FM: 2007-08; mp_arc:08-18, arXiv:0801.3568

Authors : Albert Fathi, Alessandro Giuliani, Alfonso Sorrentino

Title: Uniqueness of Invariant Lagrangian Graphs in a Homology or a Cohomology Class

Abstract: Given a smooth compact Riemannian manifold M and a Hamiltonian H on the cotangent space T*M, strictly convex and superlinear in the momentum variables, we prove uniqueness of certain "ergodic" invariant Lagrangian graphs within a given homology or cohomology class. In particular, in the context of quasi-integrable Hamiltonian systems, our result implies global uniqueness of Lagrangian KAM tori with fixed rotation vector. This result extends generically to the C0-closure of KAM tori.

Keywords: Invariant tori; quasi-integrable Hamiltonian systems; Aubry-Mather theory.

Albert Fathi,
Ecole normale superieure de Lyon,
Unite' de Mathematiques Pures et Appliquees, UMR CNRS 5669,
46, allee d'Italie, 69364 Lyon Cedex 07 France

Alessandro Giuliani,
Dipartimento di Matematica,
Universita' di Roma Tre,
L.go S. Leonardo Murialdo 1, 00146 Roma, Italy

Alfonso Sorrentino,
Department of Mathematics,
Princeton University, Princeton 08544 NJ, USA