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

TL;DR

This paper explores recurrence properties in dynamical systems extended to hyperspaces and fuzzy set spaces, providing characterizations and stronger results in metrizable spaces.

Contribution

It introduces new topological recurrence concepts for induced maps on hyperspaces and fuzzy sets, extending classical dynamical systems theory.

Findings

Characterization of topological and multiple recurrence in hyperspace and fuzzy set systems

Stronger recurrence equivalences established in completely metrizable spaces

Connections made between nonwandering, Van der Waerden systems and extended dynamics

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.