GIFZS的模糊吸引子
Fuzzy Attractors Appearing from GIFZS
摘要：
Cabrelli，Forte，Molter和Vrscay在1992年研究了迭代函数系统（简称IFS）及其分形理论的{fuzzy}版本，其思想是将经典的Hutchinson-Barnsley算子推广到度量空间的自映射，推广到模糊空间的适当自映射，这是分形理论中非常丰富和重要的部分。另一方面，Miculescu和Mihail在2008年引入了IFS理论的另一个推广--他们考虑的不是度量空间$X$的自映射，而是定义在有限笛卡儿积$X^m$上的映射。事实证明，经典的哈钦森-巴恩斯利分形理论的许多部分在这个广义的背景下都有自然的对应物。特别地，如果$X$是完备的，那么这种映射的适当压缩系统生成唯一的分形集。本文证明Cabrelli等人的\emph{fuzzyfication}思想可以自然地适用于有限笛卡儿积上的映射的情形。特别地，我们定义了广义迭代模糊函数系统（简称GIFZS）的概念，并证明了它生成唯一的模糊分形集。我们还研究了GIFs及其分形的一些基本性质，并考虑了我们的设置是否给我们一些新的模糊分形集的问题。
英文摘要：
Cabrelli, Forte, Molter and Vrscay in 1992 considered a {fuzzy} version of the theory of iterated function systems (IFSs in short) and their fractals%The idea was to extend the classical Hutchinson-Barnsley operator to selfmaps of a metric space to appropriate selfmaps of space of fuzzy , which now is quite rich and important part of the fractals theory. On the other hand, Miculescu and Mihail in 2008 introduced another generalization of the IFSs' theory - instead of selfmaps of a metric space $X$, they considered mappings defined on the finite Cartesian product $X^m$. %It turns out that many parts of the classical Hutchinson-Barnsley fractals theory have natural counterparts in this generalized setting. In particular, if $X$ is complete, then appropriately contractive systems of such maps generate unique fractal sets. In this paper we show that the \emph{fuzzyfication} ideas of Cabrelli et al. can be naturally adjusted to the case of mappings defined on finite Cartesian product. In particular, we define the notion of a generalized iterated fuzzy function system (GIFZS in short) and prove that it generates a unique fuzzy fractal set. We also study some basic properties of GIFZSs and their fractals, and consider the question whether our setting gives us some new fuzzy fractal sets.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
