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Some remarks on the Poincaré-Birkhoff theorem. (English) Zbl 1194.37069
This is a new development on the Poincaré-Birkhoff fixed point theorem. The authors introduce the notion of a ‘positive path’, defined with respect to a homeomorphism of a topological space. This is an oriented continuous curve with the property that the homeomorphism does not map any point of the curve to a preceding point. Using this construct, the authors provide an alternative proof of the Poincaré-Birkhoff theorem (rather, of a generalisation of it due to P. H. Carter). They also give a succinct but well-informed account of the history of the theorem, and connect their work to that of J. Franks.

MSC:
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37E40 Dynamical aspects of twist maps
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
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[1] F. Béguin, S. Crovisier, and F. Le Roux, Pseudo-rotations of the open annulus, Bull. Braz. Math. Soc. (N.S.) 37 (2006), no. 2, 275 – 306. · Zbl 1105.37029 · doi:10.1007/s00574-006-0013-2 · doi.org
[2] G. D. Birkhoff : Proof of Poincaré’s last geometric theorem, Trans. Amer. Math. Soc., 14 (1913), 14-22.
[3] G. D. Birkhoff : An extension of Poincaré’s last geometric theorem, Acta. Math., 47 (1925), 297-311.
[4] L. E. J. Brouwer, Beweis des ebenen Translationssatzes, Math. Ann. 72 (1912), no. 1, 37 – 54 (German). · JFM 43.0569.02 · doi:10.1007/BF01456888 · doi.org
[5] M. Brown and W. D. Neumann, Proof of the Poincaré-Birkhoff fixed point theorem, Michigan Math. J. 24 (1977), no. 1, 21 – 31. · Zbl 0402.55001
[6] Patricia H. Carter, An improvement of the Poincaré-Birkhoff fixed point theorem, Trans. Amer. Math. Soc. 269 (1982), no. 1, 285 – 299. · Zbl 0507.55002
[7] F. Dalbono and C. Rebelo, Poincaré-Birkhoff fixed point theorem and periodic solutions of asymptotically linear planar Hamiltonian systems, Rend. Sem. Mat. Univ. Politec. Torino 60 (2002), no. 4, 233 – 263 (2003). Turin Fortnight Lectures on Nonlinear Analysis (2001). · Zbl 1098.54516
[8] Wei Yue Ding, A generalization of the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc. 88 (1983), no. 2, 341 – 346. · Zbl 0522.55005
[9] John Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. of Math. (2) 128 (1988), no. 1, 139 – 151. · Zbl 0676.58037 · doi:10.2307/1971464 · doi.org
[10] John Franks, A variation on the Poincaré-Birkhoff theorem, Hamiltonian dynamical systems (Boulder, CO, 1987) Contemp. Math., vol. 81, Amer. Math. Soc., Providence, RI, 1988, pp. 111 – 117. · doi:10.1090/conm/081/986260 · doi.org
[11] Lucien Guillou, Théorème de translation plane de Brouwer et généralisations du théorème de Poincaré-Birkhoff, Topology 33 (1994), no. 2, 331 – 351 (French). · Zbl 0924.55001 · doi:10.1016/0040-9383(94)90016-7 · doi.org
[12] Lucien Guillou, A simple proof of P. Carter’s theorem, Proc. Amer. Math. Soc. 125 (1997), no. 5, 1555 – 1559. · Zbl 0863.54033
[13] Howard Jacobowitz, Periodic solutions of \?\(^{\prime}\)\(^{\prime}\)+\?(\?,\?)=0 via the Poincaré-Birkhoff theorem, J. Differential Equations 20 (1976), no. 1, 37 – 52. · Zbl 0285.34028 · doi:10.1016/0022-0396(76)90094-2 · doi.org
[14] Howard Jacobowitz, Corrigendum: The existence of the second fixed point: a correction to ”Periodic solutions of \?”+\?(\?,\?)=0 via the Poincaré-Birkhoff theorem” (J. Differential Equations 20 (1976), no. 1, 37 – 52), J. Differential Equations 25 (1977), no. 1, 148 – 149. · Zbl 0354.34043 · doi:10.1016/0022-0396(77)90187-5 · doi.org
[15] B. de Kerékjártó : The plane translation theorem of Brouwer and the last geometric theorem of Poincaré, Acta Sci. Math. Szeged, 4 (1928-29), 86-102. · JFM 54.0612.02
[16] Frédéric Le Roux, Homéomorphismes de surfaces: théorèmes de la fleur de Leau-Fatou et de la variété stable, Astérisque 292 (2004), vi+210 (French, with English and French summaries). · Zbl 1073.37046
[17] Rogério Martins and Antonio J. Ureña, The star-shaped condition on Ding’s version of the Poincaré-Birkhoff theorem, Bull. Lond. Math. Soc. 39 (2007), no. 5, 803 – 810. · Zbl 1132.54026 · doi:10.1112/blms/bdm064 · doi.org
[18] H. Poincaré : Sur un théorème de géométrie, Rend. Circ. Mat. Palermo, 33 (1912), 375-407.
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