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.
@article{bwmeta1.element.doi-10_2478_s11533-013-0360-3,
author = {Emma D'Aniello and Timothy Steele},
title = {Chaotic behaviour of the map x - o(x, f)},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {584-592},
zbl = {1305.37011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0360-3}
}
Emma D’Aniello; Timothy Steele. Chaotic behaviour of the map x ↦ ω(x, f). Open Mathematics, Tome 12 (2014) pp. 584-592. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0360-3/
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