Percival Lagrangian approach to Aubry-Mather theory

We present some streamlined proofs of some of the basic results in Aubry-Mather theory (existence of quasi-periodic minimizers, multiplicity results when there are gaps among minimizers) based on the study of hull functions. We present results in arbitrary number of dimensions We also compare the proofs and results with those obtained in other formalisms

Localized Stable Manifolds for Whiskered Tori in Coupled Map Lattices with Decaying Interaction

In this paper we consider lattice systems coupled by local interactions. We prove invariant manifold theorems for whiskered tori (we recall that whiskered tori are quasi-periodic solutions with exponentially contracting and expanding directions in the linearized system). The invariant manifolds we construct generalize the usual (strong) (un) stable manifolds and allow us to consider also non-resonant manifolds. We show that if the whiskered tori are localized near a collection of specific sites, then so are the invariant manifolds. We recall that the existence of localized whiskered tori has recently been proven for symplectic maps and flows in Fontich et. al. (submitted), but our results do not need that the systems are symplectic. For simplicity we will present first the main results for maps, but we will show tha the result for maps imply the results for flows. It is also true that the results for flows can be proved directly following the same ideas.Comment: This version corrects a bibliographical typo that appears in the version published in Annales Henri Poincare: Reference [FdlLS12] was wrongly listed by the publisher as "submitted to Jour. Diff. Equ." in the published version. Reference [FdlLS12] has not been submitted to Jour. Diff. Eq