论文标题

弱关同的迭代功能系统及其他:手册

Weakly contractive iterated function systems and beyond: A manual

论文作者

Leśniak, Krzysztof, Snigireva, Nina, Strobin, Filip

论文摘要

我们对迭代功能系统(IF)进行系统的说明,该功能系统(IFS)的弱收缩(Browder,Rakotch,拓扑)。我们表明,吸引子和渐近稳定的不变性措施以及随机迭代算法的有效性(“混乱游戏”)的有效性可以很容易地用于弱合同系统。我们表明,弱收缩IFSS的吸引子类实质上比古典IFSS的分形更宽。另一方面,我们表明,在合理的空间中,典型的紧凑型集合并不是任何弱收缩IF的吸引子。我们通过使用固定点理论的几种工具:球的几何形状,平均收缩,重新射击技术,有序集和非划分的度量来探讨打破缩放障碍的可能性和限制。从这些考虑中,可以得出结论,尽管在轻度条件下的一般迭代功能系统可以轻易地确保存在不变的集合和不变性措施,以确定吸引子的存在和独特的不变措施是一个更加困难的问题。这解释了承包系统在IFSS理论中的核心作用。

We give a systematic account of iterated function systems (IFS) of weak contractions of different types (Browder, Rakotch, topological). We show that the existence of attractors and asymptotically stable invariant measures, and the validity of the random iteration algorithm ("chaos game"), can be obtained rather easily for weakly contractive systems. We show that the class of attractors of weakly contractive IFSs is essentially wider than the class of classical IFSs' fractals. On the other hand, we show that, in reasonable spaces, a typical compact set is not an attractor of any weakly contractive IFS. We explore the possibilities and restrictions to break the contractivity barrier by employing several tools from fixed point theory: geometry of balls, average contractions, remetrization technique, ordered sets, and measures of noncompactness. From these considerations it follows that while the existence of invariant sets and invariant measures can be assured rather easily for general iterated function systems under mild conditions, to establish the existence of attractors and unique invariant measures is a substantially more difficult problem. This explains the central role of contractive systems in the theory of IFSs.

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