Dynamics and Translation Surfaces

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。低复杂性动力系统包括符号系统，如Sturmian序列和Morse序列；代数起源空间上的圆周流和其他幂等流；有理多边台球和相关系统，如平移面上的线性流和区间交换变换。在这个提案中，我们考虑了处理低复杂性动态的这些不同领域之间的关系的问题。如果我们考虑一个台球轨迹，并跟踪它击中的侧面，我们可以将结果解释为一个符号系统。我们研究以这种方式出现的符号系统。对这类系统的研究给了我们一个共同的基础，即符号技术和多边形台球技术都可以应用。固定拓扑型平移曲面的集合构成了我们认为是模空间的拓扑空间。这个空间上有一个自然的幂等流，它通常被称为圆周流。我们考虑这种流在多大程度上类似于Ratner分类定理所描述的幂等流。在物理学、生物学、经济学和许多其他领域中，人们研究随时间变化的系统数学模型的长期行为。如果这些系统是确定性的和自治的，也就是说，与环境隔离，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，出现在不同领域的系统有共同的行为元素。例如，“混沌”系统有一些共同的特征。在这个方案中，我们研究了一类我们称之为低复杂性的系统，而不是典型的“混沌”或“高复杂性”的动力系统。这一建议涉及三类低复杂性动力系统及其之间的关系。

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.