December  2014, 6(4): 549-566. doi: 10.3934/jgm.2014.6.549

A dynamical condition for differentiability of Mather's average action

1. 

IEF, Campus UFV-Florestal, Universidade Federal de Viçosa, Florestal, MG 35690-000, Brazil

2. 

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

Received  October 2013 Revised  August 2014 Published  December 2014

We prove the differentiability of Mather's average action on all rotation vectors of measures whose supports are contained in a Lipschitz Lagrangian asymptotically isolated graph, invariant by Tonelli Hamiltonians. We also show the relationship between differentiability of $\beta $ and local integrability of the Hamiltonian flow.
Citation: 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
References:
[1]

M-C. Arnaud, The tiered Aubry set for autonomous Lagrangian functions,, Ann. Inst. Fourier (Grenoble), 58 (2008), 1733.  doi: 10.5802/aif.2397.  Google Scholar

[2]

M-C. Arnaud, A particular minimization property implies $C^{0}$-integrability,, Journal of Differential Equations, 250 (2011), 2389.  doi: 10.1016/j.jde.2010.12.002.  Google Scholar

[3]

V. Bangert, Minimal geodesics,, Erg. Theory and Dynamical Systems, 10 (1999), 263.  doi: 10.1017/S014338570000554X.  Google Scholar

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P. Bernard, Existence of $C^{1,1}$ critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds,, Ann. Sci. École Norm. Sup., 40 (2007), 445.  doi: 10.1016/j.ansens.2007.01.004.  Google Scholar

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

[6]

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

[7]

D. Burago, S. Ivanov and B. Kleiner, On the structure of the stable norm of periodic metrics,, Math. Research Letters, 4 (1997), 791.  doi: 10.4310/MRL.1997.v4.n6.a2.  Google Scholar

[8]

G. Contreras and R. Iturriaga, Global Minimizers of Autonomous Lagrangians,, $22^{\circ }$ Colóquio Brasileiro de Matemática IMPA, (1999).   Google Scholar

[9]

G. Contreras, L. Macarini and G. Paternain, Periodic orbits for exact magnetic flows on surfaces,, International Mathematics Research Notices, 2004 (2004), 361.  doi: 10.1155/S1073792804205050.  Google Scholar

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G. Contreras, J. Delgado and R. Iturriaga, Lagrangian flows: The dynamics of globally minimizing orbits-II,, Bol. Soc. Brasil. Mat., 28 (1997), 155.  doi: 10.1007/BF01233390.  Google Scholar

[11]

M. J. Dias Carneiro, On minimizing measures of the action of autonomous Lagrangians,, Nonlinearity, 8 (1995), 1077.  doi: 10.1088/0951-7715/8/6/011.  Google Scholar

[12]

M. J. Dias Carneiro and A. Lopes, On the minimal action function of autonomous lagrangians associated to magnetic fields,, Annales de l'I. H. P., 16 (1999), 667.  doi: 10.1016/S0294-1449(00)88183-4.  Google Scholar

[13]

A. Fathi and A. Siconolf, Existence of $C^{1}$ critical sub-solutions of the Hamilton-Jacobi equation,, Invent. Math., 155 (2004), 363.  doi: 10.1007/s00222-003-0323-6.  Google Scholar

[14]

A. Fathi, Weak KAM Theorem and Lagrangian Dynamics Preliminary Version Number 10,, 2008., ().   Google Scholar

[15]

A. Fathi, A. Figalli and L. Rifford, On the Hausdorff dimension of the Mather quotient,, Comm. Pure Appl. Math., 62 (2009), 445.  doi: 10.1002/cpa.20250.  Google Scholar

[16]

A. Fathi, A. Giuliani and A. Sorrentino, Uniqueness of Invariant Lagrangian Graphs in a Homology or a Cohomology Class,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 8 (2009), 659.   Google Scholar

[17]

