Recurrence and Mixing in Dynamical Systems

该项目是研究动力系统的统计和混合性质。建议的工作分为三个方面：(I)(与S·瓦恩蒂共同)第一个研究领域是研究在什么条件下动力系统的“柱面邻域”具有对数正态分布的测量。此外，我们还对一大类动力系统收敛到中心极限定理的速度感兴趣，在这类动力系统中，根据返回邻域的小，允许混合被‘延迟’。我们预计我们的技术也将使我们能够根据重对数律和不变性原理对重复时间统计进行精细的分析。(Ii)(与S·维恩蒂)在过去的三年里，我们发展了一个通用的方案，可以用来确定返回时间的极限分布。我们希望在L S杨用来获得相关函数衰减率的‘塔结构’的基础上进一步发展这一方案。(Iii)(与胡一起)第三个研究领域是抛物线映射，它具有非常明显的抛物性，因此通用不变测度是无穷的。在这种情况下，传递算子收敛到一个常数，该常数完全由地图在抛物线点处的行为决定。我们想用Hilbert度规方法和更新理论的一些结果来确定收敛速度。在所考虑的三个问题中，前两个问题特别起源于Shannon的信息论工作：动力系统的特征分析是通过跟踪轨道并根据它们经过的划分元素识别得到的编码来得到的。用这种方式，‘圆柱体集合’代表一个给定的符号序列，而确定该序列的返回统计量是编码理论中的一个经典问题。Shannon和其他人已经证明了预期返回时间通常是指数的，其中速率等于度量熵(即平均信息量)。例如，返回时间的研究在数据压缩方案的实施中是有益的。一般来说，返回时间的分布的详细知识对于对可能来自实验数据或数值模拟的时间序列进行可靠的分析是必要的。所提出的研究集中于寻找返回时间在极限对数正态分布下的各种动力系统的一般条件，并提供误差项。误差项将取决于系统的动力学参数。

The project is to research statistical and mixing properties of dynamical systems. The proposed work falls into three areas as follows:(i) (jointly with S Vaienti) The first area of research is to investigate under what conditions `cylinder neighbourhoods' if a dynamical system have mesures that are lognormally distributed. We are moreover interested in the speed of convergence to the Central Limit Theorem for a wide class of dynamical systems where the mixing is allowed to be `delayed' according to the smallness of the return neighbourhood.We expect that our techniques will also enable us to do a fine analysis of the repeat time statistics in terms of the Law of the Iterated Logarithm and the Invariance Principle.(ii) (jointly with S Vaienti) In the last three years we have developed a general scheme that can be used to determine the limiting distribution of return times. We wish to carry this scheme further by basing it on the `tower construction' which was used by L S Young to obtain rates of decay for the correlation function.(iii) (jointly with H Hu) The third area of research is on parabolic maps that have very pronounced parabolicity so that the generic invariant measure is infinite. The transfer operator in this case converges to a constant which is entirely determined by the behaviour of the map at the parabolic point. We want to determine the rate of convergence using the Hilbert metric method and some results from renewal theory.Out of the three problems considered the first two in particular have their origin in Shannon's work in information theory: Dynamical systems are characteristically analysed by using the coding which is derived by tracing orbits and identifying them according to the partition elements they pass through. In this way a `cylinder set' represents a given sequence of symbols and it is a classical problem in coding theory to determine the return statistics of this sequence. Shannon and others have shown that expected return time is typically exponential where the rate is equal to the metric entropy (i.e.\ the average information content). The study of return times has for instance been beneficial in the implementation of data compression schemes.Generally a detailed knowledge of the distribution of return times is necessary to do a reliable analysis of time series that might come from experimental data or numerical simulations. The proposed research focuses on finding general conditions for a large variety of dynamical systems under which the return times are in the limit lognormally distributed and also to provide error terms. The error terms will depend on the dynamical parameters of the system.