The Geometry of Iterated Function Systems with Overlaps

具有重叠的迭代函数系统的几何

• 批准号: 2448204
• 金额: --
• 依托单位: University of St Andrews
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2448204
• 关键词: Geometry; Iterated; Function; Systems; Overlaps

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.

分形是高度不规则的几何物体，在任意精细尺度下观察时具有复杂的细节。迭代函数系统分形也许是分形的最简单的例子：这些分形是由有限数量的较小副本组成的，如由有限组收缩映射所描述的。了解较小的副本重叠的方式，以及它们重叠的程度，可以提供有关分形的拓扑结构、分形维数和其他属性的信息。现有的许多分形研究都是在没有精确重叠的情况下进行的，而重叠在定量意义上是很好的。较少的工作已经做了确切的重叠，和开发技术tunderstand的影响，这种大量的重叠的几何形状的分形是一个重要的问题。迭代函数系统也会产生具有复杂多重分形性质的自然测度。许多关于分形集几何性质的自然问题都有相应的重分形测度的推广。大部分的研究将是理论分析，虽然计算工作是必不可少的研究具体的例子，并作为一种方式，激励更好地理解的理论属性。

项目成果

Attainable forms of intermediate dimensions

中间维度的可实现形式

• DOI: 10.54330/afm.120529
• 发表时间: 2022
• 期刊: Annales Fennici Mathematici
• 作者: Banaji A

Local dimensions of self-similar measures satisfying the finite neighbour condition*

满足有限邻域条件的自相似测度的局部维数*

• DOI: 10.1088/1361-6544/ac8040
• 发表时间: 2022
• 期刊: Nonlinearity
• 影响因子: 1.7
• 作者: Hare K