分形几何中的嵌入问题

项目名称： 分形几何中的嵌入问题

项目编号： No.11471124

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 熊瑛

作者单位： 华南理工大学

项目金额： 65万元

中文摘要： 嵌入是研究几何、拓扑的重要途径，特别是保持某种几何结构的嵌入。本项目计划研究分形几何中的仿射嵌入与双Lipschitz嵌入，前者保持所在欧氏空间的线性结构，后者则在一定扭曲下保持了度量。我们将研究自相似集的仿射嵌入问题及相关的Furstenberg猜测，探讨压缩比代数性质的不同导致的几何结构的差异。这个问题源于Furstenberg对乘2和乘3动力系统的研究。Furstenberg猜测，乘2不变集的仿射像与乘3不变集的交一定会比较小，其维数不超过Mastrand定理给出的典型值。我们还将研究分形集的双Lipschitz嵌入问题，及其与双Lipschitz等价问题的关系。双Lipschitz等价问题是分形几何的重要课题，与代数数论也有密切的联系。我们希望在双Lipschitz嵌入问题上得到一些普遍的结果，以此来推动相对困难的双Lipschitz等价的研究。

中文关键词： 分形集；嵌入；仿射嵌入；双Lipschitz嵌入

英文摘要： Embedding, especially which preserves some geometric structures, is an important means in the study of geometry and topology. This project is planed to study the affine embedding and biLipschitz embedding in fractal geometry, the former preserves linear structure of Euclidean space and the latter preserves metric structure with some distortion. We want to study the affine embedding problem of self-similar sets and a related conjecture of Furstenberg, and investigate the difference of geometric structure caused by the difference algebraic properties of the ratios. This problem comes from Furstenberg's study of times2 and times3 dynamical systems. Furstenberg conjectured that the intersection of any affine image of a 2-invariant set and a 3-invariant set must be small in the sense that the dimension of the intersection does not greater than the generic value given by Mastrand theorem. We also want to study the biLipschitz embedding problem and its relationship with biLipschitz equivalence problem. BiLipschitz equivalence problem is an important topic in fractal geometry and it is closely related to algebraic number theory. We hope to obtain some general results on biLipschitz embedding problem and then promote the research of biLipschitz equivalence problem.

英文关键词： fractals;embedding;affine embedding;bi-Lipschitz embedding