Outer billiards on the manifolds of oriented geodesics of the three dimensional space forms

GODOY, YAMILE; HARRISON, MICHAEL; MARCOS SALVAI

JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES

OXFORD UNIV PRESS

Lugar: Oxford; Año: 2024

0024-6107

Let M_kappa be the three-dimensional space form of constant curvature kappa = 0, 1, -1, that is, Euclidean space 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 G_kappa of oriented complete geodesics of M_kappa, for which the billiard table is the subset of 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, G_kappa has a K"{a}hler structure associated with the Killing form of 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 R^{2n} defined in terms of the standard symplectic structure. We show that B does not preserve the fundamental symplectic form on 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.