ABSTRACT 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.

中文翻译：

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弱收缩迭代函数系统及其他：手册

摘要我们系统地说明了不同类型（Browder、Rakotch、拓扑）弱收缩的迭代函数系统（IFS）。我们表明，对于弱收缩系统，可以很容易地获得吸引子和渐近稳定不变测度的存在，以及随机迭代算法（“混沌游戏”）的有效性。我们表明弱收缩 IFS 的吸引子类本质上比经典 IFS 的分形类更宽。另一方面，我们表明，在合理的空间中，典型的紧集不是任何弱收缩 IFS 的吸引子。我们通过使用不动点理论中的几种工具来探索打破收缩障碍的可能性和限制：球的几何形状、平均收缩、重新测量技术、有序集、和非紧凑性的措施。从这些考虑可以得出，虽然对于一般迭代函数系统在温和条件下可以很容易地确保不变集和不变测度的存在，但确定吸引子和唯一不变测度的存在是一个困难得多的问题。这解释了收缩系统在 IFS 理论中的核心作用。