高维随机覆盖问题及其在动力系统中的应用

项目名称： 高维随机覆盖问题及其在动力系统中的应用

项目编号： No.11201155

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 李兵

作者单位： 华南理工大学

项目金额： 22万元

中文摘要： 随机覆盖问题源于Borel对随机级数收敛性的研究,经典模型为圆周上的随机覆盖，即考虑被圆周上的一列随机区间覆盖无穷多次的点集E（常被称为覆盖集）及其余集F，这两类集合均是随机集，人们分别从拓扑、测度、维数等方面研究这两个集合的结构和大小，并对其它性质也进行了研究，如击中概率等，现逐步与分形几何、动力系统、概率论等领域结合起来。本项目研究的是高维自仿随机覆盖、高维Possion覆盖及动力系统型覆盖等问题，将对这些模型中的随机集E和F的Hausdorff维数、Packing维数、投影及击中概率中的0-1律进行研究，希望建立随机覆盖集与自仿集、limsup型随机分形之间的联系，使用这两类分形中已有的工具更精细地描述集合E，并寻找Kahane开问题的答案，即给出用随机正方体或球覆盖时torus中每点均被覆盖的充分必要条件，同时，寻找满足指数混合性的动力系统种类和研究该类动力系统的覆盖行为。

中文关键词： 随机覆盖；Hausdorff 维数；动力系统；覆盖集；自仿

英文摘要： Random covering problem origins from the study about the convergence of the random series, due to Borel. The classical model is the random covering on the circle, that is, given a sequence of random intervals on the circle, consider the covering set E of points covered infinitely times by these intervals and its complementary set F, both of which are random sets. This project focus on the self-affine random covering in high dimension case, Poisson covering for the high dimension, dynamical covering etc. We consider the Hausdorff and packing dimensions, the projection sets and hitting probabilities of random sets E and F for these models.We try to find the relationships among random covering set, self-affine sets and limsup random fractals. By the known tools in such two kinds of fractals, we want to describe the set E in the finer way. We also hope to give the solution of Kahane's open problem, that is to say, obtaining a sufficient and necessary condition for that every point in the torus is covered infinitely often by a sequence of random cubes or balls. Meanwhile, we will try to find more kinds of dynamical systems satisfying the exponentially mixing property and study their covering behaviors.

英文关键词： Random covering；Hausdorff dimension；dynamical systems；covering set；self-affine