Smooth Dynamics, Dimension Theory, Geodesic Flows, and Mathematical Biology

在严格意义上，动力系统的不变集一般不是自相似的。然而，在与他人的合作中，PI已经表明，在一些重要的情况下，这些集合可以分解为每个具有一种比例对称的子集。具有这种结构的集合称为多重分形（MF）。该建议包括继续研究这些多重分形的精细结构，并尝试使用MF分析来提供对动力系统的新见解，并可能产生动力系统的新（物理）分类。此外，所提出的工作还涉及数学生物学中出现的各种维数理论问题，以及非负曲率与复杂测地线流动动力学之间关系的研究。对于后者，我们将研究（in）著名的（xy）^2哈密顿系统，这是一个经典的杨-米尔斯场模型，它的轨道等效于非负曲面上的测地线流。许多物理和生物系统（包括湍流、植物根系、受应力的金属碎片、大脑的核磁共振图像、云层和宇宙中的星系）似乎具有某种复杂的分形结构。数学上这样的对象被称为多重分形。在之前的工作中，首席研究员为研究一些重要的多重分形类提供了严格的数学基础。计划是将这项工作扩展到更大的系统类别，并使用这种数学分析来帮助理解潜在的物理或生物系统。PI对植物生物学的应用特别感兴趣。在另一个领域，一大类天体力学和等离子体物理学的核心物理系统可以通过首先将它们转换为称为测地线流的“几何系统”，然后研究测地线流来研究。在之前的工作中，PI表明，这些流中的一大类，有些人认为很容易理解，在数学和物理上很无聊，具有极其复杂的行为，实际上是混沌的。本文将继续研究一些具体的例子，包括一个来自规范场动力学的例子，这是粒子物理学的核心理论问题之一。

Invariant sets of dynamical systems are not generally self-similar in the strict sense. However, in work with others, the PI has shown that in some important cases, these sets can be decomposed into subsets each possessing a type of scaling symmetry. Sets which admit such structure are called multifractals (MF). This proposal involves the continuing investigation of the fine structure of these multifractals and an attempt to use the MF analysis to give new insights into dynamical systems and possibly yield a new (physical) classification of dynamical systems. In addition, the proposed work involves various problems in dimension theory which arise in mathematical biology as well as research on the relations between non-negative curvature and complicated dynamics of the geodesic flow. Regarding the latter, the (in)famous (xy)^2 Hamiltonian system, a model for a classical Yang-Mills field, which is orbit equivalent to a geodesic flow on a non-negative curved surface will be studied. Many physical and biological systems (including turbulent fluids, root systems of plants, stressed pieces of metal, NMR images of the brain, clouds, and galaxies in the universe) seem to possess some type of complicated fractal structure. Mathematically such objects are called multifractals. In previous work, the principal investigator presented a rigorous mathematical foundation for the study of some important classes of multifractals. The plan is to extend this work to larger classes of systems and to use this mathematical analysis to help understand the underlying physical or biological systems. The PI is particularly interested in applications to plant biology. In a different area, a large class of physical systems which are central to celestial mechanics and plasma physics can be studied by first transforming them to a ''geometric system'' called a geodesic flow and then studying the geodesic flow. In previous work, the PI showed that a large class of these flows, which some thought were easily understood and mathematically and physically boring, have extremely complicated behavior and are in fact chaotic. The investigations into some specific examples including an example from gauge field dynamics, which is one of the central theoretical problems in particle physics will be continued.