迭代函数系的分离条件及其应用

项目名称： 迭代函数系的分离条件及其应用

项目编号： No.11471075

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： 邓起荣

作者单位： 福建师范大学

项目金额： 60万元

中文摘要： 本项目研究的分形集是指由迭代函数系生成的非空紧集K，这是分形几何的主要研究对象之一。主要研究如下几个方面的问题。一是自仿Tile的连通性、一般自相似集和自仿集的连通性问题。二是双Lipchitz迭代函数系生成的分形集K的维数问题。包括Hausdorff维数和计盒维数相等的条件，维数的计算方法等。三是一般自相似集和自仿集的勒贝格测度的计算方法（有内点时）。四是分形集K上定义的随机过程。这里，我们主要关注由迭代函数系定义的加边树（augmented tree）上的马氏链以及相应的格林函数、Martin核和Martin边界。因为上述问题的研究都与迭代函数系的分离条件相关，我们也要进一步研究迭代函数系的分离条件的刻画。

中文关键词： 迭代函数系；分形集；分形维数

英文摘要： The fractal sets we shall study in this program are compact sets defined by iterated function systems; they are the main objects of fractal geometry. We shall mainly consider the following problems. The first aspect is the connectedness of self-affine tiles, general self-affine sets and self-similar sets. The second aspect is on the dimension problems of fractal sets generated by bi-lipchitzan iterated function systems. Including conditions implying the equality of Hausdorff dimension and box dimension and the computation of these dimensions. The third aspect is on the calculation of the Lebesgue's measure of a self-similar or self-affine set when it contains interior points. The fifth aspect is on stochastic processes defined on fractal sets. Here, we mainly consider Markoff chains defined on augmented trees for iterated function systems, the related Green's function, the Martin kernel and the Martin boundary of the Markoff chain. Since these problems are related to the separation conditions of an iterated function system, we shall also further characterize separation conditions of iterated function systems.

英文关键词： iterated function systems;fractal set;fractal dimension