Dynamics on Surfaces

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。这是一个研究低维动力系统方面的项目。拟议的研究解决了代数属性之间的联系，一个组，在一个表面上的行动，在一个区域保持的方式和可能的拓扑性质的动态的行动。一个主题是试图找到一个光滑的群作用的全局不动点，并使用该群的诱导表示到不动点处的切空间的自同构中来推断关于该作用的信息。这个项目的预期结果将推进我们对动力系统的认识，并将探索动力学和代数之间的新关系。对这类变换，特别是保面积变换的研究，有着悠久的历史，可以追溯到Henri Poincare和G。D.伯克霍夫，工作的动机是在天体力学的问题。 对于这样一个系统，“状态”是表面上的一个点，目标是了解所有状态的集合如何随时间演变。时间可以被认为是连续的（用真实的数表示）或离散的（用整数表示）。本项目处理离散的情况下，并将其推广到考虑类似的时间表示的演变，例如，由一个矩阵，而不是一个整数。这是一个长期计划的一部分，以了解动力系统的代数性质和它们所表现出的几何或拓扑行为之间的关系。有许多应用的结果在这一领域的更广泛的科学领域，特别是经典力学，天体力学和更现代的混沌理论。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This is a project to investigate aspects of low-dimensional dynamical systems. The proposed research addresses the connection between algebraic properties of a group that acts on a surface in an area-preserving way and the possible topological nature of the dynamics of the action. One theme is to try to find global fixed points for a smooth group action and use the induced representation of the group into the automorphisms of the tangent space at the fixed point to conclude information about the action. Anticipated results from this project will advance our knowledge of dynamical systems and will explore new relationships between dynamics and algebra.This proposal concerns transformations of surfaces as they evolve in time. The study of such transformations, especially area-preserving ones, has a long history going back to work of Henri Poincare and G. D. Birkhoff, work that was motivated by problems in celestial mechanics. For such a system a "state" is a point on a surface and the objective is to understand how the collection of all states evolves in time. Time can be considered as either continuous (represented by a real number) or discrete (represented by an integer). The present project deals with the discrete case and generalizes it to consider evolutions where the analogue of time is represented, for example, by a matrix rather than an integer. This is part of a long-term program to understand the relationship between the algebraic nature of dynamical systems and the geometric or topological behavior they exhibit. There are numerous applications of results in this area to broader fields of science, especially to classical mechanics, celestial mechanics, and more modern chaos theory.