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 -
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