Renormalization and rigidity in one-dimensional dynamics

一维动力学中的重正化和刚性

• 批准号: 328565-2011
• 负责人: Khanin, Konstantin
• 金额: $ 1.31万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2012
• 资助国家: 加拿大
• 起止时间: 2012-01-01 至 2013-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=508692
• 关键词: Renormalization; rigidity; dimensional; dynamics

Renormalization theory was first developed in physics in the context of quantum field theory and statistical mechanics. At the end of 70's in the pioneering work of M. Feigenbaum the renormalization ideas paved their way into the theory of dynamical systems. By now renormalization is one of the most powerful methods in the asymptotic analysis of dynamical systems. The main idea is to study long-term behaviour of dynamical systems in the rescaled coordinate system near places where trajectories return to the initial position on a smaller and smaller scales. It turns out that after such a rescaling the dynamical system exhibit universal behaviour which depends only on topological (or, combinatorial) type, and local structure of critical, or singular, points. Remarkably, in many examples such universal behaviour can be studied rigorously. Roughly speaking, it can be described in terms of fixed points of renormalization. Simple fixed points appear in the smooth case. Such fixed points are related to the theory of linearization which form an essential part of the celebrated Kolmogorov-Arnold-Moser theory. The case when dynamical system has critical, or singular, points is of special interest since it corresponds to highly nontrivial renormalization fixed points. In the case of maps with one critical point a beautiful renormalization theory was developed in 1990's by Sullivan, McMullen and Lyubich. The proposed research project aims to extend significantly our understanding of the renormalization behaviour. We intend to study systems with many singular, or critical points, and hope to reveal a rich structure of the renormalization attractor in this case. We also plan to study global universality in the case of general critical points, which is one of the central open problems in the theory of renormalization.

重整化理论最早是在量子场论和统计力学的背景下发展起来的。70年代末，在费根鲍姆的开创性工作中，S的重正化思想为动力系统理论铺平了道路。重整化是目前动力系统渐近分析中最有力的方法之一。其主要思想是在重新标度的坐标系中研究动力系统的长期行为，这些地方的轨迹在越来越小的尺度上返回到初始位置。结果表明，在这样的重新标度后，动力系统表现出仅依赖于拓扑(或组合)类型和临界点或奇异点的局部结构的普遍行为。值得注意的是，在许多例子中，这种普遍行为可以得到严格的研究。粗略地说，它可以用重整化的不动点来描述。在光滑情况下出现简单不动点。这种不动点与线性化理论有关，线性化理论是著名的科尔莫戈洛夫-阿诺德-莫泽理论的重要组成部分。动力系统存在临界点或奇异点的情况是特别有趣的，因为它对应于高度非平凡的重整化不动点。在具有一个临界点的映射的情况下，Sullivan，McMullen和Lyubich在1990年的S中发展了一个美丽的重整化理论。拟议的研究项目旨在极大地扩展我们对重整化行为的理解。我们打算研究具有许多奇点或临界点的系统，并希望在这种情况下揭示重整化吸引子的丰富结构。我们还计划在一般临界点的情况下研究全局普适性，这是重整化理论中的中心开放问题之一。