Research in One-Dimensional and Complex Dynamics and Thermodynamical Formalism

一维复杂动力学与热力学形式主义研究

• 批准号: 9701145
• 负责人: Yunping Jiang
• 金额: $ 7.5万
• 依托单位: CUNY Queens College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9701145&HistoricalAwards=false
• 关键词: Research; Dimensional; Complex; Dynamics; Thermodynamical

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.

简介 该项目第一部分的目标是通过结合热力学形式主义和Teichmuller理论的知识和技术来开发新的成果，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，圆上的齐格蒙德函数空间可以被视为泛泰希穆勒空间的切空间。人们希望转移算子的谱性质的知识可以应用于Teichmuller理论的研究，反之亦然。一个特别有趣的情况是当一个转移算子是泛Teichmuller空间的非线性映射的正切映射时。一种称为Yoccoz难题的技术也被调查作为一个替代的马尔可夫划分的研究不可重正化的二次多项式的数学性质。在该项目的第二部分中，Mandelbrot集在无限重正化点和 研究了无穷可重正化二次多项式的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。本项目研究了具有任意指数和非对称性的折叠映射的一个推广的Feigenbaum猜想。 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：在一个单参数系统中，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适数来表征。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，并为进一步的研究开发了一些新的工具。这个项目的第一部分的目标是解决一些未解决的问题，并开发与普遍性有关的新结果。更具体地说，一维和复杂的动力系统的一些物理和几何性质进行了研究。曼德尔布罗特集的发现是计算机发展的直接重要成果之一，它打破了发展某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、线和球体。但在真实的世界中，大多数几何物体都是分形的， 当一个人在显微镜下观察一个几何物体时，Mandelbrot集不仅打破了分形几何研究的视觉和心理障碍，而且承载了一大类系统的普遍分形结构。然而，对曼德布罗特集几何结构的完整理解还远未完成。 本项目的第二部分的目标是研究Mandelbrot集的拓扑结构，并研究相应的无穷重正化点的Julia集的拓扑结构。