Research in One-Dimensional and Complex Dynamics and Thermodynamical Formalism

这个项目第一部分的目标是将热力学形式论和TeichMuller理论中的知识和技术结合起来，发展新的结果，并将它们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型行列式理论。请注意，圆上的Zygmund函数空间可视为泛TeichMuller空间的切线空间。希望转移算符的谱性质的知识可以应用于TeichMuller理论的研究，反之亦然。一个特别有趣的例子是当转移算子是泛Teichmuller空间的非线性映射的切映射时。在研究不可重整化二次多项式的热力学性质时，还研究了一种称为Yoccoz拼图的方法，作为马尔可夫分拆的替代。在项目的第二部分，研究了无穷可重正化点上的Mandelbrot集和无穷可重正则二次多项式的Julia集的拓扑性质，因为这是证明Mandelbrolt集是局部连通的唯一剩余点。研究了具有任意指数和非对称性的折叠映象的广义Feigbaum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中的普遍规律：在像二次多项式族这样的单参数系统中(这样的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)，演化成混沌运动的规则运动可以用一个普适数字来表征。这种新的范式提供了丰富的数学问题。反过来，对这些问题的研究也加深了对与普遍性相关的概念的理解，为马可的进一步发展提供了一些新的工具。该项目第一部分的目标是解决一些悬而未决的问题，并取得与普遍性有关的新成果。更具体地说，研究了一维复杂动力系统的一些热力学性质和几何性质。Mandelbrot集的发现是快速计算机发展的直接重要结果之一，它打破了一些传统几何概念发展的障碍。传统上，几何学的数学研究集中在圆、线和球体等平滑对象上。但在现实世界中，当你看着显微镜看几何物体时，大多数几何物体的形状更具分形性。Mandelbrot集不仅打破了研究分形几何的视觉和心理障碍，而且在一大类系统中具有普遍的分形结构。然而，对曼德尔布罗特集几何结构的完全理解还远未完成。项目的第二部分的目的是研究Mandelbrot集的拓扑结构，并研究Julia集对应的无限可重正化点的拓扑结构。

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.