Smooth Dynamical Systems

本文主要研究低维流形上光滑反同态的动力学性质.相关的问题涉及奇异吸引子（有时称为“遍历吸引子”）的存在，同宿切线的丰度，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，莫拉和维亚纳的工作，我们知道存在面积减少的光滑参数化族，其遍历吸引子存在于参数的正测度集。 一个自然的问题是，“积极措施”是否可以用“开放”取代。“首席研究员以前的工作表明，在同宿切线的存在下，不存在具有遍历吸引子的面积减少的平面单同态的开集。本研究的一部分调查是否缺乏遍历吸引子是一个“通用”的属性。此外，将尝试证明遍历吸引子的存在，在系统中，是保持在相空间的某些部分-从而消除目前的依赖急剧收缩的领域在所有已知的技术。 当前项目中的其他问题集中在具有不同平滑度的映射的符号扩展的存在上。 该研究涉及低维映射的动力学。它与理解非线性微分方程的解（或轨道）密切相关。这样的方程出现在许多科学学科中。举个具体的例子，它们为生物学中的系统提供了模型，如昼夜节律、血液流动和神经脉冲的电动力学;在天文学中与轨道卫星的设计有关;甚至在天气预报中。在与前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，在这些系统的研究中，人们必须依赖于通过计算机数值探索获得的某些几何信息。这个项目试图提供一个数学上严格的处理有关的重要性质的方程，出现在这种情况下。这些属性的知识有助于提供科学家或工程师可以用来描述，测试和改进自然界中发生的各种事情的模型的工具。因此，目前的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.