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Topological dynamics for the endograph metric II: Extremely radical properties

Antoni López-Martínez

TL;DR

The paper analyzes the dynamics of the Zadeh extension $(\mathcal{F}(X),\hat{f})$ under the endograph metric $d_E$ and contrasts them with the classical hyperspace extension $(\mathcal{K}(X),\overline{f})$. It establishes that $d_E$ induces highly radical behavior for contractive/expansive properties, chain recurrence/transitivity/mixing, and the shadowing property, offering new resolutions to open questions and novel counterexamples. Key contributions include sharp characterizations: (i) $(\mathcal{F}_{0}(X),\hat{f})$, $(\mathcal{F}_{S}(X),\hat{f})$, $(\mathcal{F}_{E}(X),\hat{f})$ are contractive iff $f$ is constant, while expansiveness properties require $|X|=1$; (ii) chain-type properties for fuzzy-end extensions align with dense-range of $f$, yielding extreme equivalences when using the endograph metric; (iii) shadowing behavior differs between finite and full shadowing across fuzzy extensions, with contractive bases enabling full shadowing for certain fuzzy metrics. Together, these results clarify the dominant role of the endograph metric in fuzzy-dynamics and suggest directions for future work on chaos-like notions and metric- vs. topological-dynamics interactions.

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 some contractive and expansive properties, for chain recurrence, chain transitivity and chain mixing, and for the shadowing property, the endograph metric behaves in an extremely radical way. Our results not only resolve certain open questions in the existing literature, but also yield completely new outcomes concerning the chain-type notions considered and the shadowing property.

Topological dynamics for the endograph metric II: Extremely radical properties

TL;DR

The paper analyzes the dynamics of the Zadeh extension under the endograph metric and contrasts them with the classical hyperspace extension . It establishes that induces highly radical behavior for contractive/expansive properties, chain recurrence/transitivity/mixing, and the shadowing property, offering new resolutions to open questions and novel counterexamples. Key contributions include sharp characterizations: (i) , , are contractive iff is constant, while expansiveness properties require ; (ii) chain-type properties for fuzzy-end extensions align with dense-range of , yielding extreme equivalences when using the endograph metric; (iii) shadowing behavior differs between finite and full shadowing across fuzzy extensions, with contractive bases enabling full shadowing for certain fuzzy metrics. Together, these results clarify the dominant role of the endograph metric in fuzzy-dynamics and suggest directions for future work on chaos-like notions and metric- vs. topological-dynamics interactions.

Abstract

Given a dynamical system we investigate several topological dynamical properties for its Zadeh extension endowed with the endograph metric . In particular, we prove that for some contractive and expansive properties, for chain recurrence, chain transitivity and chain mixing, and for the shadowing property, the endograph metric behaves in an extremely radical way. Our results not only resolve certain open questions in the existing literature, but also yield completely new outcomes concerning the chain-type notions considered and the shadowing property.
Paper Structure (15 sections, 16 theorems, 54 equations)

This paper contains 15 sections, 16 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Notation and general background
  3. The extended systems $(\mathcal{K}(X),\overline{f})$ and $(\mathcal{F}(X),\hat{f})$
  4. Four metrics for the space $\mathcal{F}(X)$
  5. Contractions and expansive properties
  6. Definitions and main result
  7. Proof of Theorem \ref{['The:contraexpansive']}
  8. The solution to an expanding problem and a useful lemma
  9. Chain recurrence, chain transitivity and chain mixing
  10. Chain-type properties for the supremum, Skorokhod and sendograph metrics
  11. Chain recurrence and the endograph metric
  12. The shadowing property
  13. Definitions, counterexamples and equivalences
  14. Shadowing for the Skorokhod, sendograph and endograph metrics
  15. Conclusions

Key Result

Proposition 2.1

Let $(X,d)$ be a metric space, $\varepsilon\geq 0$, and let $C_1,C_2,C_3,C_4 \in \mathcal{C}(X)$. Then:

Theorems & Definitions (38)

  • Proposition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Lemma 2.5: Lopez2025_arXiv_topological-I
  • Lemma 2.6
  • proof
  • Remark 2.7
  • Theorem 3.1
  • Remark 3.2
  • ...and 28 more