Advertisement

Inventiones mathematicae

, Volume 200, Issue 1, pp 201–261 | Cite as

Generic hyperbolicity of Aubry sets on surfaces

  • G. Contreras
  • A. Figalli
  • L. Rifford
Article

Abstract

Given a Tonelli Hamiltonian of class \(C^2\) on the cotangent bundle of a compact surface, we show that there is an open dense set of potentials in the \(C^2\) topology for which the Aubry set is hyperbolic in its energy level.

Notes

Acknowledgments

Large part of this work was done while AF and GC were visiting the Mathematics Department at the University Nice Sophia Antipolis, whose warm hospitality is gratefully acknowledged. The authors would like to thank the anonymous referees for their valuable comments and suggestions to improve the quality of the paper.

References

  1. 1.
    Abraham, R., Marsden, J.E.: Foundations of Mechanics. Benjamin, London (1978)zbMATHGoogle Scholar
  2. 2.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. In: Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)Google Scholar
  3. 3.
    Arnaud, M.-C.: Le “closing lemma” en topologie \(C^1\). Mém. Soc. Math. Fr. 74, vi+120 (1998)Google Scholar
  4. 4.
    Arnaud, M.-C.: Fibrés de Green et régularité des graphes \(C^0\)-lagrangiens invariants par un flot de Tonelli. Ann. Henri Poincaré 9(5), 881–926 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Arnaud, M.-C.: The link between the shape of the Aubry-Mather sets and their Lyapunov exponents. Ann. Math. 174(3), 1571–1601 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Arnaud, M.-C.: Green bundles and related topics. In Proceedings of the International Congress of Mathematicians, ICM’10 (2010)Google Scholar
  7. 7.
    Arnaud, M.-C.: Green bundles, Lyapunov exponents and regularity along the supports of the minimizing measures. Ann. Inst. H. Poincaré Anal. Non Linéaire 29(6), 989–1007 (2012)Google Scholar
  8. 8.
    Bernard, P.: Existence of \(C^{1,1}\) critical sub-solutions of the Hamilton–Jacobi equation on compact manifolds. Ann. Sci. École Norm. Sup. 40(3), 445–452 (2007)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Bernard, P.: On the Conley decomposition of Mather sets. Rev. Mat. Iberoam. 26(1), 115–132 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Cannarsa, P., Sinestrari, C.: Semiconcave functions, Hamilton–Jacobi equations, and optimal control. In: Progress in Nonlinear Differential Equations and their Applications, vol. 58. Birkhäuser Boston Inc., Boston (2004)Google Scholar
  11. 11.
    Cannas da Silva, A.: Lectures on symplectic geometry. In: Lecture Notes in Mathematics, vol. 1764. Springer, Berlin (2001)Google Scholar
  12. 12.
    Clarke, F.H.: Optimization and nonsmooth analysis. In: Canadian Mathematical Society Series of Monographs and Advances Texts. Wiley, New York (1983)Google Scholar
  13. 13.
    Clarke, F.H.: Functional analysis, calculus of variations and optimal control. In: Graduate Texts in Mathematics, vol. 264. Springer, London (2013)Google Scholar
  14. 14.
    Contreras, G., Iturriaga, R.: Convex Hamiltonians without conjugate points. Ergod. Theory Dyn. Syst. 19(4), 901–952 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Contreras, G., Paternain, G.: Connecting orbits between static classes for generic lagrangian systems. Topology 41(4), 645–666 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Evans, L.C., Gariepy, R.F.: Measure theorem and fine properties of functions. In: Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)Google Scholar
  17. 17.
    Fathi, A.: Regularity of \(C^1\) solutions of the Hamilton–Jacobi equation. Ann. Fac. Sci. Toulouse 12(4), 479–516 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Fathi, A.: Weak KAM Theorem and Lagrangian Dynamics. Cambridge University Press, Cambridge (to appear)Google Scholar
  19. 19.
