Topological dynamics for the endograph metric I: Equivalences with other metrics

TL;DR

This work investigates how the endograph metric $d_E$ on the Zadeh fuzzification $(\mathcal{F}(X),\hat{f})$ interacts with classical dynamics by proving robust equivalences for transitivity, recurrence, Devaney chaos, and the specification property across a network of systems: $(X,f)$, $(\mathcal{K}(X),\overline{f})$, and the fuzzy extensions with respect to $d_{\infty}$, $d_{0}$, $d_{S}$, and $d_{E}$. A central technical tool, Lemma 2.3, connects endograph closeness to Hausdorff closeness of level-sets, enabling transfer of properties from hyperspaces to fuzzy spaces. The paper establishes that, for any Furstenberg family $\mathcal{A}$, the properties hold equivalently in all extended systems, including new results for point-$\mathcal{A}$-transitivity on complete metric spaces and for the endograph and sendograph metrics. These results extend and unify existing literature, resolving open questions and providing a general framework to study how fuzzification preserves or mirrors classical dynamical behaviours. The findings have potential implications for both theoretical development and applications where fuzzy dynamics are used to model uncertain or imprecise systems.

Abstract

Given a dynamical system $(X,f)$ we investigate several topological dynamical properties for its Zadeh extension $(\mathcal{F}(X),\hat{f})$ endowed with the endograph metric $d_{E}$. In particular, we prove that for topological $\mathcal{A}$-transitivity, topological $(\ell,\mathcal{A})$-recurrence, Devaney chaos, and for the specification property, the endograph metric behaves similarly to the supremum metric $d_{\infty}$, the Skorokhod metric $d_{0}$ and the sendograph metric $d_{S}$. Our results not only resolve certain open questions in the existing literature, but also yield completely new outcomes in terms of point-$\mathcal{A}$-transitivity.

Theorems & Definitions (26)

• Proposition 2.1
• Proposition 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Theorem 3.1
• proof : Proof of Theorem \ref{['The:A-tran']}
• Remark 3.2
• Theorem 3.3
• proof : Proof of Theorem \ref{['The:(l,A)-rec']}