Pointwise attractors which are not strict

TL;DR

This paper explores examples of pointwise attractors in dynamical systems that are not strict attractors, especially when a nonexpansive map is added, highlighting differences between these attractor types.

Contribution

It introduces a class of pointwise attractors, including the Sierpiński carpet, that are not strict when an additional nonexpansive map is incorporated.

Findings

Examples of pointwise attractors not being strict are provided.

Adding a nonexpansive map can change the nature of attractors.

The paper includes classical fractals like the Sierpiński carpet as examples.

Abstract

We deal with the finite family $\mathcal{F}$ of continuous maps on the Hausdorff space. A nonempty compact subset $A$ of such space is called a strict attractor if it has an open neighborhood $U$ such that $A=\lim_{n\to\infty}\mathcal{F}^n(S)$ for every nonempty compact $S\subset U$. Every strict attractor is a pointwise attractor, which means that the set $\{x\in X ; \lim_{n\to\infty}\mathcal{F}^n(x)=A\}$ contains $A$ in its interior. We present a class of examples of pointwise attractors - from the finite set to the Sierpi\'nski carpet - which are not strict when we add to the system one nonexpansive map.