Statistical and Number-Theoretical Properties of Dynamical Systems

概率论中有一些经典的结果，使人们能够预测随机现象的平均长期行为。该计划的目的是研究某类确定性现象在多大程度上可以被视为随机现象，以便了解原始系统的特性。可以用这种方法处理的确定性数列的例子出现在数论和量子力学的问题中；这些提供了这个项目的重点。在这些情况下，可以使用概率论的语言来表达本项目所解决的问题，但所需的方法具有不同的性质，并且根据应用而变化，来自动力系统理论。PI已经成功地在这些主题上取得了一些成果，并在动力学和数论之间快速发展的相互作用中发挥了积极作用。PI计划研究数论起源的确定性过程的统计性质，如Weyl指数和、乘法算术函数和连分数展开。虽然由动力系统产生的过程的几个特性是已知的，但PI的目标是研究更精细的细节并回答该领域的新问题。应该强调的是，有时极限分布与经典概率论的分布一致（这意味着感兴趣的确定性现象是一个很好的随机代理）；另一方面，通常会得到非标准的发行版。这意味着经典的概率方法不能应用，理论需要新的思想和工具。本课题研究的主要方法是动力学性质的，如均匀空间中单参数流的等分布、拉特纳理论和谱方法。PI已经成功地在这些主题上取得了一些成果，并在动力学和数论之间快速发展的相互作用中发挥了积极作用。这一建议的多面性将成为PI发起活动（如互动式研讨会、专题讨论会、讲习班）的机会，目的是普及研究课题及其应用。

There are classical results in probability theory that allow one to predict the average long term behavior of random phenomena. The aim of the proposed project is to study to what extent certain classes of deterministic phenomena can be treated as if they were random in order to understand properties of the original system. Examples of deterministic sequences of numbers that can be approached this way arise in problems of number theory and quantum mechanics; these provide the focus of this project. In these settings, it is possible to phrase the questions addressed by this project using the language of probability theory, but the methods required are of a different nature and vary according to the application, coming from the theory of dynamical systems. The PI has already successfully achieved several results on these topics, and has been active in the fast-advancing interplay between dynamics and number theory.The PI plans to study the statistical properties of deterministic processes of number theoretical origin such as Weyl exponential sums, multiplicative arithmetic functions, and continued fraction expansions. Although several properties of processes arising from dynamical systems are known, the PI aims to investigate finer details and answer novel questions in the field. It should be stressed that sometimes the limiting distributions agree with the classical ones of probability theory (which means that the deterministic phenomenon of interest is a good stochastic proxy); on the other hand, often one gets non-standard distributions. This means that the classical probabilistic methods cannot be applied and the theory requires new ideas and tools. The main methods in this project are of dynamical nature, such as equidistribution of one-parameter flows in homogeneous spaces, Ratner theory, and spectral methods. The PI has already successfully achieved several results on the topics, and has been active in the fast-advancing interplay between dynamics and number theory. The multifaceted aspects of this proposal will serve as an opportunity for the PI to initiate activities (such as interactive seminars, symposia, workshops) with the aim of popularizing the research subjects and their applications.