The Geometry of Iterated Function Systems with Overlaps

分形是高度不规则的几何物体，当在任意精细的尺度上观察时，它们具有复杂的细节。迭代函数系统分形可能是分形的最简单的例子：这些分形是由有限数量的自身较小副本组成的分形，由有限组收缩映射描述。了解较小的副本重叠的方式，以及它们重叠的程度，可以提供有关拓扑结构、分形维数和分形的其他属性的信息。许多现有的分形研究都是在没有精确重叠的情况下进行的，而且重叠在定量意义上表现良好。在精确重叠的情况下所做的工作较少，开发技术来理解这种实质性重叠对分形几何的影响是一个重要的问题。迭代函数系统也产生具有复杂多重分形性质的自然测度。关于分形集几何性质的许多自然问题，都有相应的推广到多重分形测度。尽管计算工作对于具体例子的研究和更好地理解理论性质是必不可少的，但大部分研究将是理论分析。

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.