Billiards and Hofer's Geometry

2025/07/07 by Mark Berezovik, Berezovik, Mark, Konstantin Kliakhandler +5
Mathematics · Physics and Astronomy · #37C83 #53D22 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematics and Applications #Relativity and Gravitational Theory #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2507.04767

openalex publication_date 2025/07/07 · openalex created_date 2025/10/20 · openalex updated_date 2026/07/28

Abstract

We present a link between billiards in convex plane domains and Hofer's geometry, an area of symplectic topology. For smooth strictly convex billiard tables, we prove that the Hofer distance between the corresponding billiard ball maps admits an upper bound in terms of a simple geometric distance between the tables. We use this result to show that the billiard ball map of a convex polygon lies in the completion, with respect to Hofer's metric, of the group of smooth area-preserving maps of the annulus. Finally, we discuss related connections to dynamics and pose several open problems.

Citations

• The p-widths of a polygon
• A dichotomy for the Hofer growth of area preserving maps on the sphere via symmetrization
• Higher Dimensional Birkhoff attractors (with an appendix by Maxime Zavidovique)
• Lagrangian configurations and Hamiltonian maps
• Nielsen theory, Floer homology and a generalisation of the Poincare-Birkhoff theorem
• The geometry of symplectic energy

Related

• On the role of the surface geometry in convex billiards
• Shortest closed billiard orbits on convex tables
• Caustic-Free Regions for Billiards on Surfaces of Constant Curvature
• Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane
• Semidispersing billiards with an infinite cusp. II