Statistical and geometric properties of dynamical systems

该项目主要研究光滑动力系统的统计和几何性质，特别是非一致双曲映射和流。提案中有几个相互关联的主题。当务之急是将我们对统计性质的理解扩展到非一致双曲型系统，特别是那些由Young Tower建模的非一致双曲型系统。作为该计划的一部分，研究人员将研究次上同调方程的群值可测解的正则性，以及这种正则性对最优周期轨道(可观量的Birkhoff平均在其上被优化的周期轨道)的影响。最近，研究人员在动态定义的Cantor集上上同调方程的可测解的Whitney正则性方面得到了一些结果，并提出将这一工作扩展到更高维。等变横截性理论将被推广并应用于哈密顿和可逆系统，这是一个重要的应用背景。最后，建议继续开发和应用分析双曲型流动混合的新方法。研究人员已经开发出了新的不变量，从而证明了每一种光滑的双曲线流都可以被稳定(快速)的混合流近似。这项研究的目的是为了更好地理解双曲流的指数混合，这是一个在物理学中具有相当重要意义的话题。许多物理系统，甚至是那些用确定性微分方程精确建模的物理系统，往往表现出一种明显的随机和不可预测的方式--确定性混沌现象。例子从天气系统和流体湍流到电子线路和动物种群。这种类型的混沌或复杂系统通常最好地根据系统上观测的统计特性来理解。复杂系统可能具有额外的几何结构，如逆对称或能量守恒，这可以改变物理系统的预期属性，并提供一种理解系统的手段。该提案旨在通过研究复杂系统的统计特性和开发新的几何方法来研究它们的行为，以加深我们对复杂系统的理解。研究结果将在科学文献和讲座中广泛传播。这项研究还将有助于休斯顿大学研究生的数学教育，部分是通过研究人员的讲座，部分是通过在他们的指导下从事这一学科和相关学科的学生。

Abstract proposal DMS-0600927The project primarily deals with statistical and geometric properties of smooth dynamical systems, especially non-uniformly hyperbolic maps and flows. There are several interrelated topics in the proposal. A top priority is the extension of our understanding of statistical properties, such as the large deviation principle and the almost sure invariance principle for vector valued observables, to non-uniformly hyperbolic systems, in particular those modeled by a Young Tower.As part of this program, the investigators will study the regularity of group-valued measurable solutions to sub-cohomological equations and the implications of such regularity for optimal periodic orbits (periodic orbits on which Birkhoff averages of an observable are optimized). Geometric methods play an important role in the proposal.Recently, the investigators obtained results on the Whitney regularity of measurable solutions to cohomological equations on dynamically defined Cantor sets and it is proposed to extend this work to higher dimensions. The theory of equivariant transversality will be extended and applied to Hamiltonian and reversible systems, a setting important for applications. Finally, it is proposed to continue with the development and application of new methods for the analysis of mixing for hyperbolic flows. The investigators have already developed new invariants leading to a proof that every smooth hyperbolic flow can be approximated by a stably (rapid) mixing flow. It is proposed to pursue this investigation to obtain an improved understanding of exponential mixing for hyperbolic flows a topic of considerable significance in physics.Many physical systems, even those modeled accurately by deterministic differential equations, often behave in an apparently random and unpredictable way -- the phenomenon of deterministic chaos. Examples range from weather systems and fluid turbulence to electronic circuits and animal populations. Chaotic or complex systems of this type are often best understood in terms of the statistical properties of observations on the system. Complex systems may possess extra geometric structure, such as reversing symmetries or energy conservation, which can alter the expected properties of the physical system as well as provide a means for understanding the system. The proposal aims to deepen our understanding of complex systems, by investigating their statistical properties and exploiting new geometric methods to study their behavior. Results of the research will be broadly disseminated in the scientific literature and in lectures. The research will also contribute to the mathematical education of graduate students at the University of Houston, in part by lectures of the investigators, and in part by students working on this and related subjects under their guidance.