The return time statistics for non-markovian maps

摘要提案DMS-0602202拟议的工作是在混合属性之间的连接，相关性的衰减和动力系统的统计特性。有三个主要方面：（一）我们希望得到一个更好的理解发生了什么事情的混合映射的返回时间附近的周期轨道。目前，它是已知的，一般点在极限泊松分布的返回时间。然而，在周期轨道附近，这种规律性的行为不能被假设，我们打算表明，极限分布是一个复合泊松分布，其中复合部分是伯努利和确定的“逃逸率”附近的周期轨道。(ii)研究的第二个领域是确定非一致双曲系统返回时间的极限分布。这将使我们能够分类的长期统计行为的一个相当大的类的动力系统，包括一些抛物系统，“台球型”系统，也高维地图类似的Henon地图已被广泛研究，在技术上是非常困难的办法。(iii)第三个主要领域，我们建议做的研究是有关一个著名的定理香农，麦克米伦和Breiman使用衰减率的动力学邻域来描述熵。我们想证明一个$\alpha$-混合系统的中心极限定理。这将大大扩展以前的结果，使用非常强的混合或规则性的假设，以获得分布结果。我们希望使用新的技术，使我们能够克服削弱混合性质所带来的困难，让我们注意到，在这一领域的研究是感兴趣的各种科学家，包括实验谁想要做数值模拟。关于返回时间分布的详细知识可以用来开发更可靠的方法来数值分析混沌动力系统的时间序列。所提出的研究也可以用于开发更有效的数据压缩算法。

Abstract proposal DMS-0602202The proposed work is on the connection between mixing properties, the decay of correlations and statistical properties of dynamical systems. There are three main areas:(i) We want to get a better understanding what happens to the return times for mixing maps near periodic orbits. Currently it is known that generic points have in the limit Poisson distributed return times. However near periodic orbits such regular behavior cannot be assumed and we intend to show that the limiting distribution is a compounded Poisson distribution where the compounded part is Bernoulli and determined by an `escape rate' near the periodic orbit.(ii) The second area of research is to determine the limiting distribution for return times for non-uniformly hyperbolic systems. This would allow us then to classify the long term statistical behavior for a considerably larger class of dynamical systems including some parabolic systems, `billiard type' systems and also higher dimensional maps similar to the Henon map which has been studied extensively and is technically very difficult to approach.(iii) The third major area in which we propose to do research is related to a famous theorem by Shannon, McMillan and Breiman that uses the decay rate of dynamical neighbourhoods to describe the entropy. We want to prove a Central Limit Theorem for a $\alpha$-mixing systems. This will considerably extend previous results that used very strong mixing or regularity assumptions to obtain distribution results. We want to use new techniques that should allow us to overcome the difficulties that are entailed by weakening the mixing properties.Let us note that research in this area is of interest to a wide variety of scientists including experimentalists who want to do numerical simulations. Detailed knowledge about the distribution of return times can be used to develop more reliable ways to numerically analyze time series of chaotic dynamical systems. The proposed research can also serve to develop more efficient data compression algorithms.