Parabolic dynamics

在物理学和其他科学中，相关的量通常可以被描述为沿着动力系统轨迹的沿着平均值。当系统具有一种称为“遍历性”的性质时，大时间的时间平均值可以近似为空间平均值，这更容易计算。对于这个基本思想的大多数应用，估计近似中涉及的误差是至关重要的。我们的目标是建立弱混沌，“抛物”系统的误差估计遍历性的方法。这样的系统的特征大致是附近轨道的次指数、多项式发散，而双曲、强混沌系统则表现出指数发散。对于抛物型方程组，没有一般的理论可以与双曲理论相媲美，迄今为止，研究主要集中在关键的有趣例子上。其中一些例子现在已经相当好地理解了，另一些仍然超出了现有研究方法的范围。抛物系统在物理学、几何学和数论的许多应用中都很重要。例如，在太阳系运动中出现的奇点的哈密顿动力学的简单模型中，以及在与数论有关的问题中，例如一个大整数可以写成给定数量的平方和的方式的数量，立方体等等。抛物系统研究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量子力学，它允许我们在越来越小的尺度上探索一个系统（或系统族）的精细结构。由于重整化是典型的双曲动力系统，因此可以通过双曲理论的工具（不变流形，李雅普诺夫指数等）来研究它。

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.)