Chaotic behaviour of the map x ↦ ω(x, f)

Emma D’Aniello; Timothy Steele

Open Mathematics (2014)

  • Volume: 12, Issue: 4, page 584-592
  • ISSN: 2391-5455

Abstract

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Let K(2ℕ) be the class of compact subsets of the Cantor space 2ℕ, furnished with the Hausdorff metric. Let f ∈ C(2ℕ). We study the map ω f: 2ℕ → K(2ℕ) defined as ω f (x) = ω(x, f), the ω-limit set of x under f. Unlike the case of n-dimensional manifolds, n ≥ 1, we show that ω f is continuous for the generic self-map f of the Cantor space, even though the set of functions for which ω f is everywhere discontinuous on a subsystem is dense in C(2ℕ). The relationships between the continuity of ω f and some forms of chaos are investigated.

How to cite

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Emma D’Aniello, and Timothy Steele. "Chaotic behaviour of the map x ↦ ω(x, f)." Open Mathematics 12.4 (2014): 584-592. <http://eudml.org/doc/269517>.

@article{EmmaD2014,
abstract = {Let K(2ℕ) be the class of compact subsets of the Cantor space 2ℕ, furnished with the Hausdorff metric. Let f ∈ C(2ℕ). We study the map ω f: 2ℕ → K(2ℕ) defined as ω f (x) = ω(x, f), the ω-limit set of x under f. Unlike the case of n-dimensional manifolds, n ≥ 1, we show that ω f is continuous for the generic self-map f of the Cantor space, even though the set of functions for which ω f is everywhere discontinuous on a subsystem is dense in C(2ℕ). The relationships between the continuity of ω f and some forms of chaos are investigated.},
author = {Emma D’Aniello, Timothy Steele},
journal = {Open Mathematics},
keywords = {Cantor space; Continuous self-map; ω-limit set; Chaos; continuous self-map; -limit set; chaos},
language = {eng},
number = {4},
pages = {584-592},
title = {Chaotic behaviour of the map x ↦ ω(x, f)},
url = {http://eudml.org/doc/269517},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Emma D’Aniello
AU - Timothy Steele
TI - Chaotic behaviour of the map x ↦ ω(x, f)
JO - Open Mathematics
PY - 2014
VL - 12
IS - 4
SP - 584
EP - 592
AB - Let K(2ℕ) be the class of compact subsets of the Cantor space 2ℕ, furnished with the Hausdorff metric. Let f ∈ C(2ℕ). We study the map ω f: 2ℕ → K(2ℕ) defined as ω f (x) = ω(x, f), the ω-limit set of x under f. Unlike the case of n-dimensional manifolds, n ≥ 1, we show that ω f is continuous for the generic self-map f of the Cantor space, even though the set of functions for which ω f is everywhere discontinuous on a subsystem is dense in C(2ℕ). The relationships between the continuity of ω f and some forms of chaos are investigated.
LA - eng
KW - Cantor space; Continuous self-map; ω-limit set; Chaos; continuous self-map; -limit set; chaos
UR - http://eudml.org/doc/269517
ER -

References

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