Research in One-Dimensional and Complex Dynamics and Thermodynamical Formalism

本项目第一部分的目标是结合热力学形式论和Teichmuller理论的知识和技术，开发新的结果，并将其应用于一维和复杂动力学的研究。研究了圆上Zygmund函数空间上的广义Fredholm行列式理论。注意，Zygmund函数在圆上的空间可以看作是通用Teichmuller空间的切空间。希望传递算子谱性质的知识可以应用于Teichmuller理论的研究，反之亦然。一个特别有趣的情况是当传递算子是一个泛Teichmuller空间的非线性映射的切映射时。在研究不可重整二次多项式的热力学性质时，还研究了一种称为Yoccoz谜题的技术，作为马尔科夫分割的替代。在项目的第二部分，研究了无限可重整点上的Mandelbrot集和无限可重整二次多项式的Julia集的拓扑性质，因为这是证明Mandelbrot集是局部连通的仅有的剩余点。研究了任意指数不对称折叠映射的广义Feigenbaum猜想。对一维动力学的研究导致了许多物理定律的发现。最近的一个发现是混沌现象的普遍规律：在一个单参数化的系统中，比如二次多项式族（这种系统出现在流体和天体力学、化学和生物学以及社会科学中），规则运动演变成混沌运动可以用一个普遍数来表征。这种新的范例提供了丰富的数学问题。反过来，对这些问题的研究使人们对与普遍性有关的概念有了更深入的理解，并开发了一些新的工具，以取得进一步的进展。本项目第一部分的目标是解决一些尚未解决的问题，并发展与普遍性有关的新结果。更具体地说，研究了一维和复杂动力系统的一些热力学和几何性质。Mandelbrot集合的发现，是快速计算机发展的直接重要成果之一，它打破了发展一些传统几何概念的障碍。传统上，几何的数学研究集中在光滑的物体上，如圆、线和球。但在现实世界中，当我们用显微镜观察一个几何物体时，大多数几何物体的形状都是分形的。Mandelbrot集合不仅打破了分形几何研究的视觉和心理障碍，而且承载了大量系统中普遍存在的分形结构。然而，对Mandelbrot集合的几何结构的完整理解还远远没有完成。该项目第二部分的目标是研究Mandelbrot集合的拓扑结构，并研究相应的无限可重整点的Julia集合的拓扑结构。

Abstract Jiang The goal of the first part of this project is to develop new results by combining knowledge and techniques from both Thermodynamical Formalism and Teichmuller Theory and apply them to the study of one-dimensional and complex dynamics. A generalized Fredholm Determinant Theory on the space of Zygmund functions on the circle is investigated. Notice that the space of Zygmund functions on the circle can be treated as the tangent space of the universal Teichmuller space. It is hoped that the knowledge of the spectral properties of transfer operators may be applied to the study of Teichmuller theory and vice versa. An especially interesting case is when a transfer operator is the tangent map of a non-linear map of the universal Teichmuller space. A technique called Yoccoz puzzles is also investigated as a replacement of the Markov partitions in the study of the thermodynamical properties of a non-renormalizable quadratic polynomial. In the second part of the project, the Mandelbrot set at infinitely renormalizable points and the topological properties of the Julia sets of infinitely renormalizable quadratic polynomials are investigated because these are the only remaining points to prove that the Mandelbrolt set is locally connected. A generalized Feigenbaum conjecture for folding mappings with arbitrary exponent and asymmetry is investigated in the project. The study of one-dimensional dynamics has lead to the discovery of many physical laws. One recent discovery is the universal law in chaotic phenomena: in a one-parameterized system like the family of quadratic polynomials (such a system appears in fluid and celestial mechanics, in chemistry and biology, and in social sciences), regular motions evolving into a chaotic motion can be characterized by a universal number. This new paradigm provides a rich supply of mathematical problems. In return, the study of these problems gives a deeper understanding of concepts connected with universality and develops some new tools to ma ke further progress. The goal of the first part of this project is to attack some unsolved problems and to develop new results concerned with universality. More specifically, some thermodynamical and geometric properties of one-dimensional and complex dynamical systems are investigated. The discovery of the Mandelbrot set, which was one of the direct important results from the development of a fast computer, breaks the barrier to develop some traditional geometric concepts. Traditionally, the mathematical study of geometry concentrates on smooth objects like circles, lines, and spheres. But in the real world, most geometric objects are more fractal in shape when one looks into the microscope at a geometric object. The Mandelbrot set not only breaks the visual and psychological barrier of the study of fractal geometry but also carries the universal fractal structure in a large class of systems. However, a complete understanding of the geometric structure of the Mandelbrot set is far from being finished. The goal of the second part of the project is to study the topological structure of the Mandelbrot set, and also to investigate the topological structure of the Julia set for a corresponding infinitely renormalizable point.