Statistical Behaviour of Recurrence in Dynamical Systems

摘要：提出的工作是研究动力系统的混合性质与其统计性质之间的联系。目标是开发一种通用方法，使我们能够在包括非均匀双曲动力系统在内的各种设置中获得归一化返回时间分布的结果。这可以推广到具有抛物点的正则不变测度区间映射的返回时间分布，或者在Julia集中具有临界点的有理映射的平衡状态。我们期望得到任意阶高阶回归分布的显式误差项。最终，我们希望将我们的方法扩展到L-S Young的“塔结构”，该方法最初是为了获得相关性衰减的速率而引入的。这一研究领域引起了许多科学家的兴趣。例如，想要进行数值模拟的实验学家将受益于所提出的问题，这将有助于他们开发更可靠的方法来数值分析混沌动力系统的时间序列。所提出的研究还可以用于开发更有效的数据压缩算法，因为它量化了新编码符号序列与其预测平均值发生的次数误差。

Abstract:The work proposed is to study the connection between mixing properties of dynamical systems and their statistical properties. The goal is to develop a general approach that will enable us to obtain results on the distribution of normalised return times in a wide variety of settings including non-uniformly hyperbolic dynamical systems. This extends to the return time distributions for the canonical invariant measure interval maps with parabolic points or for equilibrium states for rational maps which have critical points in the Julia set. We expect to obtain explicit error terms for the distribution of higher order returns of any order. Ultimately we hope to extend our approach to L-S Young's `tower construction' which originally was introduced to obtain rates on the decay of correlations.This area of research is of interest to a wide variety of scientist. For instance experimentalists who want to do numerical simulations will benefit from the proposed problems which will assist them in developing more reliable ways to numerically analyse time series of chaotic dynamical systems. The proposed research can also serve to develop more efficient data compression algorithms since it quantifies the error in the number of times with which a newly coded sequence of symbols occurs from its predicted average value.