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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  and
JOANA OLIVEIRA DOS SANTOS
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]
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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
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 152 , Issue 1 , January 2012 , pp. 167 - 178
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Bernard, P. and Santos, J. Oliveira dosA geometric definition of the Aubry-Mather set. J. Top. Anal. 2 (3) (2010), 385–393.Google Scholar
[2]Arnaud, M. C.Pseudographs and the Lax-Oleinik semi-group: a geometric and dynamical interpretation. Nonlinearity 24 (1) (2011), 71–79.CrossRefGoogle Scholar
[3]Arnaud, M. C.On a Theorem due to Birkhoff. Geom. Funct. Anal. 20 (6) (2010), 1307–1316.CrossRefGoogle Scholar
[4]Bialy, M. and Polterovich, L.Hamiltonian diffeomorphisms and Lagrangian distributions. Geometric and Functional Analysis 2 (2) (1992), 173–210.CrossRefGoogle Scholar
[5]Bialy, M. and Polterovich, L.Hamiltonian systems, Lagrangian tori and Birkhoff's theorem. Math. Ann. 292 (1) (1992), 619–627.Google Scholar
[6]Bernard, P.Connecting orbits of time dependent lagrangian systems. Ann. Inst. Fourier 52 (5) (2002), 1533–1568.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), 445–452.Google Scholar
[8]Bernard, P.Symplectic aspects of Mather theory. Duke Math. J. 136 (3) (2007), 401–420.Google Scholar
[9]Bernard, P.The dynamics of pseudographs in convex Hamiltonian systems. J.A.M.S. 21 (3) (2008), 615–669.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), 345–348.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), 788–809.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), 1–27.CrossRefGoogle Scholar
[16]Fathi, A. and Siconolfi, A.Existence of C 1 critical subsolutions of the Hamilton-Jacobi equation. Invent. math. 155 (2) (2004), 363–388.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), 185–228.Google Scholar
[18]Mané, R.Lagrangian flows: the dynamics of globally minimizing orbits. Bull. Brazilian Math. Soc. 28 (2) (1997), 141–153.Google Scholar
[19]Mather, J. N.Variational construction of connecting orbits. Ann. lnst. Fourier 43 (5) (1993), 1349–1386.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), 593–619.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), 685–710.CrossRefGoogle Scholar
[23]Viterbo, C. Symplectic homogenization. Arxiv preprint arXiv:0801.0206, 2007.Google Scholar