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Functional envelope of a non-autonomous discrete system

100%
Barzanouni A.
Nonautonomous Dynamical Systems
|
2017
|
tom 4
|
nr 1
98-107
EN
Let (X, F = {fn}n =0∞) be a non-autonomous discrete system by a compact metric space X and continuous maps fn : X → X, n = 0, 1, ....We introduce functional envelope (S(X), G = {Gn}n =0∞), of (X, F = {fn}n =0∞), where S(X) is the space of all continuous self maps of X and the map Gn : S(X) → S(X) is defined by Gn(ϕ) = Fn ∘ ϕ, Fn = fn ∘ fn-1 ∘ . . . ∘ f1 ∘ f0. The paper mainly deals with the connection between the properties of a system and the properties of its functional envelope.
2
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Invariant sets and Knaster-Tarski principle

76%
Leśniak K.
Open Mathematics
|
2012
|
tom 10
|
nr 6
2077-2087
EN
Our aim is to point out the applicability of the Knaster-Tarski fixed point principle to the problem of existence of invariant sets in discrete-time (multivalued) semi-dynamical systems, especially iterated function systems.
3
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Chaotic behaviour of the map x ↦ ω(x, f)

76%
D’Aniello E., Steele T.
Open Mathematics
|
2014
|
tom 12
|
nr 4
584-592
EN
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.
4
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A class of continua that are not attractors of any IFS

64%
Kulczycki M., Nowak M.
Open Mathematics
|
2012
|
tom 10
|
nr 6
2073-2076
EN
This paper presents a sufficient condition for a continuum in ℝn to be embeddable in ℝn in such a way that its image is not an attractor of any iterated function system. An example of a continuum in ℝ2 that is not an attractor of any weak iterated function system is also given.
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