Measures, Dimension, and Ergodic Theory

测量、维数和遍历理论摘要本项目是几何测量理论、分形几何和遍历理论相互作用的研究领域。迭代函数系统，俗称“混沌博弈”，为研究分形现象提供了一个方便的框架。这类系统的不变测度包括在许多概率和动态模型中出现的独立的、同分布的随机变量的无穷和的乘积，以及随机矩阵乘积的平稳测度。中心问题之一是确定这种度量何时是绝对连续的。一个密切相关的研究方向是关注吸引子的拓扑性质，特别是自相似集和自仿射集。尤其具有挑战性的是那些不均匀收缩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换和平铺动力系统，以及所有自同构的代数编码。我们对所谓的非pisot情况特别感兴趣，在这种情况下，建议使用自仿射集上定义的β变换的一些变体。要在平面上玩“混沌游戏”，应该指定一组平面变换（称为迭代函数系统），并以规定的概率随机迭代地应用其中一个变换。在某些技术条件下（称为“平均收缩”），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。这是在计算机屏幕上绘制分形的一种流行方法，但它不仅仅是一个游戏：迭代函数广泛应用于信号处理、图像压缩和仿真算法，以及数学动力系统理论。我们在电脑屏幕上看到的图像实际上是一个灰调图像，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。一个相关的研究方向是对混沌博弈过程中出现的分形物体的“动物园”进行分类。项目的第二部分涉及另一类可以通过迭代获得的对象：无限替换序列和自相似平铺。现在迭代过程包括将每个符号替换为一个符号块，或将每个瓦片替换为一小块瓦片。然后用限制对象来创建一个我们研究的迷人的动力系统。除了它们内在的美，这样的系统在物理学中也有应用。

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.