    Fathi, A., Figalli, A., Rifford, L.: On the Hausdorff dimension of the Mather quotient. Commun. Pure Appl. Math. 62(4), 445–500 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Fathi, A., Siconolfi, A.: Existence of \(C^1\) critical subsolutions of the Hamilton–Jacobi equation. Invent. math. 1155, 363–388 (2004)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Figalli, A., Rifford, L.: Closing Aubry sets I. Commun. Pure Appl. Math. (to appear)Google Scholar
  22. 22.
    Figalli, A., Rifford, L.: Closing Aubry sets II. Commun. Pure Appl. Math. (to appear)Google Scholar
  23. 23.
    Figalli, A., Rifford, L.: Aubry sets, Hamilton–Jacobi equations, and Mañé conjecture. In: Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations, vol. 599. Contemp. Math. Am. Math. Soc., Providence (2013)Google Scholar
  24. 24.
    Goroff, D.L.: Hyperbolic sets for twist maps. Ergod. Theory Dyn. Syst. 5(3), 337–339 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Herman, M.: Some open problems in dynamical systems. In: Proceedings of the International Congress of Mathematicians, vol. II, pp. 797–808 (Berlin, 1998). em Doc. Math. (Extra, vol. II) (1998)Google Scholar
  26. 26.
    Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems. In: Encyclopedia of Mathematics and its Applications, vol. 54. Cambridge University Press, Cambridge (1995)Google Scholar
  27. 27.
    Le Calvez, P.: Les ensembles d’Aubry-Mather d’un difféomorphisme conservatif de l’anneau déviant la verticale sont en général hyperboliques. C. R. Acad. Sci. Paris Sér. I Math. 306(1), 51–54 (1988)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Mañé, R.: Quasi-Anosov diffeomorphisms and hyperbolic manifolds. Trans. Am. Math. Soc. 229, 351–370 (1977)CrossRefzbMATHGoogle Scholar
  29. 29.
    Mañé, R.: Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity 9(2), 273–310 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Mañé, R.: Lagrangian flows: the dynamics of globally minimizing orbits. Bol. Soc. Bras. Math. (N.S.) 28(2), 141–153 (1997)CrossRefzbMATHGoogle Scholar
  31. 31.
    Mather, J.N.: Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207, 169–207 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Mather, J.N.: Variational construction of connecting orbits. Ann. Inst. Fourier 43, 1349–1386 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Nikolaev, I., Zhuzhoma, E.: Flows on \(2\)-dimensional manifolds. In: Lecture Notes in Mathematics, vol 1705. Springer, Berlin (1999)Google Scholar
  34. 34.
    Ratcliffe, J.G.: Foundations of hyperbolic manifolds. In: Graduate Texts in Mathematics, vol. 149. Springer, New York (1994)Google Scholar
  35. 35.
    Rifford, L.: On viscosity solutions of certain Hamilton–Jacobi equations: regularity results and generalized Sard’s theorems. Commun. Partial Differ. Equ. 33(3), 517–559 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Rifford, L.: Regularity of weak KAM solutions and Mañé’s Conjecture. Séminaire Laurent Schwartz—EDP et applications. pp. 1–22 (2011–2012)Google Scholar
  37. 37.
    Rifford, L., Ruggiero, R.: Generic properties of closed orbits of Hamiltonian flows from Mañé’s viewpoint. Int. Math. Res. Not. 22, 5246–5265 (2012)MathSciNetGoogle Scholar
  38. 38.
    Stein, E.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. In: Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton (1993)Google Scholar
  39. 39.
    Villani, C. Optimal transport, old and new. In: Grundlehren des mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338. Springer, Berlin (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.CIMATGuanajuato GTOMexico
  2. 2.Department of MathematicsThe University of Texas at AustinAustinUSA
  3. 3.Université Nice Sophia Antipolis, Laboratoire J.A. Dieudonné, UMR 7351Nice Cedex 2France

Personalised recommendations