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1998 | 155 | 2 | 161-176
Tytuł artykułu

The fixed-point property for deformations of tree-like continua

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let f be a map of a tree-like continuum M that sends each arc-component of M into itself. We prove that f has a fixed point. Hence every tree-like continuum has the fixed-point property for deformations (maps that are homotopic to the identity). This result answers a question of Bellamy. Our proof resembles an old argument of Brouwer involving uncountably many tangent curves. The curves used by Brouwer were originally defined by Peano. In place of these curves, we use rays that were originally defined by Borsuk.
Rocznik
Tom
155
Numer
2
Strony
161-176
Opis fizyczny
Daty
wydano
1998
otrzymano
1996-09-18
Twórcy
  • Department of Mathematics, California State University, Sacramento, California 95819, U.S.A., hagopian@csus.edu
Bibliografia
  • [A] M. A. Armstrong, Basic Topology, McGraw-Hill, London, 1979.
  • [B] D. P. Bellamy, A tree-like continuum without the fixed point property, Houston J. Math. 6 (1979), 1-13.
  • [Be] R. Bennett, Locally connected 2-cell and 2-sphere-like continua, Proc. Amer. Math. Soc. 17 (1966), 674-681.
  • [Bi1] R. H. Bing, Snake-like continua, Duke Math. J. 18 (1951), 653-663.
  • [Bi2] R. H. Bing, The elusive fixed point property, Amer. Math. Monthly 76 (1969), 119-132.
  • [Bo1] K. Borsuk, Sur un continu acyclique qui se laisse transformer topologiquement en lui même sans points invariants, Fund. Math. 24 (1935), 51-58.
  • [Bo2] K. Borsuk, A theorem on fixed points, Bull. Acad. Polon. Sci. 2 (1954), 17-20.
  • [Br] L. E. J. Brouwer, On continuous vector distributions on surfaces, Proc. Konink. Akad. Wetensch. (Amsterdam) 11 (1909), 850-858.
  • [C] R. W. Conn, The engineering of magnetic fusion reactors, Scientific American 249 (4) (October, 1983), 60-71.
  • [Co] H. Cook, Tree-likeness of dendroids and λ-dendroids, Fund. Math. 68 (1970), 19-22.
  • [H1] C. L. Hagopian, Fixed-point problems for disk-like continua, Amer. Math. Monthly 83 (1976), 471-473.
  • [H2] C. L. Hagopian, Uniquely arcwise connected plane continua have the fixed-point property, Trans. Amer. Math. Soc. 248 (1979), 85-104.
  • [H3] C. L. Hagopian, The fixed-point property for deformations of uniquely arcwise connected continua, Topology Appl. 24 (1986), 207-212.
  • [H4] C. L. Hagopian, Fixed points of arc-component-preserving maps, Trans. Amer. Math. Soc. 306 (1988), 411-420.
  • [H5] C. L. Hagopian, Fixed points of tree-like continua, in: Contemp. Math. 72, Amer. Math. Soc., 1988, 131-137.
  • [H6] C. L. Hagopian, A fixed-point theorem for tree-like continua, Topology Proc. 16 (1991), 57-62.
  • [H7] C. L. Hagopian, Fixed-point problems in continuum theory, in: Contemp. Math. 117, Amer. Math. Soc., 1991, 79-86.
  • [H8] C. L. Hagopian, The fixed-point property for simply-connected plane continua, Trans. Amer. Math. Soc. 348 (1996), 4525-4548.
  • [L] S. Lefschetz, Continuous transformations of manifolds, Proc. Nat. Acad. Sci. U.S.A. 9 (1923), 90-93.
  • [Le] I. W. Lewis, Continuum theory problems, Topology Proc. 8 (1983), 361-394.
  • [M] H. F. Mathis, A short proof that an isotropic antenna is impossible, Proc. Institute Radio Engineers 39 (1951), 970.
  • [Mi1] P. Minc, A tree-like continuum admitting fixed point free maps with arbitrarily small trajectories, Topology Appl. 46 (1992), 99-106.
  • [Mi2] P. Minc, A periodic point free homeomorphism of a tree-like continuum, Trans. Amer. Math. Soc. 348 (1996), 1487-1519.
  • [OR] L. G. Oversteegen and J. T. Rogers, Jr., Fixed-point-free maps on tree-like continua, Topology Appl. 13 (1982), 85-95.
  • [W] G. T. Whyburn, Analytic Topology, rev. ed., Amer. Math. Soc. Colloq. Publ. 28, Amer. Math. Soc., Providence, R.I., 1963.
  • [Y] G. S. Young, Fixed-point theorems for arcwise connected continua, Proc. Amer. Math. Soc. 11 (1960), 880-884.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-fmv155i2p161bwm
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