Displaying similar documents to “Two kinds of chaos and relations between them.”

Distributional chaos on tree maps: the star case

Jose S. Cánovas (2001)

Commentationes Mathematicae Universitatis Carolinae

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Let 𝕏 = { z ∈ ℂ : z n ∈ [ 0 , 1 ] } , n ∈ ℕ , and let f : 𝕏 → 𝕏 be a continuous map having the branching point fixed. We prove that f is distributionally chaotic iff the topological entropy of f is positive.

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

Emma D’Aniello, Timothy Steele (2014)

Open Mathematics

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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...