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

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.