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

M.-C. Arnaud

Publications Mathématiques de l'IHÉS (2009)

  • Volume: 109, page 1-17
  • ISSN: 0073-8301

Abstract

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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).

How to cite

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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 109 (2009): 1-17. <http://eudml.org/doc/274347>.

@article{Arnaud2009,
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).},
author = {Arnaud, M.-C.},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {symplectic twist map; Lipschitz map; Green bundle},
language = {eng},
pages = {1-17},
publisher = {Springer-Verlag},
title = {Three results on the regularity of the curves that are invariant by an exact symplectic twist map},
url = {http://eudml.org/doc/274347},
volume = {109},
year = {2009},
}

TY - JOUR
AU - Arnaud, M.-C.
TI - Three results on the regularity of the curves that are invariant by an exact symplectic twist map
JO - Publications Mathématiques de l'IHÉS
PY - 2009
PB - Springer-Verlag
VL - 109
SP - 1
EP - 17
AB - 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).
LA - eng
KW - symplectic twist map; Lipschitz map; Green bundle
UR - http://eudml.org/doc/274347
ER -

References

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  1. 1. M. L. Bialy, R. S. MacKay, Symplectic twist maps without conjugate points, Isr. J. Math.141 (2004), p. 235-247 Zbl1055.37060MR2063035
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  4. 4. A. Chenciner, Systèmes dynamiques différentiables. Article à l’Encyclopedia Universalis. 
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  6. 6. A. Fathi, Une interprétation plus topologique de la démonstration du théorème de Birkhoff, appendice au ch.1 de [9], pp. 39–46. 
  7. 7. P. Foulon, Estimation de l’entropie des systèmes lagrangiens sans points conjugués, Ann. Inst. Henri Poincaré Phys. Théor.57 (1992), p. 117-146 Zbl0806.58029MR1184886
  8. 8. L. W. Green, A theorem of E. Hopf, Mich. Math. J.5 (1958), p. 31-34 Zbl0134.39601MR97833
  9. 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 Zbl0448.58019
  10. 10. M. Herman, Sur les courbes invariantes par les difféomorphismes de l’anneau, Asterisque1 (1983), p. 103-104 Zbl0532.58011MR499079
  11. 11. R. Iturriaga, A geometric proof of the existence of the Green bundles, Proc. Amer. Math. Soc.130 (2002), p. 2311-2312 Zbl1067.37037MR1896413
  12. 12. P. Le Calvez, Etude topologique des applications déviant la verticale, Ens. Mat., Soc. Bras. Mat., 2 (1990). Zbl1202.37066

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