Digits, distributions, and rigidity in dynamical systems

动力系统工具和技术对于现代科学和技术是不可或缺的，因为它们能够准确地模拟从太空飞行到细胞生长再到经济冲击的各种过程。动力系统的广泛范围和由此产生的海量数据使其定量和计算方面的研究成为一个重要的研究领域。拟议的方案将在几个方面对这一领域作出贡献。研究主题可以大致分组，因为它们分别涉及数字、分布和刚性。*数字。该计划的主要长期目标是为动力系统建立一个关于数字、它们的性质和分布的全面理论。在这种情况下，一个突出的现象是一种特殊的对数分布令人惊讶地普遍存在，通常被称为本福德定律。对于后者，许多数学问题仍然存在，该计划将解决这些问题，尤其是建立在申请者之前的发现补助金下获得的结果之上。动力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产生的数字所固有的随机性。另一个重要的目标是深入分析由数字统计引起的概率，特别是关于圆的概率，目前对其离散化和(约束)逼近性质只有部分了解。这种分析将包括对新的量化概念的研究，如扭曲(相对于标度、基数或总和)和概率度量的扩散，目前尚无理论框架，但也包括对均匀分布理论中经典结果的改进。*刚性。该计划的第三个共同主题是僵化。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。许多深刚性结果已经在动力学中得到了证实。对这个现有的动物园，该计划将引入两个新物种，即数字刚性和等价刚性。这两种形式的刚性都提供了对线性系统行为的意想不到的见解，线性系统可以说是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高度相关。合在一起，它们加深了对动力系统重要数量方面的理解。根据这一计划建立的结果将对纯数学和应用数学的其他领域感兴趣，特别是数论、最优运输和统计学。它们将导致新的或更可靠的统计测试和数据可信性检查(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，这将使所有使用动力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.*** *** *** *** *** *** *** **