Attractors as a bridge from topological properties to long-term behavior in dynamical systems

TL;DR

This paper explores the relationship between topological properties and long-term behavior in dynamical systems, introducing refined notions of attractors and establishing conditions linking topological mixing and exactness.

Contribution

It refines and introduces new notations for attractors within relations, and establishes necessary and sufficient conditions connecting topological properties with attractor types.

Findings

Phase space of a topologically exact system is an attractor for its inverse.

A system is topologically mixing iff its phase space is a physical attractor.

Provided examples of non-trivial topologically mixing and exact IFSs.

Abstract

This paper refined and introduced some notations (namely attractors, physical attractors, proper attractors, topologically exact and topologically mixing) within the context of relations. We establish necessary and sufficient conditions, including that the phase space of a topologically exact system is an attractor for its inverse, and vice versa, and that a system is topologically mixing if and only if its phase space is a physical attractor. Through iterated function systems (IFSs), we illustrate classes of non-trivial topologically mixing and topologically exact IFSs. Additionally, we use IFSs to provide an example of topologically mixing system, generated by finite of homeomorphisms on a compact metric space, that is not topologically exact. These findings connect topological properties with attractor types, providing deeper insights into the long-term dynamics of such systems.