Quantitative aspects of dynamical systems

动力系统为许多应用科学提供了基本模型和强有力的工具。由于动力学的范围不断扩大，定量和计算方面现在构成了一个充满活力的研究领域。在这一领域内，拟议的研究方案将 (I)在动力系统的背景下发展本福德定律的全面理论； (本福德定律是一个有充分记录的经验观察，描述了数字数据中有效数字的特殊分布；虽然它的重要和令人惊讶的影响，例如在数值分析和统计中，最近吸引了很多关注，但许多关于本福德定律的自然数学问题仍然悬而未决。) (Ii)通过一种新的约束量化方法来研究有限数据集和动态相关的度量； (3)介绍和研究有限时间动力学的基本概念和工具，如双曲性和谱，并将它们应用于非自治分支和驱动台球。 总体而言，该计划将加深对动力系统重要数量方面的理解，进而导致更可靠的统计测试和似是而非的检查(例如，会计和选举舞弊)，以及改进的数值程序，无论动力系统作为现实世界过程的模型。

Dynamical systems provide fundamental models and powerful tools for many applied sciences. Due to the ever-expanding scope of dynamics, quantitative and computational aspects now constitute a vibrant research area. Within this area, the proposed research program will (i) develop a comprehensive theory of Benford's Law in the context of dynamical systems; (Benford's Law is a well-documented empirical observation stating a particular distribution of significant digits in numerical data; while its important and surprising implications, e.g. in numerical analysis and statistics, have recently attracted a lot of attention, many natural mathematical questions concerning Benford's Law are still open.) (ii) study finite data sets and dynamically relevant measures by means of a novel approach to constrained quantization; (iii) introduce and study fundamental concepts and tools for finite-time dynamics, e.g. hyperbolicity and spectrum, and apply them to nonautonomous bifurcations and driven billiards. Overall, the program will bring about a deepened understanding of important quantitative aspects of dynamical systems, leading in turn to more reliable statistical tests and plausibility checks (e.g. for accounting and election fraud) as well as improved numerical procedures wherever dynamical systems serve as models for real-world processes.