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).
Weak asymptotic periodicity, periodicity, $\omega$-limit set, equicontinuity, transitive system, proximal relation, regionally proximal relation
"Proximality and transitivity in relation to points that are asymptotic to themselves,"
Turkish Journal of Mathematics: Vol. 47:
7, Article 10.
Available at: https://journals.tubitak.gov.tr/math/vol47/iss7/10