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).
@article{PMIHES_2009__109__1_0,
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},
pages = {1--17},
publisher = {Springer-Verlag},
volume = {109},
year = {2009},
doi = {10.1007/s10240-009-0017-8},
mrnumber = {2511585},
zbl = {1177.53070},
language = {en},
url = {http://archive.numdam.org/articles/10.1007/s10240-009-0017-8/}
}
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 SP - 1 EP - 17 VL - 109 PB - Springer-Verlag UR - http://archive.numdam.org/articles/10.1007/s10240-009-0017-8/ DO - 10.1007/s10240-009-0017-8 LA - en ID - PMIHES_2009__109__1_0 ER -
%0 Journal Article %A Arnaud, M.-C. %T Three results on the regularity of the curves that are invariant by an exact symplectic twist map %J Publications Mathématiques de l'IHÉS %D 2009 %P 1-17 %V 109 %I Springer-Verlag %U http://archive.numdam.org/articles/10.1007/s10240-009-0017-8/ %R 10.1007/s10240-009-0017-8 %G en %F PMIHES_2009__109__1_0
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 109 (2009), pp. 1-17. doi : 10.1007/s10240-009-0017-8. http://archive.numdam.org/articles/10.1007/s10240-009-0017-8/
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