ω-Limit sets for triangular mappings

Fundamenta Mathematicae (2001)

• Volume: 167, Issue: 1, page 1-15
• ISSN: 0016-2736

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Abstract

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In 1992 Agronsky and Ceder proved that any finite collection of non-degenerate Peano continua in the unit square is an ω-limit set for a continuous map. We improve this result by showing that it is valid, with natural restrictions, for the triangular maps (x,y) ↦ (f(x),g(x,y)) of the square. For example, we show that a non-trivial Peano continuum C ⊂ I² is an orbit-enclosing ω-limit set of a triangular map if and only if it has a projection property. If C is a finite union of Peano continua then, in addition, a coherence property is needed. We also provide examples of two slightly different non-Peano continua C and D in the square such that C is and D is not an ω-limit set of a triangular map. In view of these examples a characterization of the continua which are ω-limit sets for triangular mappings seems to be difficult.

How to cite

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Victor Jiménez López, and Jaroslav Smítal. "ω-Limit sets for triangular mappings." Fundamenta Mathematicae 167.1 (2001): 1-15. <http://eudml.org/doc/282539>.

@article{VictorJiménezLópez2001,
abstract = {In 1992 Agronsky and Ceder proved that any finite collection of non-degenerate Peano continua in the unit square is an ω-limit set for a continuous map. We improve this result by showing that it is valid, with natural restrictions, for the triangular maps (x,y) ↦ (f(x),g(x,y)) of the square. For example, we show that a non-trivial Peano continuum C ⊂ I² is an orbit-enclosing ω-limit set of a triangular map if and only if it has a projection property. If C is a finite union of Peano continua then, in addition, a coherence property is needed. We also provide examples of two slightly different non-Peano continua C and D in the square such that C is and D is not an ω-limit set of a triangular map. In view of these examples a characterization of the continua which are ω-limit sets for triangular mappings seems to be difficult.},
author = {Victor Jiménez López, Jaroslav Smítal},
journal = {Fundamenta Mathematicae},
keywords = {triangular map; Peano continuum; limit set},
language = {eng},
number = {1},
pages = {1-15},
title = {ω-Limit sets for triangular mappings},
url = {http://eudml.org/doc/282539},
volume = {167},
year = {2001},
}

TY - JOUR
AU - Victor Jiménez López
AU - Jaroslav Smítal
TI - ω-Limit sets for triangular mappings
JO - Fundamenta Mathematicae
PY - 2001
VL - 167
IS - 1
SP - 1
EP - 15
AB - In 1992 Agronsky and Ceder proved that any finite collection of non-degenerate Peano continua in the unit square is an ω-limit set for a continuous map. We improve this result by showing that it is valid, with natural restrictions, for the triangular maps (x,y) ↦ (f(x),g(x,y)) of the square. For example, we show that a non-trivial Peano continuum C ⊂ I² is an orbit-enclosing ω-limit set of a triangular map if and only if it has a projection property. If C is a finite union of Peano continua then, in addition, a coherence property is needed. We also provide examples of two slightly different non-Peano continua C and D in the square such that C is and D is not an ω-limit set of a triangular map. In view of these examples a characterization of the continua which are ω-limit sets for triangular mappings seems to be difficult.
LA - eng
KW - triangular map; Peano continuum; limit set
UR - http://eudml.org/doc/282539
ER -

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