不严格的点式吸引子

Pub Date : 2024-01-01 DOI:10.1016/j.indag.2023.10.002

Magdalena Nowak

引用次数: 0

摘要

我们处理的是豪斯多夫空间 X 上连续映射的有限族 F。如果该空间的非空紧凑子集 A 有一个开放邻域 U，使得对于每个非空紧凑 S⊂U，A=limn→∞Fn(S)，则该子集称为严格吸引子。每个严格吸引子都是点式吸引子，这意味着集合{x∈X;limn→∞Fn(x)=A}的内部包含A。我们提出了一类点式吸引子的例子--从有限集到西尔潘斯基地毯--当我们在系统中加入一个非膨胀映射时，这些吸引子就不是严格的了。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Pointwise attractors which are not strict

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 $A = {lim}_{n \to \infty} F^{n} (S)$ for every nonempty compact $S \subset U$ . Every strict attractor is a pointwise attractor, which means that the set ${x \in X; {lim}_{n \to \infty} F^{n} (x) = A}$ 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.

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