Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



An improvement of the Poincaré-Birkhoff fixed point theorem

Author: Patricia H. Carter
Journal: Trans. Amer. Math. Soc. 269 (1982), 285-299
MSC: Primary 54H25; Secondary 55M25, 58F99
MathSciNet review: 637039
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If $ g$ is a twist homeomorphism of an annulus $ A$ in the plane which leaves at most one point in the interior of $ A$ fixed, then there is an essential simple closed curve in the interior of $ A$ which meets its image in at most one point; hence the annular region bounded by this simple closed curve and the inside component of the boundary of $ A$ is mapped onto either a proper subset or a proper superset of itself.

References [Enhancements On Off] (What's this?)

  • [1] Harold Abelson and Charles Stanton, Poincaré’s geometric theorem for flows, J. Differential Geometry 11 (1976), no. 1, 129–131. MR 0415685
  • [2] Richard B. Barrar, Proof of the fixed point theorems of Poincaré and Birkhoff, Canad. J. Math. 19 (1967), 333–343. MR 0210106
  • [3] G. D. Birkhoff, Proof of Poincaré's geometric theorem, Trans. Amer. Math. Soc. 14 (1913), 14-22.
  • [4] -, An extension of Poincaré's last geometric theorem, Acta Math. 47 (1925), 297-311.
  • [5] -, Dynamical systems, Amer. Math. Soc. Colloq. Publ., vol. 9, Amer. Math. Soc., Providence, R. I., 1927.
  • [6] M. Brown and W. D. Neumann, Proof of the Poincaré-Birkhoff fixed point theorem, Michigan Math. J. 24 (1977), no. 1, 21–31. MR 0448339
  • [7] P. Carter, An improvement of the Poincaré-Birkhoff Fixed Point Theorem, Dissertation, University of Florida, 1978.
  • [8] C. B. García, A fixed point theorem including the last theorem of Poincaré, Math. Programming 9 (1975), no. 2, 227–239. MR 0418069
  • [9] Robert Hermann, Some differential-geometric aspects of the Lagrange variational problem, Illinois J. Math. 6 (1962), 634–673. MR 0145457
  • [10] H. Jacobowitz, Periodic solution of $ x'' + f(x,\,t) = 0$ via the Poincaré-Birkhoff theorem, J. Differential Equations 20 (1976), 37-52.
  • [11] Howard Jacobowitz, Corrigendum: The existence of the second fixed point: a correction to “Periodic solutions of 𝑥”+𝑓(𝑥,𝑡)=0 via the Poincaré-Birkhoff theorem” (J. Differential Equations 20 (1976), no. 1, 37–52), J. Differential Equations 25 (1977), no. 1, 148–149. MR 0437857
  • [12] B. de Kerékjarto, The plane translation theorem of Brouwer and the last geometric theorem of Poincaré, Acta Sci. Math. (Szeged) 4 (1928-1929), 86-102.
  • [13] R. L. Moore, Foundations of point set theory, Revised edition. American Mathematical Society Colloquium Publications, Vol. XIII, American Mathematical Society, Providence, R.I., 1962. MR 0150722
  • [14] Walter D. Neumann, Generalizations of the Poincaré Birkhoff fixed point theorem, Bull. Austral. Math. Soc. 17 (1977), no. 3, 375–389. MR 0584597
  • [15] M. H. A. Newman, Fixed point and coincidence theorems, J. London Math. Soc. 27 (1952), 135–140. MR 0046637
  • [16] H. Poincaré, Sur un théorème de géométrie, Rend. Circ. Mat. Palermo 33 (1912), 375-407.
  • [17] Carl P. Simon and Charles J. Titus, The fixed point index of symplectic maps, Géométrie symplectique et physique mathématique (Colloq. Internat. C.N.R.S., Aix-en-Provence, 1974) Paris, 1975, pp. 19–28. Éditions Centre Nat. Recherche Sci (English, with French summary). With a question by W. Klingenberg and an answer by Simon. MR 0461565
  • [18] T. van der Walt, Fixed and almost fixed points, Mathematical Centre Tracts, vol. 1, Mathematisch Centrum, Amsterdam, 1963. MR 0205246

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54H25, 55M25, 58F99

Retrieve articles in all journals with MSC: 54H25, 55M25, 58F99

Additional Information

Keywords: Fixed point property, homeomorphism, twist homeomorphism of the annulus
Article copyright: © Copyright 1982 American Mathematical Society