Skip to main content
SpringerLink
Log in

Lyapunov Exponents of Minimizing Measures for Globally Positive Diffeomorphisms in All Dimensions

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

The globally positive diffeomorphisms of the 2n-dimensional annulus are important because they represent what happens close to a completely elliptic periodic point of a symplectic diffeomorphism where the torsion is positive definite. For these globally positive diffeomorphisms, an Aubry–Mather theory was developed by Garibaldi and Thieullen that provides the existence of some minimizing measures. Using the two Green bundles \({G_-}\) and \({G_+}\) that can be defined along the support of these minimizing measures, we will prove that there is a deep link between:

  • the angle between \({G_-}\) and \({G_+}\) along the support of the considered measure \({\mu}\);

  • the size of the smallest positive Lyapunov exponent of \({\mu}\);

  • the tangent cone to the support of \({\mu}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Arnaud, M.-C.: Type des points fixes des difféomorphismes symplectiques de \({\mathbb{T}^n \times \mathbb{R}^n}\). (French) [The type of fixed points of the symplectic diffeomorphisms of \({\mathbb{T}^n \times \mathbb{R}^n}\)] Mém. Soc. Math. France (N.S.) 48, 63 (1992)

  2. 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)

    Article  ADS  MathSciNet  Google Scholar 

  3. Arnaud M.-C.: The link between the shape of the Aubry–Mather sets and their Lyapunov exponents. Ann. Math. 174(3), 1571–1601 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Arnaud M.-C.: Lower and upper bounds for the Lyapunov exponents of twisting dynamics: a relationship between the exponents and the angle of Oseledets’ splitting. Ergod. Theory Dyn. Syst. 33(3), 693–712 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arnaud, M.-C.: Lyapunov exponents for conservative twisting dynamics: a survey. In: Proceedings of Ergodic theory workshop of the university of de Chapel Hill, Walter de Gruyter (to appear). http://arxiv.org/abs/1410.1834

  7. Aubry S., Le Daeron P.Y.: The discrete Frenkel–Kontorova model and its extensions. I. Exact results for the ground-states. Physica D 8(3), 381–422 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Bernard P.: The dynamics of pseudographs in convex Hamiltonian systems. J. Am. Math. Soc. 21(3), 615–669 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bialy M.L., MacKay R.S.: Symplectic twist maps without conjugate points. Isr. J. Math. 141, 235–247 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Birkhoff G.D.: Surface transformations and their dynamical application. Acta Math. 43, 1–119 (1920)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bochi, J., Viana, M.: Lyapunov exponents: how frequently are dynamical systems hyperbolic? In: Modern Dynamical Systems and Applications, pp. 271–297. Cambridge University Press, Cambridge (2004)

  12. Bouligand, G.: Introduction à la géométrie infinitésimale directe. Librairie Vuiberts, Paris (1932)

  13. 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)

  14. Fathi, A.: Weak KAM Theorems in Lagrangian Dynamics (book in preparation)

  15. Garibaldi E., Thieullen P.: Minimizing orbits in the discrete Aubry–Mather model. Nonlinearity 24(2), 563–611 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Golé, C.: Symplectic twist maps. Global variational techniques. In: Advanced Series in Nonlinear Dynamics, vol. 18. World Scientific Publishing Co., Inc., River Edge (2001)

  17. Gomes D.A.: Viscosity solution methods and the discrete Aubry–Mather problem. Discrete Contin. Dyn. Syst. 13(1), 103–116 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Le Calvez, P.: Les ensembles d’Aubry–Mather d’un difféomorphisme conservatif de l’anneau déviant la verticale sont en général hyperboliques. (French) [The Aubry–Mather sets of a conservative diffeomorphism of the annulus twisting the vertical are hyperbolic in general] C. R. Acad. Sci. Paris Sér. I Math. 306(1), 51–54 (1988)

  19. Mané R.: Lagrangian flows: the dynamics of globally minimizing orbits. Int. Pitman Res. Notes Math. Ser. 362, 120–131 (1996)

    MathSciNet  MATH  Google Scholar 

  20. Mather J.N.: Existence of quasiperiodic orbits for twist homeomorphisms of the annulus. Topology 21(4), 457–467 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  21. Moser, J.: Proof of a generalized form of a fixed point theorem due to G. D. Birkhoff. Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), pp. 464–494. Lecture Notes in Mathematics, vol. 597. Springer, Berlin (1977)

  22. Moser J.: Monotone twist mappings and the calculus of variations. Ergod. Theory Dyn. Syst. 6(3), 401–413 (1986)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques d’Avignon (EA 2151), Avignon Université, 84 018, Avignon, France

    M.-C. Arnaud

  2. Institut universitaire de France, Paris, France

    M.-C. Arnaud

  3. ANR-12-BLAN-WKBHJ, Lyon, France

    M.-C. Arnaud

Authors
  1. M.-C. Arnaud
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M.-C. Arnaud.

Additional information

Communicated by K. Khanin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arnaud, MC. Lyapunov Exponents of Minimizing Measures for Globally Positive Diffeomorphisms in All Dimensions. Commun. Math. Phys. 343, 783–810 (2016). https://doi.org/10.1007/s00220-016-2599-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-016-2599-6

Keywords

Search

Navigation