Proximality and transitivity in relation to points that are asymptotic to themselves

We discuss dynamical systems that exhibit at least one weakly asymptotically periodic point. In the general case we prove that the system becomes trivial (it is either a periodic point or a fixed point) provided it is equicontinuous and transitive. This result can be used to provide a simple characterization of periodic points in transitive systems. We also discuss systems whose orbits are both proximal and weakly asymptotically periodic. As a result, we obtain a more general tool to detect mutual dynamics between two close orbits which need not be bounded (or have the empty limit set).