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Outer length billiards on a large scale

Peter Albers, Lael Edwards-Costa, Serge Tabachnikov

TL;DR

The paper develops foundational aspects of the outer length billiard: it provides a generating function $H(x,x')$ and a corresponding invariant area form, proves a monotone twist, and analyzes the far-field dynamics. In the far regime, orbits concentrate on origin-centered circles and $F^2$ is well-approximated by the time-1 flow of a homogeneous Hamiltonian $H(\alpha,r)=2r$, yielding a circle-like continuous limit with speed tied to the width $w(\alpha)$. For ellipse tables, the outer length map is integrable with confocal ellipses as invariant curves and an explicit invariant measure on the invariant curve; in the large-scale limit, this measure converges to $\mu(\alpha)=\frac{2}{w_{E_1}(\alpha)}\,d\alpha$, reflecting a Poncelet-grid-type structure. The paper further shows the existence of invariant curves arbitrarily far from the table via Moser’s small twist theorem, and analyzes the asymptotic dynamics of the centers of the auxiliary circles, linking them to the symplectic polar dual and to dual ellipses in the elliptic case. Overall, these results connect outer length billiard dynamics to a Hamiltonian-infinity framework, revealing stable, circle-like behavior and deep dualities with the table’s geometry.

Abstract

We present some foundational results about the outer length billiard system, including its generating function and the invariant area form. We describe the limiting behavior of the orbits far away from the billiard table: the orbits of the map lie on the origin-centered circles, and the second iteration of the map is approximated by the flow of a Hamiltonian function whose level curves are these circles. Furthermore, the orbits "at infinity" of the centers of the auxiliary circles involved in the definition of the map lie on the curves that are polar dual to the symmetrization of the billiard table, the curves which are traced by the second iteration of the usual outer billiard map "at infinity".

Outer length billiards on a large scale

TL;DR

The paper develops foundational aspects of the outer length billiard: it provides a generating function and a corresponding invariant area form, proves a monotone twist, and analyzes the far-field dynamics. In the far regime, orbits concentrate on origin-centered circles and is well-approximated by the time-1 flow of a homogeneous Hamiltonian , yielding a circle-like continuous limit with speed tied to the width . For ellipse tables, the outer length map is integrable with confocal ellipses as invariant curves and an explicit invariant measure on the invariant curve; in the large-scale limit, this measure converges to , reflecting a Poncelet-grid-type structure. The paper further shows the existence of invariant curves arbitrarily far from the table via Moser’s small twist theorem, and analyzes the asymptotic dynamics of the centers of the auxiliary circles, linking them to the symplectic polar dual and to dual ellipses in the elliptic case. Overall, these results connect outer length billiard dynamics to a Hamiltonian-infinity framework, revealing stable, circle-like behavior and deep dualities with the table’s geometry.

Abstract

We present some foundational results about the outer length billiard system, including its generating function and the invariant area form. We describe the limiting behavior of the orbits far away from the billiard table: the orbits of the map lie on the origin-centered circles, and the second iteration of the map is approximated by the flow of a Hamiltonian function whose level curves are these circles. Furthermore, the orbits "at infinity" of the centers of the auxiliary circles involved in the definition of the map lie on the curves that are polar dual to the symmetrization of the billiard table, the curves which are traced by the second iteration of the usual outer billiard map "at infinity".

Paper Structure

This paper contains 27 sections, 24 theorems, 115 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Generating function and invariant area form
  3. Proof.
  4. Proof.
  5. Proof.
  6. Ellipses
  7. The second iteration of the outer length billiard map at infinity
  8. Proof.
  9. Proof.
  10. Proof.
  11. Proof.
  12. Proof.
  13. Proof.
  14. Proof.
  15. Proof.
  16. ...and 12 more sections

Key Result

Lemma 2.1

The partial derivatives if the generating function are as follows: where $k$ is the curvature function of ${\gamma}$, and $\varphi$ is the angle between the tangent segments $Ax$ and $Ax'$.

Figures (20)

  • Figure 1: Three billiards.
  • Figure 2: The outer length billiard.
  • Figure 3: Outer length billiard map on a segment $AB$: one has $F(C)=D$.
  • Figure 4: Left: the orbit of a point "at infinity": the table is a circle in the $L_p$-norm with $p=3/2$. Right: the orbit of the centers of the auxiliary circles "at infinity": the orbit a circle in the $L_p$-norm with $p=3$, the polar dual to the table.
  • Figure 5: To Lemma \ref{['lm:twist']}.
  • ...and 15 more figures

Theorems & Definitions (28)

  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Remark 2.4
  • Lemma 2.5
  • Remark 2.6
  • Theorem 1
  • Theorem 2
  • Lemma 4.1
  • Lemma 4.2
  • ...and 18 more