Recurrence and Mixing in Dynamical Systems

该项目是研究动力系统的统计和混合性质。拟议的工作福尔斯三个领域如下：（一）（与S Vaienti）的第一个研究领域是调查在什么条件下“圆柱邻域”，如果一个动力系统的测量是对数正态分布。此外，我们感兴趣的速度收敛到中央极限定理的广泛的一类动力系统的混合被允许“延迟”根据小的返回neighborhood.We预计，我们的技术也将使我们能够做一个很好的分析的重复时间统计方面的法律的重对数和不变原理。(ii)（与S Vaienti联合）在过去的三年里，我们已经开发了一个通用的计划，可以用来确定返回时间的极限分布。我们希望进一步进行这一计划的基础上的'塔建设'这是使用的LS杨获得衰减率的相关函数。(iii)（联合胡）第三个领域的研究是对抛物线地图有非常明显的抛物线，使通用不变措施是无限的。在这种情况下，转移算子收敛到一个常数，这个常数完全由映射在抛物点处的行为决定。我们要确定的收敛速度使用希尔伯特度量方法和一些结果更新theory.Out的三个问题考虑的前两个特别有其起源香农的工作在信息理论：动力系统的特点分析使用的编码是来自跟踪轨道和确定他们根据分区元素，他们通过。在这种方式下，一个“圆柱集”表示一个给定的符号序列，它是编码理论中的一个经典问题，以确定这个序列的返回统计。Shannon和其他人已经表明，期望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，对返回时间的研究有利于数据压缩方案的实施。通常，对返回时间分布的详细了解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。建议的研究重点是寻找一般条件下的返回时间是在极限对数正态分布的各种动力系统，并提供误差项。误差项将取决于系统的动态参数。

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.