On Lagrangian tangent sweeps and Lagrangian outer billiards

Dmitry Fuchs, Serge Tabachnikov

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)203-213
Number of pages11
JournalGeometriae Dedicata
Volume182
Issue number1
DOIs
StatePublished - Jan 1 2016

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

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