Composants of the horseshoe

Christoph Bandt

Fundamenta Mathematicae (1994)

  • Volume: 144, Issue: 3, page 231-241
  • ISSN: 0016-2736

Abstract

top
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.

How to cite

top

Bandt, Christoph. "Composants of the horseshoe." Fundamenta Mathematicae 144.3 (1994): 231-241. <http://eudml.org/doc/212026>.

@article{Bandt1994,
abstract = {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.},
author = {Bandt, Christoph},
journal = {Fundamenta Mathematicae},
keywords = {horseshoe; bucket handle continuum; tent map},
language = {eng},
number = {3},
pages = {231-241},
title = {Composants of the horseshoe},
url = {http://eudml.org/doc/212026},
volume = {144},
year = {1994},
}

TY - JOUR
AU - Bandt, Christoph
TI - Composants of the horseshoe
JO - Fundamenta Mathematicae
PY - 1994
VL - 144
IS - 3
SP - 231
EP - 241
AB - 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.
LA - eng
KW - horseshoe; bucket handle continuum; tent map
UR - http://eudml.org/doc/212026
ER -

References

top
  1. [1] J. M. Aarts and R. J. Fokkink, On composants of the bucket handle, Fund. Math. 139 (1991), 193-208. Zbl0759.54017
  2. [2] C. Bandt and K. Keller, A simple approach to the topological structure of fractals, Math. Nachr. 154 (1991), 27-39. Zbl0824.28007
  3. [3] C. Bandt and T. Retta, Topological spaces admitting a unique fractal structure, Fund. Math. 141 (1992), 257-268. Zbl0832.28011
  4. [4] D. P. Bellamy, Homeomorphisms of composants, Houston J. Math. 5 (1979), 313-318. Zbl0457.54031
  5. [5] W. Dębski and E. D. Tymchatyn, Homeomorphisms of composants in Knaster continua, Topology Proc. 12 (1987), 239-256. Zbl0674.54023
  6. [6] K. Falconer, Fractal Geometry, Wiley, New York, 1990. 
  7. [7] R. J. Fokkink, The structure of trajectories, Dissertation, Delft, 1991. Zbl0768.54028
  8. [8] S. E. Holte, Generalized horseshoe maps and inverse limits, Pacific J. Math. 156 (1992), 297-306. Zbl0723.58034
  9. [9] Z. Janiszewski, Oeuvres Choisies, PWN, Warszawa, 1962. 
  10. [10] K. Kuratowski, Théorie des continues irréductibles entre deux points I, Fund. Math. 3 (1922), 200-231. Zbl48.0211.01
  11. [11] K. Kuratowski, Topology, Vol. II, PWN, Warszawa, and Academic Press, New York, 1968. 
  12. [12] M. Misiurewicz, Embedding inverse limits of interval maps as attractors, Fund. Math. 125 (1985), 23-40. Zbl0587.58032
  13. [13] S. B. Nadler, Continuum Theory, Marcel Dekker, New York, 1992. 
  14. [14] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. Zbl0202.55202
  15. [15] W. Szczechla, Inverse limits of certain interval mappings as attractors in two dimensions, Fund. Math. 133 (1989), 1-23. Zbl0708.58012
  16. [16] W. T. Watkins, Homeomorphic classification of certain inverse limit spaces with open bonding maps, Pacific J. Math. 103 (1982), 589-601. Zbl0451.54027

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.