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Three results on the regularity of the curves that are invariant by an exact symplectic twist map

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Abstract

A theorem due to G. D. Birkhoff states that every essential curve which is invariant under a symplectic twist map of the annulus is the graph of a Lipschitz map. We prove: if the graph of a Lipschitz map h:T→R is invariant under a symplectic twist map, then h is a little bit more regular than simply Lipschitz (Theorem 1); we deduce that there exists a Lipschitz map h:T→R whose graph is invariant under no symplectic twist map (Corollary 2).

Assuming that the dynamic of a twist map restricted to a Lipschitz graph is bi-Lipschitz conjugate to a rotation, we obtain that the graph is even C 1 (Theorem 3).

Then we consider the case of the C 0 integrable symplectic twist maps and we prove that for such a map, there exists a dense G δ subset of the set of its invariant curves such that every curve of this G δ subset is C 1 (Theorem 4).

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References

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Chenciner, La dynamique au voisinage d’un point fixe elliptique conservatif: de Poincaré et Birkhoff à Aubry et Mather., Astérisque, 1983/84(121–122) (1985), 147–170 (in French). [The dynamics at the neighborhood of a conservative elliptic fixed point: from Poincare and Birkhoff to Aubry and Mather] Seminar Bourbaki.

    MathSciNet  Google Scholar 

  4. A. Chenciner, Systèmes dynamiques différentiables. Article à l’Encyclopedia Universalis.

  5. G. Contreras and R. Iturriaga, Convex Hamiltonians without conjugate points, Ergod. Theory Dyn. Syst., 19(4) (1999), 901–952.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Fathi, Une interprétation plus topologique de la démonstration du théorème de Birkhoff, appendice au ch.1 de [9], pp. 39–46.

  7. P. Foulon, Estimation de l’entropie des systèmes lagrangiens sans points conjugués, Ann. Inst. Henri Poincaré Phys. Théor., 57(2) (1992), 117–146.

    MATH  MathSciNet  Google Scholar 

  8. L. W. Green, A theorem of E. Hopf, Mich. Math. J., 5 (1958), 31–34.

    Article  MATH  Google Scholar 

  9. M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Étud. Sci. Publ. Math., 49 (1979), 5–233.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Herman, Sur les courbes invariantes par les difféomorphismes de l’anneau, Asterisque, 1 (1983), 103–104.

    MathSciNet  Google Scholar 

  11. R. Iturriaga, A geometric proof of the existence of the Green bundles, Proc. Amer. Math. Soc., 130(8) (2002), 2311–2312.

    Article  MATH  MathSciNet  Google Scholar 

  12. P. Le Calvez, Etude topologique des applications déviant la verticale, Ens. Mat., Soc. Bras. Mat., 2 (1990).

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  1. Université d’Avignon et des Pays de Vaucluse Laboratoire d’Analyse Non Linéaire et Géométrie (EA 2151), 84018, Avignon, France

    M.-C. Arnaud

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  1. M.-C. Arnaud
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Arnaud, MC. Three results on the regularity of the curves that are invariant by an exact symplectic twist map. Publ.math.IHES 109, 1–17 (2009). https://doi.org/10.1007/s10240-009-0017-8

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  • DOI: https://doi.org/10.1007/s10240-009-0017-8

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