Three results on the regularity of the curves that are invariant by an exact symplectic twist map |
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Authors: | M-C Arnaud |
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Institution: | (1) Université d’Avignon et des Pays de Vaucluse Laboratoire d’Analyse Non Linéaire et Géométrie (EA 2151), 84018 Avignon, France |
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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). |
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Keywords: | |
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