Three results on the regularity of the curves that are invariant by an exact symplectic twist map
Publications Mathématiques de l'IHÉS, Volume 109  (2009), p. 1-17

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},
     publisher = {Springer-Verlag},
     volume = {109},
     year = {2009},
     pages = {1-17},
     doi = {10.1007/s10240-009-0017-8},
     zbl = {1177.53070},
     mrnumber = {2511585},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 109 (2009) , pp. 1-17. doi : 10.1007/s10240-009-0017-8. http://www.numdam.org/item/PMIHES_2009__109__1_0/

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