Topics in Fractal Geometry, Dynamics, and Ergodic Theory

本课题研究了动力系统的维度理论和遍历平铺与置换理论中的问题。在第一个方向上，提出了研究非共形动力系统不变集的维度特征及其自然度量。这些问题与具有重叠的自相似集上的问题和Cantor集的算术和有关。第二个方向涉及非周期自仿射、它们的推广和符号类比。这两个方向之间有许多联系；例如，自仿射瓷砖往往具有分形边界，数论问题(例如，PV数的出现)在这两个领域都是突出的。要研究的问题涉及瓷砖和代换的动力学和衍射谱。数学混沌理论研究动力系统中混沌运动的出现(这是物理、生物、经济等各种现象的数学模型)。这些运动常常与各种“分形”现象联系在一起，现代维理论使用各种工具来研究这些运动及其不变集(如“奇异吸引子”)的精细结构。尽管如此，这个“分形混沌世界”的许多方面仍然是神秘的，拟议中的研究团队将进一步深入了解它们。项目的第二部分涉及到所谓的“非周期有序”，其标志是一类非凡的固体，在1980年代初被发现，被称为准晶，它表现出尖锐的衍射谱，具有禁止的10倍对称。在20世纪70年代，S在一项纯粹的数学发展中发现了彭罗斯瓷砖，它被证明是某些准晶的良好模型，但在数学和物理方面仍有许多悬而未决的问题。拟议的研究的第二部分使用遍历理论和符号动力学的技术来研究其中的一些问题。

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.