M. Herman, Inégalités "a priori" pour des tores lagrangiens invariants par des difféomorphismes symplectiques,, Inst. Hautes Études Sci. Publ. Math., 70 (1989), 47.   Google Scholar

[18]

R. Mañé, Generic properties and problems of minimizing measure of Lagrangian dynamical systems,, Nonlinearity, 9 (1996), 273.  doi: 10.1088/0951-7715/9/2/002.  Google Scholar

[19]

R. Mañé, Global Variational Methods in Conservative Dynamics,, IMPA, (1993).   Google Scholar

[20]

D. Massart, On Aubry sets and Mather's action functional,, Israel J. Math., 134 (2003), 157.  doi: 10.1007/BF02787406.  Google Scholar

[21]

D. Massart, Vertices of Mather's Beta function, II,, Ergodic Theory Dynam. Systems, 29 (2009), 1289.  doi: 10.1017/S0143385708000631.  Google Scholar

[22]

D. Massart, Aubry sets vs Mather sets in two degrees of freedom,, Cal. Var. Partial Diff. Eqns, 42 (2011), 429.  doi: 10.1007/s00526-011-0393-z.  Google Scholar

[23]

D. Massart, Stable norm of surfaces: Local structure of the unit ball at rational directions,, Geom. Funct. Anal., 7 (1997), 996.  doi: 10.1007/s000390050034.  Google Scholar

[24]

D. Massart and A. Sorrentino, Differentiability of Mather's average action and integrability on closed surfaces,, Nonlinearity, 24 (2011), 1777.  doi: 10.1088/0951-7715/24/6/005.  Google Scholar

[25]

J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian Systems,, Math. Zeitschrift, 207 (1991), 169.  doi: 10.1007/BF02571383.  Google Scholar

[26]

J. R. Munkres, Elements of Algebraic Topology,, Addison-Wesley Publ. Co., (1984).   Google Scholar

[27]

G. Paternain, L. Polterovich and K. Siburg, Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory,, Mosc. Math. J., 3 (2003), 593.   Google Scholar

[28]

A. Sorrentino, On the integrability of Tonelli Hamiltonians,, Trans. Amer. Math. Soc., 363 (2011), 5071.  doi: 10.1090/S0002-9947-2011-05492-9.  Google Scholar

show all references

References:
[1]

M-C. Arnaud, The tiered Aubry set for autonomous Lagrangian functions,, Ann. Inst. Fourier (Grenoble), 58 (2008), 1733.  doi: 10.5802/aif.2397.  Google Scholar

[2]

M-C. Arnaud, A particular minimization property implies $C^{0}$-integrability,, Journal of Differential Equations, 250 (2011), 2389.  doi: 10.1016/j.jde.2010.12.002.  Google Scholar

[3]

V. Bangert, Minimal geodesics,, Erg. Theory and Dynamical Systems, 10 (1999), 263.  doi: 10.1017/S014338570000554X.  Google Scholar

[4]

P. Bernard, Existence of $C^{1,1}$ critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds,, Ann. Sci. École Norm. Sup., 40 (2007), 445.  doi: 10.1016/j.ansens.2007.01.004.  Google Scholar

[5]

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

[6]

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

[7]

D. Burago, S. Ivanov and B. Kleiner, On the structure of the stable norm of periodic metrics,, Math. Research Letters, 4 (1997), 791.  doi: 10.4310/MRL.1997.v4.n6.a2.  Google Scholar

[8]

G. Contreras and R. Iturriaga, Global Minimizers of Autonomous Lagrangians,, $22^{\circ }$ Colóquio Brasileiro de Matemática IMPA, (1999).   Google Scholar

[9]

G. Contreras, L. Macarini and G. Paternain, Periodic orbits for exact magnetic flows on surfaces,, International Mathematics Research Notices, 2004 (2004), 361.  doi: 10.1155/S1073792804205050.  Google Scholar

[10]

