Pointwise attractors which are not strict

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 Sierpiński carpet - which are not strict when we add to the system one nonexpansive map.