McMullen地毯上自仿测度的加倍性及其相关分形问题的研究

Self-affine sets are a class of typical fractal sets. Because of the particularity of the self-affine structure, people often encounter some difficulties and challenges when they study the self-affine sets.McMullen carpets are the simplest self-affine sets on which there have been some researches. Even for such sets, there are still many unresolved problems.The project will be devoted to studying the doubling property of self-affine measures on McMullen carpets and the Lipschitz equivalence of McMullen carpets. We shall consider the following two problems: (1)Characterize the doubling property of self-affine measures on general McMullen carpets; (2)Discuss the Lipschitz equivalence of McMullen carpets. The research of these problems involves the structure of McMullen carpets, self-affine measures, doubling measures, and the Lipschitz equivalence, and so on. It is of great significance for better understanding McMullen carpets.

自仿集是一类典型的分形集。由于自仿结构的特殊性，人们围绕自仿集展开研究时通常会遇到一些困难与挑战。McMullen地毯是一类最简单的自仿集，对于它的研究已有一些结果，但是仍有许多与之相关的未解决的问题。本项目致力于研究McMullen地毯上自仿测度的加倍性质以及McMullen地毯的Lipschitz等价性，拟考虑如下两个问题：(一) 对广义McMullen地毯上自仿测度的加倍性质进行刻画；(二) 讨论McMullen地毯的Lipschitz等价性。这些问题的研究涉及到McMullen地毯的结构、自仿测度、加倍测度、Lipschitz等价性等知识。本项目的研究和探讨对进一步理解McMullen地毯具有很重要的意义。

自相似集和自仿集一直是分形几何的主要研究对象，而分形集的Lipschitz等价性问题又是分形理论的热点问题之一。目前许多研究是围绕全不连通的自相似集展开的，而对于非全不连通的情况，研究起来是很困难的。基于此现状，我们研究了一对非全不连通“分形方块（9块留6块）”之间的Lipschitz等价性，并得到了好的结果。此外，根据不确定测度的性质以及图的拓扑结构，我们还研究了拓扑网络图的匹配数问题。上述研究内容和结果既丰富了几何测度论及分形理论，也为今后的理论研究奠定了良好的基础。