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

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

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