Teorema de Poincaré–Birkhoff

Na topologia simplética e sistemas dinâmicos, o teorema de Poincaré–Birkhoff (também conhecido como teorema do ponto fixo de Poincaré–Birkhoff e último teorema geométrico de Poincaré) afirma que todo homeomorfismo que preserva área e orientação definido em uma coroa circular (anel) que faz girar as duas fronteiras em direções opostas tem pelo menos dois pontos fixos.[1]

História

O teorema de Poincaré–Birkhoff foi descoberto por Henri Poincaré, que o publicou em um artigo de 1912 intitulado "Sur un théorème de géométrie" ("Sobre um teorema de geometria"), e demonstrou-o para alguns casos especiais. O caso geral foi demonstrado por George D. Birkhoff em seu artigo de 1913 intitulado "Proof of Poincaré's geometric theorem" ("Demonstração do teorema geométrico de Poincaré").[2]

Uma demonstração construtiva, por meio do método homotópico foi apresentada por Li Yong e Lin Zhenghua em 1995.[3]

Referências

  1. ↑ Brown, M.; Neumann, W. D.; Proof of the Poincaré-Birkhoff fixed point theorem. Michigan Math. J. 24 (1977), no. 1, 21--31. doi:10.1307/mmj/1029001816.
  2. ↑ Poincaré last theorem. Encyclopedia of Mathematics. - www.encyclopediaofmath.org
  3. ↑ Li Yong and Lin Zhenghua; Transactions of the American Mathematical Society, Vol. 347, No. 6 (Jun., 1995), pp. 2111–2126

Leitura adicional

  • M. Brown; W. D. Neumann. "Proof of the Poincaré-Birkhoff fixed-point theorem". Michigan Math. J. volume 24, 1977, p. 21–31.
  • P. Le Calvez; J. Wang. "Some remarks on the Poincaré–Birkhoff theorem". Proc. Amer. Math. Soc. volume 138, No.2, 2010, p. 703–715.
  • J. Franks. "Generalizations of the Poincaré-Birkhoff Theorem", Annals of Mathematics Second Series, Vol. 128, No. 1 (Jul., 1988), pp. 139–151.

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