Smooth Dynamical Systems

本文主要研究低维流形光滑微分同态的动力学问题。相关问题涉及奇异吸引子的存在性（有时称为“遍历吸引子”）、同斜切线的丰度以及符号扩展的存在性。根据定义，遍历吸引子是非原子遍历SRB测度的支持。由于Benedicks， Carleson, Wang, Young， Mora和Viana的工作，我们知道存在光滑的减小面积的微分同胚族，其中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”来代替。先前的工作表明，在同斜切线的存在下，不存在具有遍历吸引子的面积递减的平面微分同胚的开集。本研究的一部分探讨了这种遍历吸引子的缺乏是否是一种“一般”性质。此外，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，从而消除当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上。本研究涉及到低维映射的动力学。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，他们为生物系统提供了模型，如昼夜节律、血液流动和神经冲动的电动力学；在天文学上，与轨道卫星的设计有关；甚至在天气预报方面。在与上述一般系统相关的某些特定模型中，已知相关方程不能显式求解。相反，在研究这些系统时，人们必须依靠某些几何信息，这些信息可以通过计算机的数值探索获得。这个项目试图提供一种数学上严谨的处理方法，处理在这种情况下出现的与方程相关的重要性质。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、测试和完善自然界中发生的各种事物的模型。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.