G. Contreras, J. Delgado and R. Iturriaga, Lagrangian flows: The dynamics of globally minimizing orbits-II,, Bol. Soc. Brasil. Mat., 28 (1997), 155.  doi: 10.1007/BF01233390.  Google Scholar

[11]

M. J. Dias Carneiro, On minimizing measures of the action of autonomous Lagrangians,, Nonlinearity, 8 (1995), 1077.  doi: 10.1088/0951-7715/8/6/011.  Google Scholar

[12]

M. J. Dias Carneiro and A. Lopes, On the minimal action function of autonomous lagrangians associated to magnetic fields,, Annales de l'I. H. P., 16 (1999), 667.  doi: 10.1016/S0294-1449(00)88183-4.  Google Scholar

[13]

A. Fathi and A. Siconolf, Existence of $C^{1}$ critical sub-solutions of the Hamilton-Jacobi equation,, Invent. Math., 155 (2004), 363.  doi: 10.1007/s00222-003-0323-6.  Google Scholar

[14]

A. Fathi, Weak KAM Theorem and Lagrangian Dynamics Preliminary Version Number 10,, 2008., ().   Google Scholar

[15]

A. Fathi, A. Figalli and L. Rifford, On the Hausdorff dimension of the Mather quotient,, Comm. Pure Appl. Math., 62 (2009), 445.  doi: 10.1002/cpa.20250.  Google Scholar

[16]

A. Fathi, A. Giuliani and A. Sorrentino, Uniqueness of Invariant Lagrangian Graphs in a Homology or a Cohomology Class,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 8 (2009), 659.   Google Scholar

[17]

M. Herman, Inégalités "a priori" pour des tores lagrangiens invariants par des difféomorphismes symplectiques,, Inst. Hautes Études Sci. Publ. Math., 70 (1989), 47.   Google Scholar

[18]

R. Mañé, Generic properties and problems of minimizing measure of Lagrangian dynamical systems,, Nonlinearity, 9 (1996), 273.  doi: 10.1088/0951-7715/9/2/002.  Google Scholar

[19]

R. Mañé, Global Variational Methods in Conservative Dynamics,, IMPA, (1993).   Google Scholar

[20]

D. Massart, On Aubry sets and Mather's action functional,, Israel J. Math., 134 (2003), 157.  doi: 10.1007/BF02787406.  Google Scholar

[21]

D. Massart, Vertices of Mather's Beta function, II,, Ergodic Theory Dynam. Systems, 29 (2009), 1289.  doi: 10.1017/S0143385708000631.  Google Scholar

[22]

D. Massart, Aubry sets vs Mather sets in two degrees of freedom,, Cal. Var. Partial Diff. Eqns, 42 (2011), 429.  doi: 10.1007/s00526-011-0393-z.  Google Scholar

[23]

D. Massart, Stable norm of surfaces: Local structure of the unit ball at rational directions,, Geom. Funct. Anal., 7 (1997), 996.  doi: 10.1007/s000390050034.  Google Scholar

[24]

D. Massart and A. Sorrentino, Differentiability of Mather's average action and integrability on closed surfaces,, Nonlinearity, 24 (2011), 1777.  doi: 10.1088/0951-7715/24/6/005.  Google Scholar

[25]

J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian Systems,, Math. Zeitschrift, 207 (1991), 169.  doi: 10.1007/BF02571383.  Google Scholar

[26]

J. R. Munkres, Elements of Algebraic Topology,, Addison-Wesley Publ. Co., (1984).   Google Scholar

[27]

G. Paternain, L. Polterovich and K. Siburg, Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory,, Mosc. Math. J., 3 (2003), 593.   Google Scholar

[28]

A. Sorrentino, On the integrability of Tonelli Hamiltonians,, Trans. Amer. Math. Soc., 363 (2011), 5071.  doi: 10.1090/S0002-9947-2011-05492-9.  Google Scholar

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