Recurrence in collective dynamics: From the hyperspace to fuzzy dynamical systems

TL;DR

The work builds a cohesive framework connecting recurrence notions for a base system $(X,f)$ with its induced dynamics on hyperspace $(\mathcal{K}(X),\overline{f})$ and fuzzy-extension system $(\mathcal{F}(X),\hat{f})$. It proves that, under suitable conditions, topological $(\ell,\mathcal{A})$-recurrence (including classical recurrence and multiple recurrence) of the product map $f_{(N)}$ for all $N$ is equivalent to the same recurrence for the hyperspace and fuzzy extensions, with precise equivalences across $\mathcal{K}(X)$ and $\mathcal{F}(X)$ via $\overline{f}$ and $\hat{f}$. In the complete-metric setting, the authors obtain stronger point-recurrence results, including quasi-rigidity equivalences for separable spaces, linking point-recurrence and AP-recurrence across the three frameworks. The paper also raises open questions about extending these results to convex hyperspaces, invariant measures for $\hat{f}$, and alternative hyperspace topologies, guiding future research in collective dynamics of complex systems.

Abstract

We study for a dynamical system $f:X\longrightarrow X$ some of the principal topological recurrence-kind properties with respect to the induced maps $\overline{f}:\mathcal{K}(X)\longrightarrow\mathcal{K}(X)$, on the hyperspace of non-empty compact subsets of $X$, and $\hat{f}:\mathcal{F}(X)\longrightarrow\mathcal{F}(X)$, on the space of normal fuzzy sets consisting of the upper-semicontinuous functions $u:X\longrightarrow [0,1]$ with compact support and such that $u^{-1}(\{1\})\neq\varnothing$. In particular, we characterize the properties of topological and multiple recurrence for the extended systems $(\mathcal{K}(X),\overline{f})$ and $(\mathcal{F}(X),\hat{f})$, which cover the cases of the so-called nonwandering and Van der Waerden systems. Special attention is given to the case where the underlying space is completely metrizable, for which we obtain some stronger point-recurrence equivalences.

Theorems & Definitions (21)

• Remark 2.1
• Definition 2.2
• Remark 2.3
• Proposition 2.4
• Proposition 2.5
• Lemma 2.6
• Theorem 3.1
• proof
• Remark 3.2
• Corollary 3.3