Li-Yorke chaos on fuzzy dynamical systems

TL;DR

The paper develops a comprehensive framework for Li-Yorke chaos and its variants in the setting of fuzzy dynamical systems, clarifying how chaos properties propagate along the natural suspensions $(X,f) \to (\\mathcal{K}(X),\\overline{f}) \to (\\mathcal{F}(X),\\hat{f})$. It proves that the main Ly-Yorke-type chaos notions and distributional chaos transfer from the base system to the hyperspace and then to the fuzzy extension, while the reverse implications generally require additional hypotheses. A key contribution is the introduction of Cantor-dense Li-Yorke chaos (CD-LYC) and its transfer under completeness, plus strong results for linear operators linking CD-LYC and U-LYC across these spaces, including hypercyclic operators as concrete instances. The paper also develops a proximality-sensitivity framework that clarifies when chaos on fuzzy dynamics enforces chaos on hyperspaces and vice versa, providing new insights and counterexamples that delineate the limits of these transfers. Altogether, the work unifies and extends previous fuzzy-dynamics results, offering a robust toolkit for analyzing chaotic behavior in hyperspace and fuzzy extensions and suggesting several avenues for further research in invariant measures, alternative fuzzy metrics, and broader dynamical contexts.

Abstract

Given a dynamical system $(X,f)$ we investigate how several variants of Li-Yorke chaos behave with respect to the extended systems $(\mathcal{K}(X),\overline{f})$ and $(\mathcal{F}(X),\hat{f})$, where $\overline{f}$ is the hyperextension of $f$ acting on the space $\mathcal{K}(X)$ of non-empty compact subsets of $X$, and where $\hat{f}$ denotes the Zadeh extension of $f$ acting on the space $\mathcal{F}(X)$ of normal fuzzy subsets of $X$. We first prove that the main variants of Li-Yorke chaos transfer from $(X,f)$ to $(\mathcal{K}(X),\overline{f})$ and from $(\mathcal{K}(X),\overline{f})$ to $(\mathcal{F}(X),\hat{f})$, but that the converse implications do not hold in general. However, combining the notions of proximality and sensitivity we introduce Cantor-dense Li-Yorke chaos, and we prove that this strengthened variant of chaos does transfer from $(\mathcal{F}(X),\hat{f})$ to $(\mathcal{K}(X),\overline{f})$ under natural assumptions.

Theorems & Definitions (35)

• Lemma 2.1
• proof
• Proposition 2.2
• Lemma 2.3
• Proposition 2.4
• Proposition 2.5
• Lemma 2.6: Lopez2025_arXiv_topological-I
• Lemma 2.7
• proof : Proof of Lemma \ref{['Lem:key2']}
• Theorem 3.1