Digits, distributions, and rigidity in dynamical systems

动力系统工具和技术是现代科学和技术所不可或缺的，因为它们能够对从空间飞行到细胞生长到经济冲击的各种过程进行准确建模。动力系统的广泛范围和由它们产生的大量数据使得它们的定量和计算方面的研究成为一个重要的研究领域。在这方面，拟议的方案将以几种方式作出贡献。研究主题可以大致分组，因为它们分别与数字，分布和刚性有关。数字。该计划的主要长期目标是建立一个全面的数字理论，其性质和分布，为动力系统。在这种情况下，一个突出的现象是一个特定的对数分布，通常被称为本福德定律的惊人的普遍性。关于后者，许多数学问题仍然存在，该计划将解决，尤其是在申请人以前的发现补助金下获得的结果。动态系统理论的最新进展将被用来精确地量化由熟悉的一维和多维系统生成的数字所固有的随机性。分配。另一个重要的目标是进行深入分析的概率引起的数字统计，特别是在圆上，离散化和（约束）的近似属性，目前只有部分理解。这一分析将包括研究新的定量概念，如失真（相对于尺度，基础或总和）和概率测度的传播，目前还没有理论框架，但也包括均匀分布理论中经典结果的改进。刚度该计划的第三个共同主题是刚性。非正式地说，在刚性的情况下，具有某些性质的物体比合理预期的要少得多。在动力学中已经建立了许多深度刚度的结果。对于这个现有的动物园，该计划将引入两个新的物种，即数字刚性和等效刚性。这两种形式的刚性都为线性系统的行为提供了意想不到的见解，可以说线性系统是在整个科学中广泛使用的最简单的一类动力系统。拟议方案的核心部分是高度相互关联的。它们一起提供了对动力系统重要的定量方面的深入理解。根据该计划建立的结果将感兴趣的其他领域的纯数学和应用数学，特别是数论，最佳运输和统计。它们将导致新的或更可靠的统计测试和数据可验证性检查（例如，关于会计，选举或科学数据中的潜在欺诈）以及改进的数值方案，这将使所有使用动态系统作为真实世界过程模型的科学家受益。*** *** *** *** *** *** **

Dynamical systems tools and techniques are indispensable to modern science and technology as they enable accurate modelling of processes ranging from space flight to cell growth to economic shocks. The wide scope of dynamical systems and the vast amounts of data generated by them have made the study of their quantitative and computational aspects an important research area. To this area the proposed program will contribute in several ways. The research topics can roughly be grouped as they relate, respectively, to digits, distributions, and rigidity.***Digits. The program's main long-term goal is to establish a comprehensive theory of digits, their properties and distributions, for dynamical systems. A prominent phenomenon in this context is the surprising ubiquity of one particular logarithmic distribution often referred to as Benford's Law. Regarding the latter, many mathematical questions remain which the program will address, building not least on results obtained under the applicant's previous Discovery Grant. Recent advances in the theory of dynamical systems will be utilized to precisely quantify the randomness inherent to the digits generated by familiar one- and multi-dimensional systems.***Distributions. Another important goal is to carry out an in-depth analysis of probabilities induced by digit statistics, notably on the circle, the discretization and (constrained) approximation properties of which are only partly understood at present. This analysis will include the study of novel quantitative concepts such as distortion (with respect to scale, base, or sum) and spread of probability measures for which no theoretical framework exists at present, but also refinements of classical results in uniform distribution theory.***Rigidity. A third common theme of the program is rigidity. Informally put, in case of rigidity far fewer objects with certain properties exist than might reasonably be expected. Many deep rigidity results have been established in dynamics. To this existing zoo, the program will introduce two new species, namely digit rigidity and equivalence rigidity. Both forms of rigidity provide unexpected insights into the behavior of linear systems, arguably the simplest class of dynamical systems widely used throughout the sciences.***The core parts of the proposed program are highly interrelated. Together, they provide a deepened understanding of important quantitative aspects of dynamical systems. The results established under this program will be of interest to other areas of pure and applied mathematics, notably number theory, optimal transport, and statistics. They will lead to new or more reliable statistical tests and data plausibility checks (e.g., with regard to potential fraud in accounting, election, or scientific data) as well as to improved numerical schemes that will benefit all scientists who use dynamical systems as models for real-world processes.*** *** *** *** *** *** *** **