Topics in Fractal Geometry, Dynamics, and Ergodic Theory

本课题主要研究动力系统的维数理论和遍历的平铺替换理论。第一个方向是研究非共形动力系统的不变量集的维数特征及其上的自然测度。这些问题涉及到有重叠的自相似集和康托尔集的算术和问题。第二个方向涉及非周期的自亲和，它们的概括，和符号类似物。这两个方向之间有许多联系；例如，自仿射瓷砖通常具有分形边界，以及数论问题(例如：（pv -number的出现）在这两个区域都很突出。所要研究的问题涉及到平铺和置换的动力学和衍射光谱。数学混沌理论处理的是动力系统中混沌运动的出现（这是物理学、生物学、经济学等领域各种现象的数学模型）。这些混沌运动通常与各种“分形”现象有关。现代维度理论使用各种各样的工具来研究这些运动的精细结构及其不变量集（如“奇异吸引子”）。尽管如此，这个“分形混沌世界”的许多方面仍然是神秘的，拟议的研究旨在进一步深入了解它们。项目的第二部分涉及所谓的“非周期秩序”，其标志是在20世纪80年代早期发现的一类引人注目的固体，被称为准晶体，它们表现出具有禁止的10倍对称的尖锐衍射光谱。20世纪70年代在一个纯数学的发展中发现的彭罗斯瓷砖，被证明是某些准晶体的一个很好的模型，但是在数学和物理方面仍然存在许多悬而未决的问题。本研究的第二部分运用遍历理论和符号动力学的方法来探讨其中的一些问题。

This project addresses problems in dimension theory ofdynamical systems and ergodic theory of tilings and substitutions.In the first direction, it is proposed to study the dimensioncharacteristics of invariant sets for non-conformal dynamicalsystems and natural measures on them. These questions are relatedto problems on self-similar sets with overlap and arithmetic sumsof Cantor sets. The second direction concerns non-periodic self-affinetilings, their generalizations, and symbolic analogs. There are manylinks between the two directions; for instance, self-affine tiles oftenhave fractal boundaries, and number-theoretic issues(e.g., the appearance of PV-numbers) are prominent in both areas.The problems to be studied are concerned with dynamical and diffractionspectra of tilings and substitutions.The mathematical chaos theory deals with the appearance of chaoticmotions in dynamical systems (which are mathematical models ofvarious phenomena in physics, biology, economics, etc.). Thesechaotic motions are often associated with various "fractal" phenomena.The modern dimension theory uses a wide variety of tools tostudy the fine structure of these motions and their invariant sets(such as "strange attractors"). Still, many aspects of this"fractal chaotic world" remain mysterious, and the proposed researchaims to get further insights into them. The second part of the projectis concerned with the so-called "aperiodic order," the hallmarkof which is the remarkable class of solids, discovered in theearly 1980's and called quasicrystals, which exhibit sharp diffractionspectrum with a forbidden 10-fold symmetry. Penrose tilings,discovered in the 1970's in a purely mathematical development, turnedout to be a good model for certain quasicrystals, but many open problemsremain, both on the mathematical and on the physical side.The second part of the proposed research investigates some of theseproblems using the techniques of ergodic theory and symbolic dynamics.