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

TL;DR

The paper develops a relation-centered approach to connect topological properties with long-term dynamics by refining attractors into physical and proper types and introducing topological exactness and mixing in the setting of relations. It proves that on a compact phase space $X$, $X$ is a physical attractor for a continuous relation $\phi$ if and only if $\phi$ is topologically mixing, and it shows that attractors imply equicontinuity and that $\phi^{-1}$ is topologically exact (and vice versa). Extending to iterated function systems via the Hutchinson operator, the authors characterize when an IFS is topologically mixing or exact, establish that chain transitivity with shadowing yields a physical attractor, and construct a concrete example of a mixing but not exact IFS on $\mathbb{T}^2$. The results illuminate how topological structure governs long-term behavior in both single-relations and IFS contexts, offering tools for analyzing attractor types and their implications for non-invertible dynamics.

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.

Figures (1)

• Figure 1: For some $m\in\mathbb{N}$, the action of $f^m$ on the square bounded by $x=x_1$, $x=x_2$, $y=y_1$ and $y=y_2$ determined by the red region.

Theorems & Definitions (17)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Proposition 2.1
• proof
• proof : Proof of Theorem \ref{['topexactattractor']} (i)
• proof : Proof of Theorem \ref{['contersen']}
• Theorem 3.1: Theorem 3.7 Nia
• proof : Proof of Theorem \ref{['attfixed']}