The Geometry of Iterated Function Systems with Overlaps

分形图是高度不规则的几何物体，当以任意精细的尺度观察时，它们具有复杂的细节。迭代函数系统分形图可能是分形图最简单的例子：它们是由有限数量的自身的较小副本组成的分形图，由一组有限的收缩映射来描述。了解较小的复制品重叠的方式和程度，可以提供有关分形图的拓扑、分维和其他属性的信息。现有的许多关于分形学的研究都是处理没有精确重叠的情况，而重叠在定量意义上表现得很好。在存在精确重叠的情况下所做的工作较少，而开发技术来了解这种重大重叠对分形图几何的影响是一个重要的问题。迭代函数系统也产生了具有复杂多重分形性的自然度量。许多关于分形集的几何性质的自然问题都相应地推广到多重分形度。大部分研究将是理论分析，尽管计算工作对于研究具体的例子和作为更好地理解理论性质的一种方式是必不可少的。

Fractals are highly irregular geometric objects which have intricate detail when viewed at arbitrarily fine scales. Iterated functionsystem fractals are perhaps the simplest examples of fractals: these are fractals that are composed of a finite number of smallercopies of themselves, as described by a finite set of contraction maps. Understanding the ways in which the smaller copiesoverlap, and the extent to which they do, provides information about the topology, fractal dimension, and other properties of thefractals. Much of the existing research on fractals treat cases where there are no exact overlaps and the overlaps are wellbehavedin a quantitative sense. Less work has been done in the presence of exact overlaps, and developing techniques tounderstand the impact of such substantial overlaps on the geometry of the fractals is an important question. Iterated functionsystems also give rise to natural measures with complex multifractal properties. Many natural questions concerning thegeometric properties of the fractal sets have corresponding generalizations to multifractal measures. Much of the research willbe theoretical analysis, though computational work is essential for the study of specific examples and as a way of motivatingbetter understanding of the theoretical properties.