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On the Number of Mather Measures of Lagrangian Systems

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Abstract

In 1996, Ricardo Ricardo Mañé discovered that Mather measures are in fact the minimizers of a “universal” infinite dimensional linear programming problem. This fundamental result has many applications, of which one of the most important is to the estimates of the generic number of Mather measures. Mañé obtained the first estimation of that sort by using finite dimensional approximations. Recently, we were able, with Gonzalo Contreras, to use this method of finite dimensional approximation in order to solve a conjecture of John Mather concerning the generic number of Mather measures for families of Lagrangian systems. In the present paper we obtain finer results in that direction by applying directly some classical tools of convex analysis to the infinite dimensional problem. We use a notion of countably rectifiable sets of finite codimension in Banach (and Frechet) spaces which may deserve independent interest.

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References

  1. Aronszajn N.: Differentiability of Lipschitzian mappings between Banach spaces. Stud. Math. 57, 147–190 (1976)

    MATH  MathSciNet  Google Scholar 

  2. Bangert V.: Minimal measures and minimizing closed normal one-currents. Geom. Funct. Anal. 9(3), 413–427 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Benyamini, Y., Lindenstrauss, J.: Geometrical Nonlinear Functional Analysis, vol. 1. AMS Colloquium Publications, 48, 2000

  4. Bernard, P., Contreras, G.: A generic property of families of Lagrangian systems. Ann. Math. 167(3) (2008)

  5. Bernard P.: Young measures, superposition, and transport. Indiana Univ. Math. J. 57(1), 247–276 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bernard P.: Connecting orbits of time dependent Lagrangian systems. Ann. Inst. Fourier 52, 1533–1568 (2002)

    MATH  MathSciNet  Google Scholar 

  7. Bernard P.: The dynamics of pseudographs in convex Hamiltonian systems. J. AMS 21(3), 625–669 (2008)

    MathSciNet  Google Scholar 

  8. Bernard P.: Existence of C 1,1 critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds. Ann. Sci. E. N. S. 40(3), 445–552 (2007)

    MATH  MathSciNet  Google Scholar 

  9. Borisovich Y.G., Zvyagin V.C., Sapranov Y.I.: Non-linear Fredholm maps and the Leray-Schauder theory. Russ. Math. Surv. 32(4), 3–54 (1977)

    Article  MATH  Google Scholar 

  10. Cheng C.-Q., Yan J.: Existence of diffusion orbits in a priori unstable Hamiltonian systems. J. Differ. Geom. 67(3), 457–517 (2004)

    MATH  MathSciNet  Google Scholar 

  11. Cheng C.-Q., Yan J.: Arnold diffusion in Hamiltonian systems: the a priori unstable case. J. Diff. Geom. 82, 229–277 (2009)

    MATH  MathSciNet  Google Scholar 

  12. Christensen J.P.R.: On sets of Haar measure zero in abelian Polish groups. Israel J. Math. 13, 255–260 (1972)

    Article  MathSciNet  Google Scholar 

  13. Csörnyei M.: Aronszajn null and Gaussian null sets coincide. Israel J. Math. 111, 191–201 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Eliashberg, Y., Mishachev, N.: Introduction to the h-principle. Graduate Studies in Math 48, AMS (2002)

  15. Fathi, A.: Weak KAM Theorem in Lagrangian Dynamics. (to appear)

  16. Fathi A., Siconolfi A.: Existence of C 1 critical subsolutions of the Hamilton-Jacobi equation. Invent. Math. 155(2), 363–388 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. Hunt, B.R., Sauer, T., Yorke, J.A.: Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces. Bull. Am. Math. Soc. (N.S.) 27(2), 217–238 (1992); with an addendum in Bull. Am. Math. Soc. (N.S.) 28(2), 306–307 (1993)

  18. Mañé R.: On the minimizing measures of Lagrangian dynamical systems. Nonlinearity 5(3), 623–638 (1992)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. Mañé R.: Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity 9(2), 273–310 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Mankiewicz P.: On the differentiability of Lipschitz mappings in Fréchet spaces. Stud. Math. 45, 15–29 (1973)

    MATH  MathSciNet  Google Scholar 

  21. Mather J.N.: Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207, 169–207 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  22. Mather J.N.: Variational construction of connecting orbits. Ann. Inst. Fourier 43, 1349–1368 (1993)

    MATH  MathSciNet  Google Scholar 

  23. Phelps R.R.: Gaussian null sets and differentiability of Lipschitz map on Banach spaces. Pacific J. Math. 77(2), 523–531 (1978)

    MATH  MathSciNet  Google Scholar 

  24. Zajíček L.: On the points of multiplicity of monotone operators. Comment. Math. Univ. Carolinae 19(1), 179–189 (1978)

    MATH  MathSciNet  Google Scholar 

  25. Zajíček L.: On Lipschitz and d. c. surfaces of finite codimension in a Banach space. Czech. Math. J. 58, 849–864 (2008)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. CEREMADE, CNRS, Pl. du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Patrick Bernard

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  1. Patrick Bernard
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Correspondence to Patrick Bernard.

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Communicated by L. Ambrosio

Membre de l’IUF.

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Bernard, P. On the Number of Mather Measures of Lagrangian Systems. Arch Rational Mech Anal 197, 1011–1031 (2010). https://doi.org/10.1007/s00205-009-0289-7

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  • DOI: https://doi.org/10.1007/s00205-009-0289-7

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