Three results on the regularity of the curves that are invariant by an exact symplectic twist map
Arnaud, M.-C.
Publications Mathématiques de l'IHÉS, Tome 110 (2009), p. 1-17 / Harvested from Numdam
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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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     author = {Arnaud, M.-C.},
     title = {Three results on the regularity of the curves that are invariant by an exact symplectic twist map},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     volume = {110},
     year = {2009},
     pages = {1-17},
     doi = {10.1007/s10240-009-0017-8},
     mrnumber = {2511585},
     zbl = {1177.53070},
     language = {en},
     url = {http://dml.mathdoc.fr/item/PMIHES_2009__109__1_0}
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Arnaud, M.-C. Three results on the regularity of the curves that are invariant by an exact symplectic twist map. Publications Mathématiques de l'IHÉS, Tome 110 (2009) pp. 1-17. doi : 10.1007/s10240-009-0017-8. http://gdmltest.u-ga.fr/item/PMIHES_2009__109__1_0/

1. M. L. Bialy, R. S. Mackay, Symplectic twist maps without conjugate points, Isr. J. Math. 141 (2004), p. 235-247 | MR 2063035 | Zbl 1055.37060

2. G. D. Birkhoff, Surface transformations and their dynamical application, Acta Math. 43 (1920), p. 1-119 | JFM 47.0985.03 | MR 1555175

3. A. Chenciner, La dynamique au voisinage d'un point fixe elliptique conservatif: de Poincaré et Birkhoff à Aubry et Mather., Astérisque 1983/84 (1985), p. 147-170 | Numdam | MR 768958 | Zbl 0582.58013

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

5. G. Contreras, R. Iturriaga, Convex Hamiltonians without conjugate points, Ergod. Theory Dyn. Syst. 19 (1999), p. 901-952 | MR 1709426 | Zbl 1044.37046

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 (1992), p. 117-146 | Numdam | MR 1184886 | Zbl 0806.58029

8. L. W. Green, A theorem of E. Hopf, Mich. Math. J. 5 (1958), p. 31-34 | MR 97833 | Zbl 0134.39601

9. M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Étud. Sci. Publ. Math. 49 (1979), p. 5-233 | Numdam | Zbl 0448.58019

10. M. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau, Asterisque 1 (1983), p. 103-104 | MR 499079 | Zbl 0532.58011

11. R. Iturriaga, A geometric proof of the existence of the Green bundles, Proc. Amer. Math. Soc. 130 (2002), p. 2311-2312 | MR 1896413 | Zbl 1067.37037

12. P. Le Calvez, Etude topologique des applications déviant la verticale, Ens. Mat., Soc. Bras. Mat., 2 (1990). | Zbl 1202.37066