Density of thin film planar billiard reflection pseudogroup in Hamiltonian symplectomorphism pseudogroup

Alexey Glutsyuk (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

Reflections from planar curves act by symplectomorphisms on the space of oriented lines with respect to the canonical symplectic form. We consider an arbitrary planar curve $\gamma$ that is either a germ, or a strictly convex closed curve. In the case of a germ we show that reflections from its small deformations and their inverse transformations generate a pseudogroup that is dense in the pseudogroup of symplectomorphisms between simply connected subdomains of an appropriate domain in the space of oriented lines. In the case of a global strictly convex closed curve we prove a similar density statement in the pseudogroup of Hamiltonian diffeomorphisms between subdomains of the phase cylinder: the space of oriented lines intersecting the given curve transversally.

Domaines: Systèmes dynamiques [math.DS]

Alexey Glutsyuk. Density of thin film planar billiard reflection pseudogroup in Hamiltonian symplectomorphism pseudogroup. 2020. ⟨hal-03026432v1⟩