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

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Fathi, Albert
Giuliani, Alessandro
Sorrentino, Alfonso
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Abstract
Description
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 rotation vector $\rho$. This result extends generically to the $C^0$-closure of KAM tori.
Comment: 20 pages. Version published on Ann. Sc. Norm. Super. Pisa Cl. Sci.(5) Vol. 8, no. 4, 659-680, 2009
Keywords
Mathematics - Dynamical Systems, Mathematical Physics, Mathematics - Symplectic Geometry, 37J50, 37J40
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