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Chaotic behaviour of the map \(x\mapsto\omega(x,f)\). (English) Zbl 1305.37011

Let \(2^{\mathbb{N}}\) be the Cantor space, \(\mathcal{C}(2^{\mathbb{N}})\) be the space of all continuous self-maps of \(2^{\mathbb{N}}\) endowed with the sup metric, and \(\mathcal{K}(2^{\mathbb{N}})\) be the hyperspace of all nonempty closed subsets of \(2^{\mathbb{N}}\) endowed with the Hausdorff metric. The main goal of the paper is to study the map \(\omega_f\) that maps each point \(x \in 2^{\mathbb{N}}\) to its \(\omega\)-limit set \(\omega(x,f)\in \mathcal{K}(2^{\mathbb{N}})\). It is shown that the set of all \(f \in \mathcal{C}(2^{\mathbb{N}})\) such that \(\omega_f\) is everywhere discontinuous on a subsystem is dense in \(\mathcal{C}(2^{\mathbb{N}})\). On the other hand, for a generic \(f \in \mathcal{C}(2^{\mathbb{N}})\), the map \(\omega_f : 2^{\mathbb{N}} \to \mathcal{K}(2^{\mathbb{N}})\) is continuous. The authors give also a direct proof of the result of N. C. Bernardes jun. and U. B. Darji [Adv. Math. 231, No. 3–4, 1655–1680 (2012; Zbl 1271.37028)] that a generic \(f \in \mathcal{C}(2^{\mathbb{N}})\) has no Li-Yorke pair. A few related examples are discussed at the end of the paper.

MSC:

37B99 Topological dynamics
54H20 Topological dynamics (MSC2010)
37B10 Symbolic dynamics

Citations:

Zbl 1271.37028
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References:

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