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December  2012, 32(12): 4183-4194. doi: 10.3934/dcds.2012.32.4183

A generic property of exact magnetic Lagrangians

Mário Jorge Dias Carneiro 1, and  Alexandre Rocha 2,

1. 

ICEX, Universidade Federal de Minas Gerais, Belo Horizonte, MG 30161-970, Brazil
2. 

CAF, Universidade Federal de Vicosa, Florestal, MG 35690-000, Brazil

Received  June 2011 Revised  May 2012 Published  August 2012

We prove that for the set of Exact Magnetic Lagrangians the pro-perty “There exist finitely many static classes for every cohomology class" is generic. We also prove some dynamical consequences of this property.
Keywords: exact magnetic Lagrangian, ergodic minimizing measure, generic property, Mather's theory, static class..
Mathematics Subject Classification: Primary: 37J50, 37J05; Secondary: 70H0.
Citation: Mário Jorge Dias Carneiro, Alexandre Rocha. A generic property of exact magnetic Lagrangians. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4183-4194. doi: 10.3934/dcds.2012.32.4183
References:
[1]

P. Bernard and G. Contreras, A generic property offamilies of Lagrangian systems, Annals of Mathematics, 167 (2008), 1099-1108. doi: 10.4007/annals.2008.167.1099.

[2]

P. Bernard, On the Conley decomposition of Mather sets, Rev. Mat. Iberoamericana, 26 (2010), 115-132. doi: 10.4171/RMI/596.

[3]

G. Contreras and R. Iturriaga, "Global Minimizers of Autonomous Lagrangians," 22 Colóquio Brasileiro deMatemática, 1999.

[4]

G. Contreras and G. Paternain, Connecting orbitsbetween static classes for generic Lagrangian systems, Topology, 41 (2002), 645-666. doi: 10.1016/S0040-9383(00)00042-2.

[5]

A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics," Cambridge Studies in Advanced Mathematics, 2010.

[6]

R. Mañé, Generic properties and problems ofminimizing measure of Lagrangian dynamical systems, Nonlinearity, 9 (1996), 273-310.

[7]

J. Mather, Action minimizing invariant measuresfor positive definite Lagrangian Systems, Math. Zeitschrift, 207 (1991), 169-207. doi: 10.1007/BF02571383.

[8]

M. Paternain and G. Paternain, Critical Values ofautonomous Lagrangian systems, Comment. Math. Helvetici, 72 (1997), 481-499. doi: 10.1007/s000140050029.

show all references

References:
[1]

P. Bernard and G. Contreras, A generic property offamilies of Lagrangian systems, Annals of Mathematics, 167 (2008), 1099-1108. doi: 10.4007/annals.2008.167.1099.

[2]

P. Bernard, On the Conley decomposition of Mather sets, Rev. Mat. Iberoamericana, 26 (2010), 115-132. doi: 10.4171/RMI/596.

[3]

G. Contreras and R. Iturriaga, "Global Minimizers of Autonomous Lagrangians," 22 Colóquio Brasileiro deMatemática, 1999.

[4]

G. Contreras and G. Paternain, Connecting orbitsbetween static classes for generic Lagrangian systems, Topology, 41 (2002), 645-666. doi: 10.1016/S0040-9383(00)00042-2.

[5]

A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics," Cambridge Studies in Advanced Mathematics, 2010.

[6]

R. Mañé, Generic properties and problems ofminimizing measure of Lagrangian dynamical systems, Nonlinearity, 9 (1996), 273-310.

[7]

J. Mather, Action minimizing invariant measuresfor positive definite Lagrangian Systems, Math. Zeitschrift, 207 (1991), 169-207. doi: 10.1007/BF02571383.

[8]

M. Paternain and G. Paternain, Critical Values ofautonomous Lagrangian systems, Comment. Math. Helvetici, 72 (1997), 481-499. doi: 10.1007/s000140050029.

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