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

100%

D’Aniello E. , Steele T.

Open Mathematics

2014

tom 12

nr 4

584-592

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.

A note on ω-limit set of a tree map

100%

Sun T. , Xi H. , Peng X.

Fasciculi Mathematici

2011

tom Nr 46

141--145

Let T be a tree and f : T →T be continuous. Denote by P(f) and ω(x, f) the set of periodic points of f and w-limit set of x under f respectively. Write ᴧ(f) = UxϵTω(x,f). In this paper, we show that if ...[wzór], then ω(x, f) is an infinite minimal set.