American Institute of Mathematical Sciences

Advanced
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 & Continuous Dynamical Systems - A, 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.  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

show all references

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

[1]

Radu Saghin. On the number of ergodic minimizing measures for Lagrangian flows. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 501-507. doi: 10.3934/dcds.2007.17.501
[2]

Jon Chaika, Howard Masur. There exists an interval exchange with a non-ergodic generic measure. Journal of Modern Dynamics, 2015, 9: 289-304. doi: 10.3934/jmd.2015.9.289
[3]

Kaizhi Wang. Action minimizing stochastic invariant measures for a class of Lagrangian systems. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1211-1223. doi: 10.3934/cpaa.2008.7.1211
[4]

Mads R. Bisgaard. Mather theory and symplectic rigidity. Journal of Modern Dynamics, 2019, 15: 165-207. doi: 10.3934/jmd.2019018
[5]

Ian D. Morris. Ergodic optimization for generic continuous functions. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 383-388. doi: 10.3934/dcds.2010.27.383
[6]

Mario Jorge Dias Carneiro, Rafael O. Ruggiero. On the graph theorem for Lagrangian minimizing tori. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6029-6045. doi: 10.3934/dcds.2018260
[7]

Oliver Jenkinson. Every ergodic measure is uniquely maximizing. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 383-392. doi: 10.3934/dcds.2006.16.383
[8]

Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757
[9]

Thierry de la Rue. An introduction to joinings in ergodic theory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 121-142. doi: 10.3934/dcds.2006.15.121
[10]

Ugo Bessi. Viscous Aubry-Mather theory and the Vlasov equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 379-420. doi: 10.3934/dcds.2014.34.379
[11]

Hans Koch, Rafael De La Llave, Charles Radin. Aubry-Mather theory for functions on lattices. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 135-151. doi: 10.3934/dcds.1997.3.135
[12]

Alexandre Rocha, Mário Jorge Dias Carneiro. A dynamical condition for differentiability of Mather's average action. Journal of Geometric Mechanics, 2014, 6 (4) : 549-566. doi: 10.3934/jgm.2014.6.549
[13]

Alfonso Sorrentino. Computing Mather's $\beta$-function for Birkhoff billiards. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5055-5082. doi: 10.3934/dcds.2015.35.5055
[14]

Jinjun Li, Min Wu. Generic property of irregular sets in systems satisfying the specification property. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 635-645. doi: 10.3934/dcds.2014.34.635
[15]

Sangtae Jeong, Chunlan Li. Measure-preservation criteria for a certain class of 1-lipschitz functions on Zp in mahler's expansion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3787-3804. doi: 10.3934/dcds.2017160
[16]

Fabio Camilli, Annalisa Cesaroni. A note on singular perturbation problems via Aubry-Mather theory. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 807-819. doi: 10.3934/dcds.2007.17.807
[17]

Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683
[18]

Aihua Fan, Lingmin Liao, Jacques Peyrière. Generic points in systems of specification and Banach valued Birkhoff ergodic average. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1103-1128. doi: 10.3934/dcds.2008.21.1103
[19]

Jessica Hyde, Daniel Kelleher, Jesse Moeller, Luke Rogers, Luis Seda. Magnetic Laplacians of locally exact forms on the Sierpinski Gasket. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2299-2319. doi: 10.3934/cpaa.2017113
[20]

Rémi Leclercq. Spectral invariants in Lagrangian Floer theory. Journal of Modern Dynamics, 2008, 2 (2) : 249-286. doi: 10.3934/jmd.2008.2.249

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences