Outer billiards in the spaces of oriented geodesics of the three-dimensional space forms (Q6561005)

Classical (planar) \textit{inner} billiard maps are standard tools to describe dynamical systems of particles moving in the interior of a closed convex curve \(\gamma\) in \(\mathbb{R}^2\). The dual picture is provided by \textit{outer} billiard maps, which are defined on the exterior of \(\gamma\), see [\textit{J. Moser}, Stable and random motions in dynamical systems. With special emphasis on celestial mechanics. Hermann Weyl Lectures. The Institute for Advanced Study. Princeton, NJ: Princeton University Press (1973; Zbl 0271.70009)]. A major generalisation of those objects appeared in the 90's in \textit{S. Tabachnikov}'s work [Adv. Math. 115, No. 2, 221--249 (1995; Zbl 0846.58038)], where the canonical symplectic structure \(\omega\) on \(\mathbb{R}^{2n}\) is used to define outer billiard maps on the exterior of quadratically convex closed hypersurfaces in \(\mathbb{R}^{2n}\); moreover, S. Tabachnikov proved that those billiard maps are symplectomorphisms.\N\NThis paper develops a different generalisation of the planar outer billiard map, replacing \(\mathbb{R}^2\) with the space \(\mathcal{G}_{\kappa}\). This denotes the 4-dimensional manifold of oriented geodesics on the 3-dimensional space form \(M_{\kappa}\) of constant curvature \(\kappa = 0, +1, -1\) (so that \(M_{\kappa}\) is, respectively, the Euclidean space \(\mathbb{R}^3\), the sphere \(\mathbb{S}^3\), or the hyperbolic space \(\mathbb{H}^3\)).\N\NThe space \(\mathcal{G}_{\kappa}\) exhibits a rich geometry, as noticed first by \textit{N. J. Hitchin} [Commun. Math. Phys. 83, 579--602 (1982; Zbl 0502.58017)] and then by several other authors. Indeed, it is naturally equipped with a complex structure \(\mathcal{J}\) and with a metric \(g_{\times}\) (induced by the cross product on \(M_{\kappa}\)), which is furthermore compatible with \(\mathcal{J}\), hence yielding a Kähler structure \(\omega_{\times}\) for every \(\kappa\). Moreover, for \(\kappa = \pm 1\), there is another metric \(g_K\) (induced by the Killing form on \(\mathrm{Iso} (M_{\kappa})\)), again compatible with \(\mathcal{J}\), therefore yielding another Kähler structure \(\omega_K\). For \(\kappa = 0\) there is no additional symplectic form, but \(\mathcal{G}_0\) is instead equipped with a Poisson structure \(\mathcal{P}\).\N\NThe main definitions and statements concerning the outer billiard maps on \(\mathcal{G}_{\kappa}\) are provided in Sections 1 and 2. In particular, the authors fix a strictly convex closed surface \(S\) in \(M_\kappa\) and define a bijection \(B: \mathcal{U} \to \mathcal{U}\), for \(\mathcal{U} \subset \mathcal{G}_\kappa\) an appropriate open submanifold (the billiard table) built using \(S\). However, the smoothness of \(B\) is not automatic, as shown by an explicit counterexample, and depends on the choice of \(S\).\N\NThe rest of the paper is therefore devoted to the proofs of further properties of the outer billiard map \(B\), in the stronger case when \(S\) is quadratically convex.\N\begin{itemize}\N\item Section 3: \(B\) is a diffeomorphism;\N\item Section 4: \(B\) can be described using the Kahler structure \((g_K, \mathcal{J}, \omega_K)\), analogously to Tabachnikov's construction;\N\item Section 5: \(B\) is a symplectomorphism with respect to \(\omega_K\) (for \(\kappa =\pm 1\)) and a Poisson diffeomorphism with respect to \(\mathcal{P}\) (for \(\kappa = 0\)), but it does not preserve \(\omega_\times\) (for any \(\kappa\)).\N\end{itemize}\N\NThe paper ends with an outlook of the dynamical properties of the outer billiard map \(B\) in the case \(\kappa = -1\), focussing in particular on a notion of holonomy for periodic points.