On Lagrangian tangent sweeps and Lagrangian outer billiards
- 62 Downloads
Given a Lagrangian submanifold in linear symplectic space, its tangent sweep is the union of its (affine) tangent spaces, and its tangent cluster is the result of parallel translating these spaces so that the foot point of each tangent space becomes the origin. This defines a multivalued map from the tangent sweep to the tangent cluster, and we show that this map is a local symplectomorphism (a well known fact, in dimension two). We define and study the outer billiard correspondence associated with a Lagrangian submanifold. Two points are in this correspondence if they belong to the same tangent space and are symmetric with respect to its foot pointe. We show that this outer billiard correspondence is symplectic and establish the existence of its periodic orbits. This generalizes the well studied outer billiard map in dimension two.
KeywordsSymplectic space Lagrangian subspace Symplectic correspondence Outer billiard Periodic orbit
The authors are grateful to ICERM for the inspiring and welcoming atmosphere. The second author was supported by NSF Grant DMS-1510055. Many thanks to the referee for the useful criticism.
- 2.Arnold, V., Gusein-Zade, S., Varchenko, A.: Singularities of Differentiable Maps. Classification of Critical Points, Caustics and Wave Fronts. Birkhäuser/Springer, New York (2012)Google Scholar
- 7.Moser, J.: Stable and random motions in dynamical systems. Ann. Math. Stud. 77. Princeton University Press, Princeton, NJ (1973)Google Scholar
- 9.Tabachnikov, S.: Billiards. SMF “Panoramas et Syntheses”, No 1 (1995)Google Scholar