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Billiards and Hofer's Geometry

Mark Berezovik, Konstantin Kliakhandler, Yaron Ostrover, Leonid Polterovich

TL;DR

This work establishes a quantitative bridge between billiard dynamics in convex planar domains and Hofer's geometry on symplectic diffeomorphisms. By proving the Lipschitz-type bound $d_H(\psi_K,\psi_L) \le 4\, d_B(K,L)$ for smooth strictly convex tables, the authors connect a dynamical invariant to a simple geometric distance between tables. They further show that convex polygons, though non-smooth, have billiard maps that sit in the Hofer completion of the group of smooth area-preserving maps, via an explicit polygon-approximation scheme extending to a natural $S^1$-equivariant map into the completion. The paper discusses dynamical consequences, links to displacement energy, and open problems, including extensions to broader convex geometries and the potential to encode dynamics through barcodes in the Hofer completion framework.

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.

Billiards and Hofer's Geometry

TL;DR

This work establishes a quantitative bridge between billiard dynamics in convex planar domains and Hofer's geometry on symplectic diffeomorphisms. By proving the Lipschitz-type bound for smooth strictly convex tables, the authors connect a dynamical invariant to a simple geometric distance between tables. They further show that convex polygons, though non-smooth, have billiard maps that sit in the Hofer completion of the group of smooth area-preserving maps, via an explicit polygon-approximation scheme extending to a natural -equivariant map into the completion. The paper discusses dynamical consequences, links to displacement energy, and open problems, including extensions to broader convex geometries and the potential to encode dynamics through barcodes in the Hofer completion framework.

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.

Paper Structure

This paper contains 7 sections, 8 theorems, 93 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A comparison theorem
  4. Approximation of polygons
  5. Comparison of the metric TEXT with other geometric distances
  6. Discussion and open problems
  7. Appendix

Key Result

Theorem 3.1

Any two billiard ball maps can be joined by a Hamiltonian path, and hence they belong to the set $\mathcal{B}\mathcal{M}$ defined by def:BM. Furthermore, for any two billiard tables $\alpha$, $\beta$ and corresponding billiard ball maps $\psi_\alpha$, $\psi_\beta$ the following inequality holds:

Figures (4)

  • Figure 1: Obtaining the curve $\beta$ ($0 < t < 1/2$).
  • Figure 2: Approximation of a polygon.
  • Figure 3: Local picture on an edge.
  • Figure 4: Local picture near a corner.

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Definition 2.10
  • ...and 20 more