Unexpected Linearly Stable Orbits in $3$-Dimensional Billiards

TL;DR

This paper constructs linearly stable periodic orbits in 3D billiards with focusing boundary components, revealing fundamental differences from planar billiards and challenging existing construction methods.

Contribution

It demonstrates the existence of stable orbits in 3D billiards with focusing boundaries placed arbitrarily far apart, highlighting new phenomena not present in 2D billiards.

Findings

Stable periodic orbits exist in 3D billiards with focusing components.

Construction methods from 2D billiards do not directly extend to 3D.

Focusing boundary components can be arbitrarily far apart in 3D settings.

Abstract

In this work, we construct linearly stable periodic orbits in $3$-dimensional domains with boundaries containing focusing components (small pieces of a sphere) where we place these components arbitrarily far apart. It demonstrates that we cannot directly implement the construction methods of planar billiards with focusing boundaries and chaotic properties into $3$-dimensional billiards.