Topological dynamics for the endograph metric I: Equivalences with other metrics

TL;DR

This paper explores the relationships between the endograph metric and other metrics in the context of topological dynamical properties of Zadeh extensions, providing new insights and resolving open questions.

Contribution

It establishes equivalences between the endograph metric and other metrics for key dynamical properties, offering new results on point-$ ext{A}$-transitivity.

Findings

Endograph metric behaves similarly to other metrics for various dynamical properties.

Resolved open questions regarding metric equivalences in topological dynamics.

Introduced new outcomes for point-$ ext{A}$-transitivity.

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 topological $\mathcal{A}$-transitivity, topological $(\ell,\mathcal{A})$-recurrence, Devaney chaos, and for the specification property, the endograph metric behaves similarly to the supremum metric $d_{\infty}$, the Skorokhod metric $d_{0}$ and the sendograph metric $d_{S}$. Our results not only resolve certain open questions in the existing literature, but also yield completely new outcomes in terms of point-$\mathcal{A}$-transitivity.