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On Birkhoff Theorems for Lagrangian invariant tori with closed orbits

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Abstract

We show that a C 1 torus that is homologous to the zero section, invariant by the geodesic flow of a symmetric Finsler metric in T 2, and possesses closed orbits is a graph of the canonical projection. This result, together with the result obtained by Bialy in 1989 for continuous invariant tori without closed orbits of symmetric Finsler metrics in T 2, shows that the second Birkhoff Theorem holds for C 1 Lagrangian invariant tori of symmetric Finsler metrics in the two torus. We also study the first Birkhoff Theorem for continuous invariant tori of Finsler metrics in T 2 and give some sufficient conditions for a continuous minimizing torus with closed orbits to be a graph of the canonical projection.

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Authors and Affiliations

  1. Dep. de Matemática - ICEx, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brasil

    M. J. Dias Carneiro

  2. PUC-Rio Dep. de Matemática, Pontificia Universidade Católica do Rio, de Janeiro, Rua Marqués de São Vicente 225, Gávea, Rio de Janeiro, Brasil

    Rafael O. Ruggiero

Authors
  1. M. J. Dias Carneiro
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  2. Rafael O. Ruggiero
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Correspondence to Rafael O. Ruggiero.

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Partially supported by CNPq, FAPERJ, TWAS

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Carneiro, M., Ruggiero, R. On Birkhoff Theorems for Lagrangian invariant tori with closed orbits. manuscripta math. 119, 411–432 (2006). https://doi.org/10.1007/s00229-005-0619-5

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  • DOI: https://doi.org/10.1007/s00229-005-0619-5

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