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Non-Birkhoff periodic orbits in symmetric billiards

Casper Oelen, Bob Rink, Mattia Sensi

TL;DR

This work identifies a quantitative criterion based on the boundary length $L$ and curvature $\kappa$ along a $\mathbb{D}_n$-symmetric Birkhoff orbit that guarantees the existence of non-Birkhoff periodic orbits with prescribed minimal period, rotation number, and spatiotemporal symmetry. The authors develop a monotone discrete variational framework, employ a gradient-flow approach to the length functional, and leverage Aubry–Mather-type ideas to locate stationary points under symmetry constraints, including detailed Hessian analysis. They prove the existence of infinitely many non-Birkhoff orbits near circular and elliptical billiards, and in particular provide criteria that yield new non-Birkhoff orbits in $\mathbb{D}_n$-symmetric billiards as well as in $\mathbb{D}_2$-symmetric cases, with several explicit constructions and a numerical toolkit. The results significantly extend understanding of non-Birkhoff dynamics in symmetric convex billiards and suggest broad applicability to other monotone variational problems with discrete symmetry.

Abstract

We study non-Birkhoff periodic orbits in symmetric convex planar billiards. Our main result provides a quantitative criterion for the existence of such orbits with prescribed minimal period, rotation number, and spatiotemporal symmetry. We exploit this criterion to find sufficient conditions for a symmetric billiard to possess infinitely many non-Birkhoff periodic orbits. It follows that arbitrarily small analytical perturbations of the circular billiard have non-Birkhoff periodic orbits of any rational rotation number and with arbitrarily long periods. We also generalize a known result for elliptical billiards to other $\mathbb{D}_2$-symmetric billiards. Lastly, we provide Matlab codes which can be used to numerically compute and visualize the non-Birkhoff periodic orbits whose existence we prove analytically.

Non-Birkhoff periodic orbits in symmetric billiards

TL;DR

This work identifies a quantitative criterion based on the boundary length and curvature along a -symmetric Birkhoff orbit that guarantees the existence of non-Birkhoff periodic orbits with prescribed minimal period, rotation number, and spatiotemporal symmetry. The authors develop a monotone discrete variational framework, employ a gradient-flow approach to the length functional, and leverage Aubry–Mather-type ideas to locate stationary points under symmetry constraints, including detailed Hessian analysis. They prove the existence of infinitely many non-Birkhoff orbits near circular and elliptical billiards, and in particular provide criteria that yield new non-Birkhoff orbits in -symmetric billiards as well as in -symmetric cases, with several explicit constructions and a numerical toolkit. The results significantly extend understanding of non-Birkhoff dynamics in symmetric convex billiards and suggest broad applicability to other monotone variational problems with discrete symmetry.

Abstract

We study non-Birkhoff periodic orbits in symmetric convex planar billiards. Our main result provides a quantitative criterion for the existence of such orbits with prescribed minimal period, rotation number, and spatiotemporal symmetry. We exploit this criterion to find sufficient conditions for a symmetric billiard to possess infinitely many non-Birkhoff periodic orbits. It follows that arbitrarily small analytical perturbations of the circular billiard have non-Birkhoff periodic orbits of any rational rotation number and with arbitrarily long periods. We also generalize a known result for elliptical billiards to other -symmetric billiards. Lastly, we provide Matlab codes which can be used to numerically compute and visualize the non-Birkhoff periodic orbits whose existence we prove analytically.

Paper Structure

This paper contains 15 sections, 32 theorems, 116 equations, 16 figures.

Table of Contents

  1. Introduction
  2. A monotone recurrence relation
  3. The Birkhoff property
  4. Spatiotemporal symmetry
  5. Symmetric periodic billiard sequences
  6. The gradient flow of the length functional
  7. Symmetric periodic Birkhoff sequences
  8. The Hessian of the periodic action
  9. Technical preliminaries for the proof of the main theorem
  10. Non-Birkhoff orbits with dihedral symmetry
  11. Non-Birkhoff orbits in billiards with $\mathbb{D}_2$-symmetry
  12. Billiards with infinitely many non-Birkhoff periodic orbits
  13. Conclusion and discussion
  14. Appendix: convexity of Limaçon-type curves
  15. Appendix: structure of the numerical code

