei Irie, Irie, Kei
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1403.1953

openalex publication_date 2014/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study periodic billiard trajectories on a compact Riemannian manifold with boundary, by applying Morse theory to Lagrangian action functionals on the loop space of the manifold. Based on the approximation method due to Benci-Giannoni, we prove that nonvanishing of relative homology of a certain pair of loop spaces implies the existence of a periodic billiard trajectory. We also prove a parallel result for path spaces. We apply those results to show the existence of short billiard trajectories and short geodesic loops. We also recover two known results on the length of a shortest periodic billiard trajectory on a convex body: Ghomi's inequality, and Brunn-Minkowski type inequality due to Artstein-Ostrover.

Citations

• Symplectic homology of disc cotangent bundles of domains in Euclidean space

Related

• On the lower bounds for the number of periodic billiard trajectories in manifolds embedded in Euclidean space
• Topology of billiard problems, II
• Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional billiards
• Lower bounds for the number of closed billiard trajectories of period 2 and 3 in manifolds embedded in Euclidean space
• Periodic trajectories in 3-dimensional convex billiards