Parabolic dynamics

抛物线动力学

• 批准号: 326748-2006
• 负责人: Forni, Giovanni
• 金额: $ 2.48万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=362551
• 关键词: Parabolic; dynamics

In physics and other sciences, relevant quantities can often be described as averages along the trajectories of a dynamical system. When the system has a property called ``ergodicity'', time averages for large times can be approximated by space averages, which are much easier to compute. For most applications of this fundamental idea, it is crucial to estimate the error involved in the approximation. Our goal is to develop methods for establishing ergodicity with error estimates for weakly chaotic, ``parabolic'' systems. Such systems are roughly characterized by subexponential, polynomial divergence of nearby orbits, in contrast with hyperbolic, strongly chaotic systems which exhibit exponential divergence. For parabolic systems there is no general theory comparable to the hyperbolic theory and the research has focused so far on key interesting examples. Some of the examples are now rather well understood, others are still beyond the reach of available methods of study. Parabolic systems are important in many applications to physics, to geometry and number theory. For instance, in simple models of Hamiltonian dynamics in the presence of singularities, of the type which occur in the motion of the Solar System, and in questions related to number theory, such as the number of ways a large integer can be written as a sum of a given number of squares, cubes and so on. A fundamental tool in the study of parabolic systems is the idea of a renormalization dynamics. The renormalization is like a powerful lense that allows us to explore the fine structure of a system (or family of systems) at smaller and smaller scales. Since the renormalization is typically a hyperbolic dynamical system, it can be studied by the tools of hyperbolic theory (invariant manifolds, Lyapunov exponents, etc.)

在物理学和其他科学中，相关的量通常可以被描述为动力系统轨迹上的平均值。当系统具有一种称为“遍历性”的特性时，大时间的时间平均值可以用空间平均值来近似，这更容易计算。对于这一基本思想的大多数应用，估计近似所涉及的误差是至关重要的。我们的目标是为弱混沌的“抛物型”系统建立具有误差估计的遍历性的方法。与呈现指数发散的双曲强混沌系统相比，这类系统的粗略特征是附近轨道的次指数、多项式发散。对于抛物系统，没有可与双曲理论相媲美的一般理论，到目前为止，研究主要集中在关键的有趣的例子上。其中一些例子现在已经很好地理解了，其他的仍然超出了现有的研究方法的范围。抛物系统在物理学、几何学和数论的许多应用中都是重要的。例如，在存在奇点的哈密顿动力学的简单模型中，在太阳系运动中出现的那种类型，以及与数论有关的问题中，例如大整数可以用几种方式写成给定数量的平方、立方体的和，等等。研究抛物系统的一个基本工具是重整化动力学的思想。重整化就像一个强大的透镜，它允许我们在越来越小的尺度上探索一个系统(或一族系统)的精细结构。由于重整化是典型的双曲动力系统，可以用双曲理论(不变流形、Lyapunov指数等)来研究它。