Outer billiards in the spaces of oriented geodesics of the three dimensional space forms
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| Format: | Preprint |
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2021
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| _version_ | 1866915188546994176 |
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| author | Godoy, Yamile Harrison, Michael Salvai, Marcos |
| author_facet | Godoy, Yamile Harrison, Michael Salvai, Marcos |
| contents | Let $M_{κ}$ be the three-dimensional space form of constant curvature $κ=0,1,-1$, 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 $M_{κ}$. We define an outer billiard map $B$ on the four dimensional space $\mathcal{G}_{κ}$ of oriented complete geodesics of $M_{κ}$, for which the billiard table is the subset of $\mathcal{G}_{κ}$ consisting of all oriented geodesics not intersecting $S$. We show that $B$ is a diffeomorphism when $S$ is quadratically convex.
For $κ=1,-1$, $\mathcal{G}_{κ}$ has a Kähler structure associated with the Killing form of $\operatorname{Iso}(M_{κ})$. 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 $\mathcal{G}_{κ}$ associated with the cross product on $M_{κ}$, for $κ=0,1,-1$.
We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_01679 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Outer billiards in the spaces of oriented geodesics of the three dimensional space forms Godoy, Yamile Harrison, Michael Salvai, Marcos Dynamical Systems Differential Geometry Primary: 37C83, 53C29, 53D22. Secondary: 53A35, 53C22, 53C35 Let $M_{κ}$ be the three-dimensional space form of constant curvature $κ=0,1,-1$, 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 $M_{κ}$. We define an outer billiard map $B$ on the four dimensional space $\mathcal{G}_{κ}$ of oriented complete geodesics of $M_{κ}$, for which the billiard table is the subset of $\mathcal{G}_{κ}$ consisting of all oriented geodesics not intersecting $S$. We show that $B$ is a diffeomorphism when $S$ is quadratically convex. For $κ=1,-1$, $\mathcal{G}_{κ}$ has a Kähler structure associated with the Killing form of $\operatorname{Iso}(M_{κ})$. 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 $\mathcal{G}_{κ}$ associated with the cross product on $M_{κ}$, for $κ=0,1,-1$. We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points. |
| title | Outer billiards in the spaces of oriented geodesics of the three dimensional space forms |
| topic | Dynamical Systems Differential Geometry Primary: 37C83, 53C29, 53D22. Secondary: 53A35, 53C22, 53C35 |
| url | https://arxiv.org/abs/2110.01679 |