Topological dynamics of Zadeh's extension on the space of upper semi-continuous fuzzy sets

TL;DR

This paper explores the topological dynamics of Zadeh's extension on upper semi-continuous fuzzy sets, establishing equivalences between various dynamical properties of the original system and its extension.

Contribution

It provides new characterizations of dynamical properties of Zadeh's extension on fuzzy sets, linking them to properties of the original system.

Findings

Zadeh's extension is transitive if and only if the original system is weakly mixing.

The extension is mildly mixing when the original system has this property.

Equicontinuity and uniform rigidity of the extension correspond to specific properties of the base system.

Abstract

In this paper, some characterizations about transitivity, mildly mixing property, $\mathbf{a}$-transitivity, equicontinuity, uniform rigidity and proximality of Zadeh's extensions restricted on some invariant closed subsets of the space of all upper semi-continuous fuzzy sets with the level-wise metric are obtained. In particular, it is proved that a dynamical system is weakly mixing (resp., mildly mixing, weakly mixing and $\mathbf{a}$-transitive, equicontinuous, uniformly rigid) if and only if the Zadeh's extension is transitive (resp., mildly mixing, $\mathbf{a}$-transitive, equicontinuous, uniformly rigid).