Pointwise attractors which are not strict

Abstract

We deal with the finite family $F$ of continuous maps on the Hausdorff space $X$. A nonempty compact subset $A$ of such space is called a strict attractor if it has an open neighborhood $U$ such that $A={lim}_{n\to \infty }{F}^n\left(S\right)$ for every nonempty compact $S\subset U$. Every strict attractor is a pointwise attractor, which means that the set $\{x\in X;{lim}_{n\to \infty }{F}^n\left(x\right)=A\}$ 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.