Measures, Dimension, and Ergodic Theory

测度、维度和遍历理论

• 批准号: 0355187
• 负责人: Boris Solomyak
• 金额: --
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0355187&HistoricalAwards=false
• 关键词: Measures; Dimension; Ergodic; Theory

DMS 0355187B SolomyakUniversity of WashingtonMEASURES, DIMENSION, AND ERGODIC THEORYABSTRACTThis project is in the area of interaction between geometricmeasure theory, fractal geometry, and ergodic theory.Iterated function systems, popularly known as the ``Chaos Game,''provide a convenient framework for the study of fractal phenomena.Invariant measures for such systems includeinfinite sums of products of idependent, identically-distributed random variables, which arise in many probablistic and dynamical models, and stationary measures for random matrix products. One of the central problems is to determine when such a measure is absolutely continuous. A closely related line of research is concerned with topological properties of attractors, in particular, self-similar and self-affine sets. Especially challenging are the systems that arenot uniformly contracting and those which have a substantial``overlap.'' On the ergodic theory side, the project deals withsubstitution and tiling dynamical systems without discrete spectrum,as well as with algebraic coding of toral automorphisms. Theso-called non-Pisot case is of particular interest to us, whereit is proposed to use some variants of the beta-transformation defined on self-affine sets.To play the ``Chaos Game'' on the plane, one should specify a family ofplanar tranformations (called an iterated function system)and iteratively apply one of them at random, with prescribedprobabilities. Under certain technical conditions (called``contracting-on-average'') the emerging picture will almost surely``converge'' to a set called the attractor of the iterated functionsystem. This is a popular method to draw fractals on the computer screen, but it is more than a game: iterated functions are widely used in signal processing, image compression, and simulation algorithms,as well as in mathematical dynamical systems theory.The picture that we see on a computer screen is actually a greytone image, which is an approximation of a measure,or probability distribution. An important problem is to decide when this distribution has a density. A related line of researchis to classify the ``zoo'' of fractal objects which arise in the courseof the Chaos Game. The second part of the project is concerned withanother class of objects which can be obtained by iteration:infinite substitutive sequences and self-similar tilings. Now theiteration procedure involves replacing each symbolby a block of symbols, or each tile by a patch of tiles.The limiting object is then used to create a fascinatingdynamical system which we study. Apart from their intrinsic beauty,such systems have found applications in physics.

DMS0355187B Solomyak华盛顿大学测度、维数和遍历理论摘要本项目研究的是几何测度理论、分形几何和遍历理论之间的相互作用。迭代函数系统，俗称“混沌博弈”，为分形现象的研究提供了一个方便的框架。这种系统的不变测度包括独立的乘积的无穷和，同分布随机变量，出现在许多概率和动力学模型，和随机矩阵产品的平稳措施。 其中一个核心问题是确定这样的测度何时是绝对连续的。一个密切相关的研究线是关注吸引子的拓扑性质，特别是自相似和自仿射集。特别具有挑战性的是那些没有统一承包的系统和那些有大量“重叠”的系统。在遍历理论方面，该项目涉及没有离散谱的替代和平铺动力系统，以及与代数编码的toral自同构。我们对所谓的非Pisot情况特别感兴趣，其中建议使用定义在自仿射集上的Beta变换的一些变体。为了在平面上玩“混沌游戏”，应该指定一族平面变换（称为迭代函数系统）并以规定的概率随机迭代应用其中之一。在某些技术条件下（称为“平均收缩”），新出现的图像几乎肯定会“收敛”到一个称为迭代函数系统吸引子的集合。这是一种在计算机屏幕上绘制分形的流行方法，但它不仅仅是一种游戏：迭代函数广泛用于信号处理，图像压缩和模拟算法，以及数学动力系统理论。我们在计算机屏幕上看到的图像实际上是一个灰度图像，它是一个近似的度量或概率分布。一个重要的问题是确定这个分布何时具有密度。 一个相关的研究路线是对混沌游戏过程中出现的分形对象的“动物园”进行分类。该项目的第二部分关注的是另一类可以通过迭代得到的对象：无限替代序列和自相似平铺。现在迭代过程包括用一个符号块替换每个符号，或者用一片瓦片替换每个瓦片，然后用限制对象创建一个我们研究的迷人的动力系统。除了它们内在的美，这些系统在物理学中也有应用。