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Bouncing Outer Billiards

Andrey Gogolev, Levi Keck, Kevin Lewis

Abstract

We introduce a new class of billiard-like system, ``bouncing outer billiards" which are 3-dimensional cousins of outer billiards of Neumann and Moser. We prove that bouncing outer billiard on a smooth convex body has at least four 1-parameter families of fixed points. We also fully describe dynamics of bouncing outer billiard on a line segment. Finally we carry out numerical experiments suggesting very complicated (non-ergodic) behavior for several shapes including the square and an ellipse.

Bouncing Outer Billiards

Abstract

We introduce a new class of billiard-like system, ``bouncing outer billiards" which are 3-dimensional cousins of outer billiards of Neumann and Moser. We prove that bouncing outer billiard on a smooth convex body has at least four 1-parameter families of fixed points. We also fully describe dynamics of bouncing outer billiard on a line segment. Finally we carry out numerical experiments suggesting very complicated (non-ergodic) behavior for several shapes including the square and an ellipse.

Paper Structure

This paper contains 16 sections, 6 theorems, 48 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Fixed points
  3. Bouncing on a Line Segment
  4. Parameterizing the Dynamics
  5. A Second Invariant
  6. Invariant Ellipses
  7. Twist Dynamics
  8. Periodic Orbits for the Billiard on the Segment
  9. Numerical Simulations
  10. Bouncing on Parabola Arcs
  11. Bouncing on the Square
  12. Bouncing on an Ellipse
  13. In $d > 0$ Case:
  14. In $d < 0$ Case:
  15. In $d > 0$ Case:
  16. ...and 1 more sections

Key Result

Theorem 2.1

For any convex $S$ with $C^3$ boundary the associated billiard map has uncountably many fixed points, which come in at least four local 1-parameter families.

Figures (15)

  • Figure 1: Bouncing Outer Billiards Dynamics
  • Figure 2: Fixed Point Lemma Setup
  • Figure 3: Several Invariant Ellipses with Height One
  • Figure 4: "M" and "W" period 4 orbits
  • Figure 5: Family of Orbits of Period 7
  • ...and 10 more figures

Theorems & Definitions (14)

  • Remark 1.1
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['fixedpointtheorem']}.
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • Remark 4.1
  • ...and 4 more