高维Frobenius问题和自相似集的Lipschitz等价

This research intends to study higher dimensional Frobenius problem and applies this results to the study of Lipschitz equivalence of self-similar sets. One-dimensional Frobenius problem is a classical problem of Number Theory. In our works[17,18], we introduced higher dimensional Frobenius problem, and studied its directional growth function and rigidity under coplanar condition, and applied this results to the study of Lipschitz equivalence of self-similar sets.. We will continue to investigate higher dimensional Frobenius problem under non-coplanar condition. We will estimate the vertices of maximal saturated cone and multiplicity function and its application. We will study the relation between directional growth function and maximal entropy under linear constraints, and study the rigidity. This results will be applied to the study of Lipschitz equivalence of self-similar sets under non-coplanar condition.. In addition, we also plan to study the Lipschitz equivalence of a simple class of planar self-affine sets.

本项目拟研究高维Frobenius问题及其在自相似集的Lipschitz等价上的应用。 一维Frobenius问题是一个经典的数论问题，我们在前期工作[17,18]中引入了高维Frobenius问题，在共面条件下研究了其方向增长函数和刚性，并把这些结果应用于自相似集的Lipschitz等价。. 本项目拟继续研究不共面情形下的高维Frobenius问题，包括极大锥的顶点估计，重数估计及应用，方向增长函数和线性约束条件熵的关系以及刚性问题。我们将把这些研究应用于不共面情形的Lipschitz等价的Falconer-Marsh问题。. 此外，本项目也计划在比较简单的情形下，研究自仿集的Lipschitz等价问题。

本项目研究了自相似集间的Lipschitz等价性， Mcmullen自仿集间的Lipschitz等价性，和分形集的拓扑结构三个方面的问题。（1）我们在论文[Zhang ，J. Math. Anal. Appl.(2017)]上中研究了一类平面自仿集的拓扑结构与性质，由此给出拓扑分类.我们在[Yang- Zhang, Submitted to Discret Comput. Geom.(2019)]中研究了凸多面体锥C的平移Tiling问题和自仿Tile的拓扑性质，刻画了其具有平移Tiling性质的条件.我们在[Rao-Yang- Zhang,Characterization of (Z^+)^n-tiling ，Submitted to Acta Arith(2019)]研究(Z^+)^n-Tiling 的刻画问题.（2）自相似集的Lipschitz等价问题起源于G.David和S.Semmes(1997)和Falconer-Marsh (Mathematica,1992)的一系列工作，我们在论文[Zhu-Yang, J. Math. Anal. Appl.(2018)]中通过引入有限状态自动机和角分离性质，证明了一类具有两态邻居自动化的自相似集Lipschitz等价当且仅当它们具有相同的Hausdorff维数. (3)我们研究Bedford-McMullen垫的Lipschitz等价问题，我们在预印本[Rao-Yang-Zhang， Bi-Lipschitz classification of Bedford-McMullen carpets (I)]中证明了自仿垫上的均匀Bernoulli测度的重分形谱是一个Lip-不变量,且上述测度的加倍性质也是Lip-不变的，我们解决了满足加倍性的全不连通的正则自仿垫和满足King的分离条件的非正则自仿垫的的Lipschitz等价问题，我们在预印本[Rao-Yang-Zhang, Bi-Lipschitz classification of Bedford-McMullen carpets (II)]中证明了满足垂直分离条件的自仿垫和一个半符号空间Lipschitz等价，从而解决了满足垂直分离条件的自仿垫的Lipschitz等价问题，进一步证明了两自仿垫的分布列在某种情况下交换顺序也是Lipchitz 等价的.