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1994 | 144 | 3 | 231-241
Tytuł artykułu

Composants of the horseshoe

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The horseshoe or bucket handle continuum, defined as the inverse limit of the tent map, is one of the standard examples in continua theory as well as in dynamical systems. It is not arcwise connected. Its arcwise components coincide with composants, and with unstable manifolds in the dynamical setting. Knaster asked whether these composants are all homeomorphic, with the obvious exception of the zero composant. Partial results were obtained by Bellamy (1979), Dębski and Tymchatyn (1987), and Aarts and Fokkink (1991). We answer Knaster's question in the affirmative. The main tool is a very simple type of symbolic dynamics for the horseshoe and related continua.
Słowa kluczowe
Rocznik
Tom
144
Numer
3
Strony
231-241
Opis fizyczny
Daty
wydano
1994
otrzymano
1993-02-02
poprawiono
1993-08-25
poprawiono
1993-11-30
Twórcy
  • FR Mathematik/Informatik, Arndt-Universität, 17487 Greifswald, Germany
Bibliografia
  • [1] J. M. Aarts and R. J. Fokkink, On composants of the bucket handle, Fund. Math. 139 (1991), 193-208.
  • [2] C. Bandt and K. Keller, A simple approach to the topological structure of fractals, Math. Nachr. 154 (1991), 27-39.
  • [3] C. Bandt and T. Retta, Topological spaces admitting a unique fractal structure, Fund. Math. 141 (1992), 257-268.
  • [4] D. P. Bellamy, Homeomorphisms of composants, Houston J. Math. 5 (1979), 313-318.
  • [5] W. Dębski and E. D. Tymchatyn, Homeomorphisms of composants in Knaster continua, Topology Proc. 12 (1987), 239-256.
  • [6] K. Falconer, Fractal Geometry, Wiley, New York, 1990.
  • [7] R. J. Fokkink, The structure of trajectories, Dissertation, Delft, 1991.
  • [8] S. E. Holte, Generalized horseshoe maps and inverse limits, Pacific J. Math. 156 (1992), 297-306.
  • [9] Z. Janiszewski, Oeuvres Choisies, PWN, Warszawa, 1962.
  • [10] K. Kuratowski, Théorie des continues irréductibles entre deux points I, Fund. Math. 3 (1922), 200-231.
  • [11] K. Kuratowski, Topology, Vol. II, PWN, Warszawa, and Academic Press, New York, 1968.
  • [12] M. Misiurewicz, Embedding inverse limits of interval maps as attractors, Fund. Math. 125 (1985), 23-40.
  • [13] S. B. Nadler, Continuum Theory, Marcel Dekker, New York, 1992.
  • [14] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817.
  • [15] W. Szczechla, Inverse limits of certain interval mappings as attractors in two dimensions, Fund. Math. 133 (1989), 1-23.
  • [16] W. T. Watkins, Homeomorphic classification of certain inverse limit spaces with open bonding maps, Pacific J. Math. 103 (1982), 589-601.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-fmv144i3p231bwm
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