Outer billiards in the spaces of oriented geodesics of the three dimensional space forms

TL;DR

This paper extends the concept of outer billiards to three-dimensional space forms, analyzing the map on the space of geodesics and revealing its symplectic properties and dynamical features, especially in hyperbolic space.

Contribution

It introduces a new outer billiard map on geodesic spaces of 3D space forms, proving its diffeomorphism and symplectic properties, and initiates dynamical analysis in hyperbolic space.

Findings

B is a diffeomorphism for quadratically convex surfaces.

B is a symplectomorphism with respect to the fundamental form.

B does not preserve the standard symplectic form.

Abstract

Let $M_{\kappa }$ be the three-dimensional space form of constant curvature $\kappa =0,1,-1$, that is, Euclidean space $\mathbb{R}^{3}$, the sphere $S^{3} $, or hyperbolic space $H^{3}$. Let $S$ be a smooth, closed, strictly convex surface in $M_{\kappa }$. We define an outer billiard map $B$ on the four dimensional space $\mathcal{G}_{\kappa }$ of oriented complete geodesics of $M_{\kappa }$, for which the billiard table is the subset of $\mathcal{G}_{\kappa }$ consisting of all oriented geodesics not intersecting $S$. We show that $B$ is a diffeomorphism when $S$ is quadratically convex. For $\kappa =1,-1$, $\mathcal{G}_{\kappa }$ has a K\"{a}hler structure associated with the Killing form of $\operatorname{Iso}(M_{\kappa })$. We prove that $B$ is a symplectomorphism with respect to its fundamental form and that $B$ can be obtained as an analogue to the construction of Tabachnikov of the outer billiard in $\mathbb{R}^{2n}$ defined in terms of the standard symplectic structure. We show that $B$ does not preserve the fundamental symplectic form on $\mathcal{G}_{\kappa }$ associated with the cross product on $M_{\kappa }$, for $\kappa =0,1,-1$. We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points.