Low Dimensional Dynamics

DMS-0555463提案摘要：本项目研究低维动力系统的各个方面。对这种变换的研究，特别是对保留变换的研究，可以追溯到庞加莱和G。D.伯克霍夫。特别是，该项目考虑了曲面上的光滑群作用，以及群的代数性质和该作用所表现出的动力学之间的关系。这一建议所涉及的一个相关问题是二维动力系统的全局不动点的存在性以及它与系统的代数性质的关系。这一领域的结果在更广泛的科学领域有许多应用，特别是经典力学和现代混沌理论。这项拟议的研究的新奇之处在于，它解决了动态系统的代数性质(称为“群作用”)与它们所表现出的几何或拓扑行为之间的关系。这一提议的预期结果将促进我们对动力系统的了解，并将探索动力学和代数之间的新关系。

Abstract for proposal DMS-0555463:This project investigates aspects of low dimensional dynamicalsystems. The study of such transformations, especially areapreserving ones, has a long history going back to Poincare andG. D. Birkhoff. In particular the project considers smooth groupactions on surfaces and the relation between the algebraic propertiesof the group and the dynamics the action exhibits. A related questionaddressed by this proposal is the question of the existence of globalfixed points for two-dimensional dynamical systems and how thisrelates to the algebraic properties of the system.This project investigates aspects of dynamical systems onsurfaces. There are numerous applications of results in this area tobroader fields of science, especially to classical mechanics and moremodern chaos theory. The novelty of the proposed research is that itaddresses the relationship between the algebraic nature of dynamicalsystems called ``group actions'' and the geometric or topologicalbehavior they exhibit. Anticipated results from this proposal willadvance our knowledge of dynamical systems and will explore newrelationships between dynamics and algebra.