Rigidity, Universality, and Complexity in Dynamics

我的研究主题是研究由简单动力学产生的复杂行为。动力学的一个关键主题是研究从动力系统的复杂行为中产生的普遍规律。这种普遍性在物理学中是众所周知的。它们可以被理解为一种自组织机制，产生于混沌行为。也许其中最著名的例子是一维动力系统中的feigenbaum型普遍性，这在过去三十年中得到了广泛的研究。这些研究建立了一个新的领域，被称为重整化理论，它彻底改变了动力学的研究。在一系列的论文中，我完全解决了一维普遍性的两个主要情况之一（临界圆图的兰福德普遍性）。与普遍性密切相关的是刚性现象。它假定具有相同普遍性质的动力系统是由坐标的平滑变化联系起来的。我的建议概述了一维动力学中普遍性和刚性的一些核心问题的研究计划。众所周知，一个由简单动力学定律指导的物理系统是很难建模和预测的。一个著名的例子是洛伦兹系统，它描述了一个非常简化的天气模型。即使模型初始条件的微小差异也可能导致天气预报的巨大差异（所谓的蝴蝶效应）。然而，人们普遍认为，虽然个别的预测可能会有所不同，但动态的总体情况是相同的，并且可以进行数值模拟。在洛伦兹例子中，全球天气模式是由洛伦兹蝴蝶吸引子描述的，这是一个定义良好的数学对象。这很容易计算（的确，这幅画已经出名了）。在最近的一系列工作中，我们已经测试了一个直观的假设，即一个简单动力系统的吸引子可以在计算机上建模。令人惊讶的是，对于一个典型的家庭，即所谓的茱莉亚家庭，这通常是错误的。这项工作开启了对实践者、理论动力学家和计算机科学家都很重要的研究篇章。我的建议为进一步研究这些具有挑战性的问题提出了一个计划。

The main theme of my research is the study of complex behaviour generated by simple dynamics. One of the key themes in Dynamics is the study of universal laws which emerge from the complex behaviour of dynamical systems. Such universalities are well-known in physics. They can be understood as a self-organization mechanism, which arises from chaotic behaviour. Perhaps the most famous examples of them are Feigenbaum-type universalities in one-dimensional dynamical systems, which have been intensively studied in the last three decades. These studies founded a new field, known as renormalization theory, which revolutionized the study of dynamics. In a series of papers I completely resolved one of the two main cases of one-dimensional universalities (the Lanford's universality for critical circle maps). Closely related to universality is the phenomenon of rigidity. It postulates that dynamical systems which exhibit the same universal properties are related by a smooth change of coordinates. My proposal outlines a program of study of some of the central questions of universality and rigidity in one-dimensional dynamics.It is a well-known fact that a physical system guided by simple dynamical laws can be difficult to model and predict. A famous example of this is the Lorenz system, which describes a very simplified weather model. Even tiny differences in the initial conditions of the model may lead to a widely diverging weather predictions (a so-called butterfly effect). Nevertheless, it is generally believed that while individual predictions may differ, the overall picture of the dynamics remains the same, and may be simulated numerically. In the Lorenz example, the global weather patterns are described by the Lorenz butterfly attractor, which is a well-defined mathematical object. It is easily computed (indeed, the picture has become famous). In a series of recent works, we have tested the intuitive assumption that the attractor of a simple dynamical system can be modeled on a computer. Surprisingly, for an archetypical family, known as Julia sets, this is generally false. This work opens a chapter of research which is important for practitioners, as well as for theoretical dynamicists and computer scientists. My proposal lays out a plan of further study of these challenging questions.