Statistical properties of dynamical systems: large deviations, extremes, return time statistics and dynamical Borel-Cantelli lemmas.

该项目旨在提高我们对非一致双曲动力系统的某些关键统计的理解：返回时间统计，极值统计，动态Borel-Cantelli引理和大偏差。虽然在描述均匀双曲系统的这些统计特性方面已经取得了进展，但均匀双曲性很少适用于物理系统，基于非均匀双曲性的模型更现实。返回时间统计量化了物理系统在某个时间段内转变为或返回到某个状态的概率。极值统计给出了系统上观测值时间序列连续极大值的极限律。Borel-Cantelli引理是建立随机过程几乎必然性态的基本工具。我们的目的是扩展我们的理解，这些统计特性的非均匀双曲系统，包括多项式相关衰减率。另一个目标是获得大偏差估计的动力系统的功能不连续性（或奇点）或产生乘法上循环，如矩阵值functions.Some物理系统是如此复杂，虽然他们可能被建模的数学方程，他们是最好的描述从统计的观点。例如，知道位于近海的飓风影响海岸上某些城镇的概率是有用的信息，通常无法提供更确定的信息。然而，这是一个主要的问题，经典的统计假设的完全随机性（独立的连续观察）很少满足现实模型的物理系统，因为连续的观察是高度相关的。 极值理论估计观测到一定量级事件的可能性，而大偏差估计离群值或罕见事件的概率。该项目旨在通过为广泛的物理现实数学模型提供极值理论，大偏差和其他统计数据的理解来改善复杂物理系统的预测。公共信息机构还将继续努力改进各级科学教育，并通过外联活动向公众提供信息。

This project seeks to improve our understanding of certain key statistics of non-uniformly hyperbolic dynamical systems: return time statistics, extreme value statistics, dynamical Borel-Cantelli lemmas and large deviations. While there has been progress in describing these statistical properties for uniformly hyperbolic systems, uniform hyperbolicity seldom holds for physical systems and models based on non-uniform hyperbolicity are more realistic. Return time statistics quantify probabilities of a physical system transforming into or returning to a state in a certain period of time. Extreme value statistics give limit laws for successive maxima of a time series of observations on a system. Borel-Cantelli lemmas are a fundamental tool in establishing the almost-sure behavior of stochastic processes. The aim is to extend our understanding of these statistical properties to non-uniformly hyperbolic systems including those with polynomial correlation decay rate. A further goal is to obtain large deviation estimates for functions on dynamical systems with discontinuities (or singularities) or which give rise to multiplicative cocycles, such as matrix valued functions.Some physical systems are so complex that, although they may be modeled by mathematical equations, they are best described from the viewpoint of statistics. For example, knowing the probabilities of a hurricane located offshore affecting certain towns on the coast is useful information and more certain information usually cannot be given. However, it is a major problem that the classical statistical assumption of complete randomness (independence of successive observations) is seldom satisfied for realistic models of physical systems since successive observations are highly correlated. Extreme value theory estimates the likelihood of observing an event of a certain magnitude, while large deviations estimates the probability of an outlier or rare event. This project aims to improve predictions of complex physical systems by providing an understanding of extreme value theory, large deviations and other statistics for a wide class of physically realistic mathematical models. The PI will also continue to work towards improving science education at all levels and by informing the public through outreach activities.