Composants of the horseshoe

Christoph Bandt

Fundamenta Mathematicae (1994)

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

Abstract

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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

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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

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