Bouncing Outer Billiards

TL;DR

This paper introduces bouncing outer billiards, a new 3D billiard-like system, proves the existence of multiple fixed point families on convex bodies, and explores complex dynamics through numerical experiments.

Contribution

It presents the concept of bouncing outer billiards, proves fixed point existence on convex bodies, and analyzes their dynamics, extending outer billiards theory to a new class.

Findings

At least four families of fixed points on smooth convex bodies

Full description of dynamics on a line segment

Numerical evidence of complex, non-ergodic behavior for various shapes

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.