Dynamics on Two-Dimensional Surfaces

这是一个研究低维动力系统方面的建议。最重要的长期项目是更好地理解C^r一般区域的拓扑描述，保留r为1的曲面的微分同态。此外，作者希望研究比整数或实数在曲面上更一般的群的作用，特别是保域作用。弗兰克斯教授将继续他对地表地图动力学的研究。对这种转换的研究，尤其是对保护区域的转换的研究，历史悠久，可以追溯到英格托尔和g.d.伯克霍夫。这一领域的结果在经典力学以及更现代的混沌动力学中有许多应用。提出的研究将解决有关区域保存二维离散动力系统中周期性行为存在的问题，以及此类系统动力学的拓扑描述。

This is a proposal to investigate aspects of low dimensional dynamicalsystems. The most important long term project is to betterunderstand a topological description of C^r generic area preservingdiffeomorphisms of surfaces where r 1. In addition the proposerhopes to investigate the action of more general groups than the integers or the reals on surfaces, and in particular area preserving actions.Professor Franks will continue his investigation of thedynamics of surface maps. The study of such transformations,especially area preserving ones, has a long history going back toPoincare and G. D. Birkhoff. There are numerous applications ofresults in this area to classical mechanics as well as more modernchaotic dynamics. The proposed research will address questionsconcerning the existence of periodic behavior in area preservingtwo-dimensional discrete dynamical systems and a topologicaldescription of the dynamics of such systems.