2011
DOI: 10.1007/s00526-011-0393-z
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Aubry sets vs. Mather sets in two degrees of freedom

Abstract: Abstract. Let L be an autonomous Tonelli Lagrangian on a closed manifold of dimension two. Let C be the set of cohomology classes whose Mather set consists of periodic orbits, none of which is a fixed point. Then for almost all c in C, the Aubry set of c equals the Mather set of c.

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Cited by 7 publications
(6 citation statements)
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“…Mather [51] has developed a theory in any dimension, closely related to the present article, of minimizing measures for a Tonelli's Lagrangian, that is, 1-periodic superlinear strictly convex Lagrangian L : T M × T 1 → R. In dimension 1, Moser [52] has proved the equivalence between two approaches, monotone twist map versus Tonelli's Lagrangian, showing that such a monotone twist map can always be seen as the time one map of some 1-periodic smooth Hamiltonian H : T * M ×T 1 → R. As noticed by Herman [38], there are interesting connections between configurations with minimal energy and Lagrangian tori invariant under symplectic diffeormorphisms of the cotangent bundle of the d-dimensional torus. Massart has generalized several results of the Aubry-Mather theory in [45,46,47].…”
Section: Introductionmentioning
confidence: 88%
“…Mather [51] has developed a theory in any dimension, closely related to the present article, of minimizing measures for a Tonelli's Lagrangian, that is, 1-periodic superlinear strictly convex Lagrangian L : T M × T 1 → R. In dimension 1, Moser [52] has proved the equivalence between two approaches, monotone twist map versus Tonelli's Lagrangian, showing that such a monotone twist map can always be seen as the time one map of some 1-periodic smooth Hamiltonian H : T * M ×T 1 → R. As noticed by Herman [38], there are interesting connections between configurations with minimal energy and Lagrangian tori invariant under symplectic diffeormorphisms of the cotangent bundle of the d-dimensional torus. Massart has generalized several results of the Aubry-Mather theory in [45,46,47].…”
Section: Introductionmentioning
confidence: 88%
“…Let us make some comments about the assumption in Theorem 1 concerning the equivalence between the Aubry-Mather set and the Mather set. This condition is of topological nature, as observed in [28] for surfaces. Particularly important for us are the results of section 4 in [28] where it is considered the case where the Mather set is a single periodic orbit: if γ separates M (see case 1.2) then the two sets are equal.…”
Section: Introductionmentioning
confidence: 59%
“…Particularly important for us are the results of section 4 in [28] where it is considered the case where the Mather set is a single periodic orbit: if γ separates M (see case 1.2) then the two sets are equal. In fact, the coincidence of the Aubry-Mather set and the Mather set is generic in homology (see Theorem 3 in [28]).…”
Section: Introductionmentioning
confidence: 99%
“…Then the radial face of β containing h 0 has at least one non-zero extremal point th 0 with t ∈ ]0, +∞[. Furthermore if the tangent cone to β at h 0 is contains no plane, then neither does the tangent cone to β at th 0 (see [Mt,Lemma 17]). Since I R (th 0 ) = I R (h 0 ), we may, for the purpose of proving Theorem 5, assume that h 0 itself is an extremal point of β.…”
Section: Proof Of Theoremmentioning
confidence: 97%