Topological dynamics for the endograph metric II: Extremely radical properties

TL;DR

This paper explores how the endograph metric induces radical topological dynamical properties in Zadeh extensions, resolving open questions and revealing new behaviors in chain recurrence, transitivity, mixing, and shadowing.

Contribution

It demonstrates that the endograph metric causes extreme behaviors in various dynamical properties of Zadeh extensions, providing new insights and answers to open problems.

Findings

Endograph metric leads to radical properties in chain recurrence and transitivity.

The metric causes extreme behaviors in shadowing property.

Results resolve open questions and introduce new phenomena in dynamical 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 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.