Smooth Dynamical Systems

本文主要研究低维流形的光滑微分同胚的动力学问题。相关问题涉及奇怪吸引子(有时称为“遍历吸引子”)的存在，同宿切的丰富性，以及符号扩张的存在。根据定义，遍历吸引子是非原子遍历SRB测度的支撑。由于Benedicks、Carleson、Wang、Young、Mora和Viana的工作，已知存在光滑的面积递减微分同胚族，其参数的正测度集存在遍历吸引子。一个自然的问题是，“积极措施”能否被“开放”取代。前人的工作表明，在存在同宿切的情况下，不存在具有遍历吸引子的面积递减平面微分同胚的开集。本研究的一部分是调查这种缺乏遍历吸引子是否是一种“一般”性质。此外，还将尝试证明在相空间的某些部分保持面积的系统中遍历吸引子的存在性，从而消除目前所有已知技术中对面积急剧收缩的依赖。当前项目中的其他问题集中在具有不同平滑程度的地图的符号扩展的存在。这项研究涉及到低维映射的动力学。它与了解非线性微分方程解(或轨道)密切相关。这样的方程出现在许多科学学科中。举几个具体的例子，它们提供了生物系统的模型，如昼夜节律、血液流动和神经冲动的电动力学；与轨道卫星设计有关的天文学；甚至天气预报。在与上述一般系统相关的一些特定模型中，已知相关方程不能显式求解。相反，在研究这些系统时，人们必须依赖于某些几何信息，这些信息是通过计算机的数值探索而获得的。这个项目试图对在这种情况下出现的与方程相关的重要性质提供数学上的严格处理。这些属性的知识有助于提供工具，科学家或工程师可以使用这些工具来描述、测试和改进自然界中发生的各种事物的模型。因此，本研究有可能为实质性的科学进步提供基础。

This research is primarily concerned with the dynamics of smooth diffeomorphisms of low- dimensional manifolds. Relevant questions deal with the existence of strange attractors (sometimes called "ergodic attractors"), the abundance of homoclinic tangencies, and the existence of symbolic extensions. By definition, an ergodic attractor is the support of a nonatomic ergodic SRB measure. Due to the work of Benedicks, Carleson, Wang, Young, Mora, and Viana it is known that there are smooth parametrized families of area-decreasing diffeomorphisms for which ergodic attractors exist for positive measure sets of parameters. A natural question is whether "positive measure" can be replaced with "open." Previous work of the principal investigator has shown that, in the presence of homoclinic tangencies, there are NO open sets of area-decreasing planar diffeomorphisms with ergodic attractors. Part of the present research investigates whether this lack of ergodic attractors is a "generic" property. In addition, attempts will be made to prove the existence of ergodic attractors in systems that are area-preserving in some parts of the phase space--thus removing the current dependence on sharp contraction of area in all known techniques. Other problems in the current project focus on the existence of symbolic extensions for maps with varying levels of smoothness. This research involves the dynamics of low-dimensional mappings. It is intimately related to understanding the solutions (or orbits) of nonlinear differential equations. Such equations arise in many scientific disciplines. To cite particular examples, they provide models for systems in biology such as circadian rhythms, blood flow, and the electrodynamics of nerve impulses; in astronomy in connection with the design of orbiting satellites; and even in weather prediction. In some specific models related to the foregoing general systems, it is known that the relevant equations cannot be solved explicitly. Instead, in the study of these systems one must rely on certain geometric information that is made available through numerical exploration with computers. This project attempts to provide a mathematically rigorous treatment of important properties related to equations that arise in such situations. The knowledge of these properties helps to provide tools that the scientist or engineer can use to describe, test, and refine models for a variety of things that occur in nature. As such, the present research has the potential to provide foundations for substantial scientific advancement.