Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-06T09:46:05.039Z Has data issue: false hasContentIssue false

A geometric definition of the Mañé-Mather set and a Theorem of Marie-Claude Arnaud.

Published online by Cambridge University Press:  19 October 2011

PATRICK BERNARD
Affiliation:
Université Paris-Dauphine, ceremade, umr cnrs 7534, Pl. du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16, France e-mail: [email protected], [email protected]
JOANA OLIVEIRA DOS SANTOS
Affiliation:
Université Paris-Dauphine, ceremade, umr cnrs 7534, Pl. du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16, France e-mail: [email protected], [email protected]

Abstract

We study some properties of Lipschitz exact Lagrangian manifolds isotopic to the zero section. We prove that if such a manifold is invariant under an optical Hamiltonian, then it must be a Lipschitz graph. This extends a recent result of Marie–Claude Arnaud. We also obtain a new geometric description of the Mañé–Mather invariant set.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bernard, P. and Santos, J. Oliveira dosA geometric definition of the Aubry-Mather set. J. Top. Anal. 2 (3) (2010), 385393.Google Scholar
[2]Arnaud, M. C.Pseudographs and the Lax-Oleinik semi-group: a geometric and dynamical interpretation. Nonlinearity 24 (1) (2011), 7179.CrossRefGoogle Scholar
[3]Arnaud, M. C.On a Theorem due to Birkhoff. Geom. Funct. Anal. 20 (6) (2010), 13071316.CrossRefGoogle Scholar
[4]Bialy, M. and Polterovich, L.Hamiltonian diffeomorphisms and Lagrangian distributions. Geometric and Functional Analysis 2 (2) (1992), 173210.CrossRefGoogle Scholar
[5]Bialy, M. and Polterovich, L.Hamiltonian systems, Lagrangian tori and Birkhoff's theorem. Math. Ann. 292 (1) (1992), 619627.Google Scholar
[6]Bernard, P.Connecting orbits of time dependent lagrangian systems. Ann. Inst. Fourier 52 (5) (2002), 15331568.Google Scholar
[7]Bernard, P.Existence of C 1,1 critical sub-solutions of the Hamilton–Jacobi equation on compact manifolds. Ann. Sci. l'Ecole Norm. Sup 40 (3) (2007), 445452.Google Scholar
[8]Bernard, P.Symplectic aspects of Mather theory. Duke Math. J. 136 (3) (2007), 401420.Google Scholar
[9]Bernard, P.The dynamics of pseudographs in convex Hamiltonian systems. J.A.M.S. 21 (3) (2008), 615669.Google Scholar
[10]Bernard, P. The Lax–Oleinik semigroup: a Hamiltonian viewpoint. Lecture notes.Google Scholar
[11]Chaperon, M.Lois de conservation et géométrie symplectique. Comptes rendus de l'Académie des sciences. Série 1, Mathématique 312 (4) (1991), 345348.Google Scholar
[12]Clarke, F. H. Optimization and nonsmooth analysis. Society for Industrial Mathematics (1990).Google Scholar
[13]Contreras, G., Iturriaga, R., Paternain, G. P. and Paternain, M.Lagrangian graphs, minimizing measures and Mañé's critical values. Geom. Funct. Anal. 8 (5) (1998), 788809.Google Scholar
[14]Fathi, A.Weak KAM theorem in Lagrangian dynamics. ghost book.Google Scholar
[15]Fathi, A. and Maderna, E.Weak KAM theorem on non compact manifolds. NoDEA: Nonlinear Diff. Eq. Appl. 14 (1) (2007), 127.CrossRefGoogle Scholar
[16]Fathi, A. and Siconolfi, A.Existence of C 1 critical subsolutions of the Hamilton-Jacobi equation. Invent. math. 155 (2) (2004), 363388.CrossRefGoogle Scholar
[17]Fathi, A. and Siconolfi, A.PDE aspects of Aubry–Mather theory for quasiconvex Hamiltonians. Calc. Var. Partial Differential Equations 22 (2) (2005), 185228.Google Scholar
[18]Mané, R.Lagrangian flows: the dynamics of globally minimizing orbits. Bull. Brazilian Math. Soc. 28 (2) (1997), 141153.Google Scholar
[19]Mather, J. N.Variational construction of connecting orbits. Ann. lnst. Fourier 43 (5) (1993), 13491386.CrossRefGoogle Scholar
[20]Paternain, G. P., Polterovich, L. V. and Siburg, K. F.Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory. Moscow Math. J. 3 (2) (2003), 593619.Google Scholar
[21]Rockafellar, R. T.Convex Analysis, second edition (Princeton University Press, 1972).Google Scholar
[22]Viterbo, C.Symplectic topology as the geometry of generating functions. Math. Ann. 292 (1) (1992), 685710.CrossRefGoogle Scholar
[23]Viterbo, C. Symplectic homogenization. Arxiv preprint arXiv:0801.0206, 2007.Google Scholar