Renormalization and rigidity in one-dimensional dynamics

重整化理论最初是在量子场论和统计力学的背景下发展起来的。70年代末，M.费根鲍姆的重整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。其主要思想是研究动力系统的长期行为在重新标度的坐标系附近的地方，轨迹返回到初始位置上一个越来越小的尺度。事实证明，经过这样的重新缩放的动力系统表现出普遍的行为，这只取决于拓扑（或组合）类型，和局部结构的关键，或奇异点。值得注意的是，在许多例子中，这样的普遍行为可以被严格地研究。粗略地说，它可以用重整化的不动点来描述。简单的不动点出现在光滑的情况下。这种不动点与线性化理论有关，线性化理论是著名的Kolmogorov-Arnold-Moser理论的重要组成部分。当动力系统有临界点或奇异点时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20世纪90年代发展了一个漂亮的重整化理论。拟议的研究项目旨在显着扩展我们的理解重整化行为。我们打算研究具有许多奇异点或临界点的系统，并希望在这种情况下揭示重整化吸引子的丰富结构。我们还计划研究一般临界点情形下的整体普适性，这是重整化理论中的中心开放问题之一。

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.