Key Result

Theorem 1.1

Let $m,n\in \mathbb{N}$ be co-prime, with $1\leq m \leq n-1$, and let $\Gamma$ be a $\mathbb{D}_n$-symmetric, $C^2$-smooth, strictly convex simple closed curve. Let $Z\in \Gamma^{\mathbb{Z}}$ be one of its two $\mathbb{D}_n$-symmetric Birkhoff periodic orbits with rotation number $\frac{m}{n}$. Deno then $\Gamma$ admits a non-Birkhoff periodic orbit with minimal period $p$, rotation number $\frac{

Figures (16)

  • Figure 1: Visualization of Theorem \ref{['thm:sample']} in four distinct symmetric convex billiards. (a)$\mathbb{D}_1$-symmetric $(6,3)$-periodic non-Birkhoff orbit in a $\mathbb{D}_2$-symmetric billiard -- note that the billiard trajectory is traversed in two directions throughout each period; (b)$\mathbb{D}_3$-symmetric $(15,5)$-periodic non-Birkhoff orbit in a $\mathbb{D}_3$-symmetric billiard; (c)$\mathbb{D}_4$-symmetric $(12,3)$-periodic non-Birkhoff orbit in a $\mathbb{D}_4$-symmetric billiard; (d)$\mathbb{D}_5$-symmetric $(245,98)$-periodic non-Birkhoff orbit in a close-to-circular $\mathbb{D}_5$-symmetric billiard.
  • Figure 2: $\mathbb{D}_3$-symmetric billiard $\Gamma$ and part of a billiard sequence $z=(\ldots, z_i, z_{i+1}, \ldots)\in \Gamma^{\mathbb{Z}}$. The angle of incidence at $z_i$ is denoted $\phi_i$, and the angle of reflection at $z_i$ is denoted $\theta_i$. We have that $z$ is a billiard orbit precisely when $\phi_i=\theta_i$ for all $i\in \mathbb{Z}$.
  • Figure 3: Symmetric billiards with Birkhoff periodic orbits and non-Birkhoff periodic orbits: (a)$\mathbb{D}_4$-symmetric billiard with two $(4,1)$-periodic Birkhoff orbits, the "short" one in red and the "long" one in cyan; (b)$\mathbb{D}_4$-symmetric billiard with $(12,3)$-periodic non-Birkhoff orbit: note that $z_9\prec z_1 \prec z_2$ holds but $z_{10} \prec z_{2} \prec z_{3}$ does not; (c)$\mathbb{D}_5$-symmetric billiard with two $(5,2)$-periodic Birkhoff orbits, the "short" one in red and the "long" one in cyan; (d)$\mathbb{D}_5$-symmetric billiard with $(15,6)$-periodic non-Birkhoff orbit: note that $z_6\prec z_{1} \prec z_2$ holds but $z_7 \prec z_{2} \prec z_{3}$ does not.
  • Figure 4: Aubry diagrams of a Birkhoff sequence (solid line) and five of its integer translates (dashed); recall Remark \ref{['rem:Bir_no_touch']}.
  • Figure 5: Symmetric billiard orbits have symmetric Aubry diagrams by Lemma \ref{['liftlemma']}. (a) Convex $\mathbb{D}_2$-symmetric billiard of Limaçon-type (see Example \ref{['ex:limacon']}) with parameter $\alpha=0.1995<\alpha^*(2)= 0.2$. possessing a $(2,1)$-periodic Birkhoff orbit (red) and a $(6,3)$-periodic non-Birkhoff orbit (blue), both satisfying $S(z_i)=z_{3+i}$ and $R(z_i)=z_{7-i}$; (b) Aubry diagrams of lifts of these orbits satisfying $x_i+x_{3+i}=i$ and $x_{7-i}-x_i=\frac{1}{2}-i$; (c) Convex $\mathbb{D}_3$-symmetric billiard of Limaçon-type (see Example \ref{['ex:limacon']}) with parameter $\alpha=0.09<\alpha^*(3)= 0.1$, possessing a $(3,1)$-periodic Birkhoff orbit (red) and a $(6,2)$-periodic non-Birkhoff orbit (blue), both satisfying $S(z_i)=z_{7-i}$; (d) Aubry diagrams of lifts of these orbits satisfying $x_i+x_{7-i}=0$.
  • ...and 11 more figures

Theorems & Definitions (86)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 3.1
  • Definition 3.2
  • ...and 76 more