Fractals and Tilings

分形和平铺

• 批准号: 0654408
• 负责人: Boris Solomyak
• 金额: $ 17.25万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0654408&HistoricalAwards=false
• 关键词: Fractals; Tilings

This project is devoted to several problems in fractal geometry and the theory of tilings and substitutions. Bernoulli convolutions are, perhaps, the simplest examples of self-similar measures with overlaps. They appeared in many different branches of mathematics and science, including signal processing, number theory and harmonic analysis.This project will investigate novel phenomena related to their multifractal spectrum, which are linked to expansions in non-integer bases. It will also address random analogs of Bernoulli convolutions, such as "branching random walks with exponentially decreasing steps."In the area of tilings and tiling dynamical systems, it is proposed to complete the characterization of expansion maps for self-affine tilings, which should lead to advances in problems on associated dynamical systems. This program was started by Thurston and Kenyon, and it has connections with geometric dynamics, theory of laminations, and rigidity theory.Another question which will be addressed is about the nature of the continuous spectral component of substitution dynamical systems.Many of the sets which appear in modern mathematics have complicated structure and cannot be described analytically. Mandelbrot introduced the term "fractal" for such sets and demonstrated that they can be useful for modeling many natural phenomena. On the other hand, mathematicians such as Peano, Cantor and Weierstrass, investigated mathematical fractals (without using this term) since the 19th century.The simplest and most basic among fractal objects are self-similar sets and measures. "Self-similarity" means, roughly speaking, that "zooming in"into the object one can see similar pictures at arbitrarily small scales.Also important are fractalsdefined probabilistically, which exhibit statistical self-similarity; they often serve as more realistic models in applications. This project will address several delicate questions on the structure of fractal sets.The second part of the proposal has to do with tilings and substitutions; it has links to fractals, but also to combinatorics and discrete geometry.Self-similar tilings, such as the Penrose tilings, have been used as models for quasicrystals, and the associated dynamical systems and their spectral properties turned out to be relevant for solid-state physics.

这个项目致力于研究分形几何中的几个问题，以及拼接和置换理论。伯努利卷积也许是具有重叠的自相似度量的最简单的例子。它们出现在许多不同的数学和科学分支中，包括信号处理、数论和调和分析。这个项目将研究与它们的多重分形谱相关的新现象，这些现象与非整数基的展开有关。它还将解决与Bernoulli卷积类似的随机问题，例如“具有指数递减步长的分支随机游动”。在瓦片和瓦片动力系统领域，有人建议完成自仿射瓦片的扩展映射的刻画，这将导致相关动力系统问题的进展。这个程序是由瑟斯顿和凯尼恩发起的，它与几何动力学、层合理论和刚性理论有关。另一个要解决的问题是关于替代动力系统的连续谱分量的性质。现代数学中出现的许多集合都具有复杂的结构，不能用解析的方式描述。曼德尔布洛特为这类集合引入了“分形”一词，并证明它们可用于对许多自然现象进行建模。另一方面，自19世纪以来，皮亚诺、康托和魏尔斯特拉斯等数学家研究了数学分形学(不使用这一术语)。最简单和最基本的分形体是自相似集和测度。“自相似”指的是，粗略地说，“放大”一个人可以在任意小的尺度上看到相似的图片。同样重要的是概率地定义的分形图，它表现出统计上的自相似性；它们通常在应用中用作更真实的模型。这个项目将解决几个关于分形集结构的微妙问题。提案的第二部分与分形和置换有关；它与分形学有联系，但也与组合学和离散几何有关。自相似分块，如彭罗斯分块，已被用作准晶的模型，相关的动力系统及其光谱特性被证明与固体物理有关。