The three versions of distributional chaos☆
Section snippets
Introduction and preliminaries
Let f be a map from a compact metric space (M, ρ) into itself. For any pair (x, y) of points in M and any positive integer n, define a distribution function byThen is a non-decreasing function, for t ⩽ 0, and for t greater than the diameter of M. PutThen Φxy is called the lower distribution function, and the upper distribution function of x and y.
Proof of the main results
Theorem 1 There is a continuous map F of a compact metric space M which is DC3 but not chaotic in the sense of Li and Yorke. Proof Let X be the set {0, 1}∞ of sequences of two symbols, equipped with the metric ρX of pointwise convergence. Let Y be the subset of the complex plane consisting of the unit circle, and a singleton {2}; i.e., z ∈ Y if and only if either ∣z∣ = 1 or z = 2. Finally, let M = X × Y, and let ρ be the max metric on M given by ρ((x1, y1), (x2, y2)) = max{ρX(x1, x2), ∣y1 − y2∣}. For any (x, y) ∈ M we let F(x, y
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2024, Ergodic Theory and Dynamical SystemsGenerating Chaos with Saddle-Focus Homoclinic Orbit
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2023, arXiv
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The research was supported, in part, by projects 201/03/1153 from the Grant Agency of Czech Republic, MSM192400002 from the Czech Ministry of Education, and PI-8/00807/FS/01 from fundación Séneca (Comunidad Autonóma de Murcia).