A C1 Arnol'd-Liouville theorem

Marie-Claude Arnaud (1) , Jinxin Xue (2)

1 LMA - EA2151 Laboratoire de Mathématiques d'Avignon
2 University of Chicago

In this paper, we prove a version of Arnol'd-Liouville theorem for C 1 commuting Hamiltonians. We show that the Lipschitz regularity of the foliation by invariant Lagrangian tori is crucial to determine the Dynamics on each Lagrangian torus and that the C 1 regularity of the foliation by invariant Lagrangian tori is crucial to prove the continuity of Arnol'd-Liouville coordinates. We also explore various notions of C 0 and Lipschitz integrability.

Keywords: (C0-)Poisson commutativity; Hamiltonian; foliation; Arnol'd- Liouville theorem; Lagrangian submanifolds; generating functions; symplectic homeomorphisms; complete integrability

Domains: Mathematics [math]; Dynamical Systems [math.DS]; Mathematics [math]; Symplectic Geometry [math.SG]

Marie-Claude Arnaud, Jinxin Xue. A C1 Arnol'd-Liouville theorem. Asterisque, 2020, Quelques aspects de la théorie des systèmes dynamiques: un hommage à Jean-Christophe Yoccoz.II, 416, pp.1-31. ⟨hal-01422530v2⟩