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。所提出的研究解决了以保面积的方式作用于曲面的群的代数性质与作用的动力学可能的拓扑性之间的联系。一个主题是试图找到光滑群行动的全局不动点，并利用群的诱导表示在该固定点处切空间的自同构中得出关于该行动的信息。这个项目的预期结果将促进我们对动力学系统的了解，并将探索动力学和代数之间的新关系。这项建议涉及曲面随时间演化的变换。对这种变换的研究，特别是面积保持变换，有着悠久的历史，可以追溯到亨利·庞加莱和G·D·伯克霍夫的工作，他们的工作是受到天体力学问题的推动。对于这样的系统，“状态”是表面上的一个点，目标是了解所有状态的集合是如何随时间演变的。时间可以被认为是连续的(用实数表示)或离散的(用整数表示)。本项目处理离散情况，并将其推广到考虑时间模拟的演变，例如，用矩阵而不是整数来表示。这是一个长期计划的一部分，目的是了解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。这一领域的结果被大量应用于更广泛的科学领域，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.