Transactions of the American Mathematical Society

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ISSN 1088-6850(e) ISSN 0002-9947(p)

On the dynamics of homology-preserving homeomorphisms of the annulus

Author(s): Marc Bonino
Journal: Trans. Amer. Math. Soc. 361 (2009), 1903-1923.
MSC (2000): Primary 37E30; Secondary 37C25, 37E45
Posted: November 4, 2008
MathSciNet review: 2465823
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Abstract: We consider the homeomorphisms of the compact annulus $\mathbb{A} = \mathbb{S}^1 \times [-1,1]$ isotopic to the symmetry $S_{\mathbb{A}}$ which interchanges the two boundary components. We prove that if such a homeomorphism is, in some sense, conservative and twisted, then it possesses a periodic orbit of period exactly two. This can be regarded as a counterpart of the Poincaré-Birkhoff theorem in the isotopy class of $S_{\mathbb{A}}$ .

References:

[1]: S. Addas-Zanata, Some extensions of the Poincaré-Birkhoff theorem to the cylinder and a remark on mappings of the torus homotopic to Dehn twists, Nonlinearity 18 (2005), 2243-2260. MR 2164740 (2006h:37064)
[2]: G. D. Birkhoff, An extension of Poincaré's last geometric theorem, Acta Math. 47 (1925), 297-311.
[3]: M. Bonino, A Brouwer-like theorem for orientation reversing homeomorphisms of the sphere, Fund. Math. 182 (2004), 1-40. MR 2100713 (2005m:37096)
[4]: M. Brown and W.D. Neumann, Proof of the Poincaré-Birkhoff fixed point theorem, Michigan Math. J. 24 (1977), 21-31. MR 0448339 (56:6646)
[5]: P.H. Carter, An improvement of the Poincaré-Birkhoff fixed point theorem, Trans. Amer. Math. Soc. 269 (1982), 285-299. MR 637039 (84h:54041)
[6]: D.B.A. Epstein, Pointwise periodic homeomorphisms, Proc. London Math. Soc. 42, No. 3, (1981), 415-460. MR 614729 (83e:57011)
[7]: J. Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. Math. 128 (1988), 139-151. MR 951509 (89m:54052)
[8]: J. Franks, Recurrence and fixed points of surface homeomorphisms, Ergod. Th. and Dyn. Syst. $8^\ast$ (1988), 99-107. MR 967632 (90d:58124)
[9]: J. Franks, A variation on the Poincaré-Birkhoff Theorem, Contemporary Mathematics 81 (1988), 111-116. MR 986260 (90e:58095)
[10]: J. Franks, Erratum to ``Generalizations of the Poincaré-Birkhoff theorem'', Ann. Math. 164 (2006), 1097-1098. MR 2259255 (2007g:54056)
[11]: M.J. Greenberg, Lectures on algebraic topology, W. A. Benjamin, Inc., New York-Amsterdam (1967). MR 0215295 (35:6137)
[12]: L. Guillou, Théorème de translation plane de Brouwer et généralisations du théorème de Poincaré-Birkhoff, Topology 33 (1994), 331-351. MR 1273787 (95h:55003)
[13]: L. Guillou, A simple proof of P. Carter's theorem, Proc. Amer. Math. Soc. 125 (1997), 1555-1559. MR 1372031 (97g:54055)
[14]: L. Guillou, Free lines for homeomorphisms of the open annulus, Trans. Amer. Math. Soc. 360 (2008), 2191-2204.
[15]: B. de Kerékjártó, Topology (I), Springer, Berlin (1923).
[16]: B. de Kerékjártó, The plane translation theorem of Brouwer and the last geometric of Poincaré, Acta Sci. Math. Szeged 4 (1928-1929), 86-102.
[17]: P. Le Calvez et A. Sauzet, Une démonstration dynamique du théorème de translation de Brouwer, Exposition. Math. 14 (1996), 277-287. MR 1409005 (97e:54043)
[18]: M.H.A. Newman, Elements of the topology of plane sets of points, Cambridge University Press (1964). MR 0044820 (13:483a)
[19]: H. Poincaré, Sur un théorème de géométrie, Rend. Circ. Math. Palermo 33 (1912), 375-407.
[20]: A. Sauzet, Applications des décompositions libres à l'étude des homéomorphismes de surfaces, Thèse (2001), Université Paris 13.
[21]: H.E. Winkelnkemper, A generalization of the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc. 102 (1988), 1028-1030. MR 934887 (89e:55006)

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