The Geometry of Iterated Function Systems with Overlaps

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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.