Fractals and Tilings

本课题主要研究分形几何中的几个问题，以及平铺和替换理论。伯努利卷积可能是最简单的具有重叠的自相似测度的例子。它们出现在数学和科学的许多不同分支中，包括信号处理、数论和谐波分析。该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扩展有关。它还将处理伯努利卷积的随机类比，例如“以指数递减的步骤进行分支随机行走”。在平铺和平铺动力系统领域，提出了完成自仿射平铺展开映射的表征，这将导致相关动力系统问题的进展。这个项目是由瑟斯顿和凯尼恩发起的，它与几何动力学、层合理论和刚度理论有联系。另一个要讨论的问题是关于取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具有复杂的结构，不能用解析的方法来描述。Mandelbrot为这样的集合引入了术语“分形”，并证明了它们可以用于建模许多自然现象。另一方面，自19世纪以来，皮亚诺、康托尔和韦尔斯特拉斯等数学家研究了数学分形（没有使用这个术语）。分形对象中最简单和最基本的是自相似集和测度。“自相似”的意思是，粗略地说，把物体“放大”，人们可以在任意小的尺度上看到相似的图片。同样重要的是概率定义的分形，它表现出统计上的自相似性；它们通常在应用程序中充当更现实的模型。这个项目将解决关于分形集结构的几个微妙问题。提案的第二部分是关于平铺和替换；它与分形有关，但也与组合学和离散几何有关。自相似的铺层，如彭罗斯铺层，已经被用作准晶体的模型，相关的动力系统和它们的光谱特性被证明与固态物理有关。

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.