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197
Propagation in Hamiltonian dynamics and relative symplectic homology
, 2003
"... The main result asserts the existence of noncontractible periodic orbits for compactly supported timedependent Hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating Hamiltonian is sufficiently large over the zero section. The proof ..."
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Cited by 59 (4 self)
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The main result asserts the existence of noncontractible periodic orbits for compactly supported timedependent Hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating Hamiltonian is sufficiently large over the zero section. The proof is based on Floer homology and on the notion of a relative symplectic capacity. Applications include results about propagation properties of sequential Hamiltonian systems, periodic orbits on hypersurfaces, Hamiltonian circle actions, and smooth Lagrangian skeletons in Stein manifolds.
The dynamics of pseudographs in convex Hamiltonian Systems
 J. Amer. Math. Soc
"... abstract. We study the evolution, under convex Hamiltonian flows on cotangent bundles of compact manifolds, of certain distinguished subsets of the phase space. These subsets are generalizations of Lagrangian graphs, we call them pseudographs. They emerge in a natural way from Fathi’s weak KAM theor ..."
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Cited by 56 (12 self)
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abstract. We study the evolution, under convex Hamiltonian flows on cotangent bundles of compact manifolds, of certain distinguished subsets of the phase space. These subsets are generalizations of Lagrangian graphs, we call them pseudographs. They emerge in a natural way from Fathi’s weak KAM theory. By this method, we find various orbits which connect prescribed regions of the phase space. Our study is inspired by works of John Mather. As an application, we obtain the existence of diffusion in a large class of a priori unstable systems and provide a solution to the large gap problem. We hope that our method will have applications to more examples. Résumé. Nous étudions l’évolution, par le flot d’un Hamiltonien convexe sur une variété compacte, de certains ensembles de l’espace des phases. Nous appelons pseudographes ces ensembles, qui sont des généralisations de graphes Lagrangiens apparaissant de manière naturelle dans la théorie KAM faible de Fathi. Par cette méthode, nous trouvons diverses orbites qui joignent des domaines donnés de l’espace des phases. Notre étude s’inspire de travaux de John Mather. Nous obtenons l’existence de diffusion dans une large classe de systèmes à priori instables comme application de cette méthode, qui permet de résoudre le probleme de l’écart entre les tores invariants. Nous espérons que la méthode s’appliquera à d’autres
Optimal mass transportation and Mather theory
 JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
, 2005
"... We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz ..."
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Cited by 54 (4 self)
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We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz laminations. We prove the existence of an optimal transport map when the transported measure is absolutely continuous. We explain the relations with Mather’s minimal measures.
Existence of diffusion orbits in a priori unstable Hamiltonian systems
 J. Differential Geom
"... Abstract. Under open and dense conditions we show that Arnold diffusion orbits exist in a priori unstable and timeperiodic Hamiltonian systems with two degrees of freedom. 1, Introduction and Results By the KAM (Kolmogorov, Arnold and Moser) theory we know that there are many invariant tori in near ..."
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Cited by 53 (3 self)
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Abstract. Under open and dense conditions we show that Arnold diffusion orbits exist in a priori unstable and timeperiodic Hamiltonian systems with two degrees of freedom. 1, Introduction and Results By the KAM (Kolmogorov, Arnold and Moser) theory we know that there are many invariant tori in nearly integrable Hamiltonian systems with arbitrary n degrees of freedom. These tori are of n dimension and occupy a nearly full Lebesgue measure set in the phase space. As an important consequence, all orbits are stable
Convex Hamiltonians without Conjugate Points
 Th and Dynam. Sys
"... this paper are disconjugate orbits of the Hamiltonian flow, i.e. orbits without conjugate points. Let ß : T ..."
