Statistical Behaviour of Recurrence in Dynamical Systems

翻译后摘要：提出的工作是研究动力系统的混合特性和它们的统计特性之间的连接。我们的目标是开发一种通用的方法，使我们能够获得各种设置，包括非均匀双曲动力系统的归一化返回时间的分布结果。这扩展到返回时间分布的规范不变的测量区间映射的抛物点或平衡状态的合理的地图，其中有临界点的Julia集。我们期望得到任何阶的高阶收益率分布的显式误差项。最终，我们希望我们的方法，以L-S杨的'塔建设'，最初介绍了获得率的衰减的相关性。这一领域的研究是感兴趣的各种各样的科学家。例如，想要进行数值模拟的实验者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的符号序列从其预测的平均值出现的次数的误差。

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.