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

Publications Mathématiques de l'IHÉS (2009)

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

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topArnaud, 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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