自仿集的拓扑结构和李普希兹等价

项目名称： 自仿集的拓扑结构和李普希兹等价

项目编号： No.11301322

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 罗军

作者单位： 重庆大学

项目金额： 23万元

中文摘要： 本项目主要研究自仿分形集上的拓扑结构以及特殊情况下的李普希兹等价问题。具体包括：（1）对于自仿迭代函数系统，我们在其符号空间上建立一个双曲图，研究它的双曲边界，我们将证明在弱分离条件下，该双曲边界与自仿集同胚；（2）研究Bedford-McMullen 地毯的拓扑结构，我们用周期性延拓的方法，给出它们拓扑分类的完整刻画；（3）研究一类局部具有类圆盘结构的连通自仿tile的拓扑特征；（4）在（1）的框架下，我们进一步研究自仿集的李普希兹等价问题，我们将借鉴之前对自相似集的李普希兹等价问题的处理方法，拟得到自仿集上的一般结论。我们希望本项目能够对自仿集的拓扑结构的研究做出有用的贡献，并对自仿集上的李普希兹等价进行初步的探索，为分形几何找到新的研究途径。

中文关键词： 自仿集；自仿测度；连通性；类圆盘性；双曲图

英文摘要： This research mainly study the topological structure of self-affine sets and their Lipschitz equivalence under some special case. The main contents are as follows: (1) for the self-affine iterated function system, we define a hyperbolic graph on the symbolic space, and study its hyperbolic boundary, we will prove that, under the weak separation condition, the hyperbolic boundary is homeomorphic to the self-affine set; (2) study the topological structure of Bedford-McMullen carpets, by using a periodic extension, we can provide a complete characterization of the topological classification; (3) study the topological characterization of a class of self-affine tiles with local disk-like structure; (4) under the setting of (1), we will go further to study the Lipschitz equivalence problem of self-affine sets, by using the previous method on self-similar sets, we try to obtain some general results on self-affine sets. We hope our research can make a useful contribution to the study on the topological structure of self-affine sets, make a first attempt to explore the Lipschitz equivalence of self-affine sets, and initiate a new study on the fractal geometry.

英文关键词： self-affine set；self-affine measure；connectedness；disk-likeness；hyperbolic graph