Dynamics and Translation Surfaces

动力学和平移表面

• 批准号: 0901521
• 负责人: John Smillie
• 金额: --
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2013-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901521&HistoricalAwards=false
• 关键词: Dynamics; Translation; Surfaces

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Low complexity dynamical systems include symbolic systems such as Sturmian sequences and the Morse sequence; horocycle flows and other unipotent flows on spaces of algebraic origin; rational polygonal billiards and related systems such as linear flows on translation surfaces and interval exchange transformations. In this proposal we consider questions which deal with the relations between these different areas of low complexity dynamics. If we consider a billiard trajectory and keep track of the sides that it hits we can interpret the result as a symbolic system. We investigate symbolic systems that arise this way. The study of such systems gives us a common ground in which symbolic techniques and polygonal billiard techniques can both be applied. The collection of translation surfaces of a fixed topological type forms a topological space which we think of as a moduli space. This space has a natural unipotent flow on it which is usually called the horocycle flow. We consider to what extent does this flow behave like the unipotent flows described by Ratner's classification theorem. In particular we consider the extent to which techniques that work in the homogeneous case do and do not work in the moduli space case.In physics, biology, economics and many other fields one studies the long term behavior of mathematical models of systems that change with time. If these systems are deterministic and autonomous, that is to say isolated from their environment, then they fall in the domain of the field of dynamical systems. A fundamental discovery of this field is that there are common elements of behavior for systems that arise in different areas. There are for example common features to "chaotic" systems. In this proposal we study a class of systems which we call low complexity by contrast with typical dynamical systems which are "chaotic" or "high complexity". This proposal deals with three families of low complexity dynamical systems and the relations between them.

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。低复杂度动力系统包括符号系统，例如Sturmian序列和莫尔斯序列;代数起源空间上的单循环流和其他单幂流;有理多边形台球和相关系统，例如平移表面上的线性流和区间交换变换。在这个建议中，我们考虑的问题，处理这些不同领域的低复杂度动态之间的关系。如果我们考虑一个台球轨迹，并跟踪它击中的边，我们可以将结果解释为一个符号系统。我们研究的是以这种方式出现的符号系统。这种系统的研究给了我们一个共同的基础，其中符号技术和多边形台球技术都可以应用。一个固定拓扑类型的平移曲面的集合形成了一个拓扑空间，我们将其视为模空间。这个空间上有一个自然的幂幺流，通常称为horocycle流。我们考虑在何种程度上这个流的行为像由拉特纳的分类定理所描述的单幂流。特别是，我们考虑在何种程度上的技术，工作在均匀的情况下做，不工作在模空间的情况下。在物理学，生物学，经济学和许多其他领域的研究长期行为的数学模型的系统，随时间变化。如果这些系统是确定性的和自治的，也就是说与它们的环境隔离，那么它们就属于动力系统领域。这个领域的一个基本发现是，在不同领域出现的系统有共同的行为要素。例如，“混沌”系统有一些共同的特征。在这个建议中，我们研究了一类系统，我们称之为低复杂性的对比，与典型的动力系统是“混沌”或“高复杂性”。本文讨论了三类低复杂度动力系统以及它们之间的关系。