Low Dimensional Dynamics

DMS-0555463提案摘要：本项目研究低维动态系统的各个方面。 对这类变换的研究，特别是保面积变换的研究，有着悠久的历史，可以追溯到Poincare和G. D.伯克霍夫 特别是该项目认为光滑groupactions表面和代数属性之间的关系组和动态的行动展品。 这个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问题，以及这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，特别是经典力学和更现代的混沌理论中有许多应用。 这项研究的新奇在于，它解决了被称为“群作用”的动力系统的代数性质与它们所表现出的几何或拓扑行为之间的关系。 从这个建议预期的结果将推进我们的知识的动力系统，并将探索新的动力学和代数之间的关系。

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.