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Cited by 52 (16 self)
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this paper are disconjugate orbits of the Hamiltonian flow, i.e. orbits without conjugate points. Let ß : T
Effective Hamiltonians and Averaging for Hamiltonian Dynamics II
"... Abstract. This paper, building upon ideas of Mather, Moser, Fathi, E and others, applies PDE methods to understand the structure of certain Hamiltonian flows. The main point is that the “cell”or “corrector”PDE, introduced and solved in a weak sense by Lions, Papanicolaou and Varadhan in their study ..."
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Cited by 50 (27 self)
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Abstract. This paper, building upon ideas of Mather, Moser, Fathi, E and others, applies PDE methods to understand the structure of certain Hamiltonian flows. The main point is that the “cell”or “corrector”PDE, introduced and solved in a weak sense by Lions, Papanicolaou and Varadhan in their study of periodic homogenization for Hamilton–Jacobi equations, formally induces a canonical change of variables, in terms of which the dynamics are trivial. We investigate to what extent this observation can be made rigorous in the case that the Hamiltonian is strictly convex in the momenta, given that the relevant PDE does not usually in fact admit a smooth solution. 1. Introduction. This is the first of a projected series of papers that develop PDE techniques to understand certain aspects of Hamiltonian dynamics with many degrees of freedom. 1.1. Changing variables.
Weak KAM theorem on non compact manifolds
, 2002
"... In this paper, we consider a time independent C2 Hamiltonian, satisfying the usual hypothesis of the classical Calculus of Variations, on a noncompact connected manifold. Using the LaxOleinik semigroup, we give a proof of the existence of weak KAM solutions, or viscosity solutions, for the associa ..."
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Cited by 45 (7 self)
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In this paper, we consider a time independent C2 Hamiltonian, satisfying the usual hypothesis of the classical Calculus of Variations, on a noncompact connected manifold. Using the LaxOleinik semigroup, we give a proof of the existence of weak KAM solutions, or viscosity solutions, for the associated HamiltonJacobi Equation. This proof works also in presence of symmetries. We also study the role of the amenability of the group of symmetries to understand when the several critical values that can be associated with the Hamiltonian coincide. 1
Lagangian flow: The dynamics of globally minimizing orbits
 II. Bol. Soc. Brasil. Math. (NS
, 1997
"... ..."
Connecting orbits of time dependent Lagrangian systems
 Ann. Inst. Fourier
"... Résumé: On donne une généralisation à la dimension supérieure des résultats obtenus par Birkhoff et Mather sur l’existence d’orbites errant dans les zones d’instabilité des applications de l’anneau déviant la verticale. Notre généralisation s’inspire fortement de celle proposée par Mather dans [7]. ..."
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Cited by 36 (16 self)
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Résumé: On donne une généralisation à la dimension supérieure des résultats obtenus par Birkhoff et Mather sur l’existence d’orbites errant dans les zones d’instabilité des applications de l’anneau déviant la verticale. Notre généralisation s’inspire fortement de celle proposée par Mather dans [7]. Elle présente cependant l’avantage de contenir effectivement l’essentiel des resultats de Birkhoff et Mather sur les difféomorphismes de l’anneau. Abstract: We generalize to higher dimension results of Birkhoff and Mather on the existence of orbits wandering in regions of instability of twist maps. This generalization is strongly inspired by the one proposed by Mather in [7]. However, its advantage is that it contains most of the results of Birkhoff and Mather on twist maps. A very natural class of problems in dynamical systems is the existence of orbits connecting prescribed regions of phase space. There are several important open questions in this line, like the one posed by Arnold: Is a generic Hamiltonian system transitive on its energy shells? Birkhoff’s theory of regions of instability of twists maps, recently extended by Mather using variational methods and by Le Calvez, provide very relevant results in that direction. In short, these works establish the existence, for a certain class of mappings of the annulus,
Generic properties of families of Lagrangian systems
 Ann. of Math
"... Abstract. We prove that a generic lagrangian has finitely many minimizing measures for every cohomology class. ..."
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Cited by 26 (3 self)
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Abstract. We prove that a generic lagrangian has finitely many minimizing measures for every cohomology class.