A generic property of exact magnetic Lagrangians

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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

Abstract
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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 & Continuous Dynamical Systems - A, 2012, 32 (12) : 4183-4194. doi: 10.3934/dcds.2012.32.4183

References:

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References:

[1]	P. Bernard and G. Contreras, A generic property offamilies of Lagrangian systems,, Annals of Mathematics, 167 (2008), 1099. doi: 10.4007/annals.2008.167.1099. Google Scholar
[2]	P. Bernard, On the Conley decomposition of Mather sets,, Rev. Mat. Iberoamericana, 26 (2010), 115. doi: 10.4171/RMI/596. Google Scholar
[3]	G. Contreras and R. Iturriaga, "Global Minimizers of Autonomous Lagrangians,", 22 Colóquio Brasileiro deMatemática, (1999). Google Scholar
[4]	G. Contreras and G. Paternain, Connecting orbitsbetween static classes for generic Lagrangian systems,, Topology, 41 (2002), 645. doi: 10.1016/S0040-9383(00)00042-2. Google Scholar
[5]	A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics,", Cambridge Studies in Advanced Mathematics, (2010). Google Scholar
[6]	R. Mañé, Generic properties and problems ofminimizing measure of Lagrangian dynamical systems,, Nonlinearity, 9 (1996), 273. Google Scholar
[7]	J. Mather, Action minimizing invariant measuresfor positive definite Lagrangian Systems,, Math. Zeitschrift, 207 (1991), 169. doi: 10.1007/BF02571383. Google Scholar
[8]	M. Paternain and G. Paternain, Critical Values ofautonomous Lagrangian systems,, Comment. Math. Helvetici, 72 (1997), 481. doi: 10.1007/s000140050029. Google Scholar

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