A topological characterization of omega-limit sets on dynamical systems

TL;DR

This paper explores topological properties of omega-limit sets in dynamical systems, establishing key results about their structure, connectedness, and quasi-attracting nature.

Contribution

It provides new topological characterizations of omega-limit sets, including their connectedness and quasi-attracting properties, in set-valued dynamical systems.

Findings

Closure and orbital functions are idempotent in set-valued systems

Compact limit sets of connected sets are connected

Omega-limit sets of compact sets are quasi-attracting

Abstract

In this article, we deal with several notions in dynamical systems. Firstly, we prove that both closure function and orbital function are idempotent on set-valued dynamical systems. And we show that the compact limit set of a connected set is also connected. Furthermore, we prove that the $\Omega$-limit set of a compact set is quasi-attracting.