Transfer operators and emergent dynamics in hyperbolic systems

动力系统是研究随时间演化的现象的数学领域。它与分析、数论、概率论和几何等其他数学领域有着深刻的联系。许多有趣的动力系统本质上是混沌的。这意味着它们对初始条件表现出敏感的依赖性，它们的长期行为不能通过跟踪它们的轨道来预测。因此，从概率的角度来研究这类系统的行为是很自然的。一种用来推断混沌动力系统统计方面的工具叫做传递算子。这样的算子描述了在动力学演化下分布如何随时间变化。更重要的是，它的光谱数据编码了重要的信息，如相关衰减率、极端事件和罕见事件，以及潜在动力学的统计数据。混沌光滑双曲动力系统的一个基本类别被称为Anosov。在过去的15年中，转移算子技术在光滑、“理想化”（单点）、Anosov系统的统计方面产生了令人印象深刻的结果，并与其他数学领域建立了显著的新联系，即与半经典分析，这是数学分析中的一个现代主题。然而，对于耦合ansov系统，传递算子技术还没有开创，因此显然，对于更一般的具有奇点的耦合分段双曲系统来说，传递算子技术还没有开创。这种耦合系统在工程、物理和生物科学中自然地以网络模型的形式出现，在研究非平衡态热力学中具有极其重要的意义。这使得传递算子的有效方法和遍历理论在提供能够产生紧急动力学的复杂系统的行为的统计见解方面显得不足：动态量，如质量和传热的逸出，是大系统中组件之间相互作用的结果。在这个项目中，我们的目标是通过开发新的传递算子技术来理解“大动力系统”的紧急动态量和宏观统计特性，从而实现光滑遍历理论和双曲动力学的新技术，这些系统的微观动力学是具有奇点的分段双曲系统。

Dynamical systems is a field of mathematics that is concerned with studying phenomena that evolve in time. It has deep connections with other areas of mathematics such as analysis, number theory, probability and geometry. Many interesting dynamical systems are chaotic in nature. This means they exhibit sensitive dependence on initial conditions and their long-term behaviour cannot be predicted by following their orbits. Thus, it is natural to study the behaviour of such systems form a probabilistic point of view. An instrumental tool to infer statistical aspects of a chaotic dynamical system is called the transfer operator. Such an operator describes how distributions change over time under the evolution of the dynamics. More importantly, its spectral data encode remarkable information, such as rates of correlation decay, extremes and rare events, on the statistics of the underlying dynamics. A fundamental class of chaotic smooth hyperbolic dynamical systems is called Anosov. In the past fifteen years transfer operator techniques produced impressive results on the statistical aspects of smooth, 'idealised' (single site), Anosov systems and made remarkable new connections with other areas of mathematics, namely with semiclassical analysis, a modern topic in mathematical analysis. However, transfer operator techniques are not yet pioneered for coupled Anosov systems, and hence obviously, not for the more general coupled piecewise hyperbolic systems with singularities. Such coupled systems appear naturally as network models in engineering, physical and biological sciences, and are of paramount importance in studying nonequilibrium thermodynamics. This leaves the fruitful approach of transfer operators, and ergodic theory in general, short on providing statistical insights on the behaviour of such complex systems that are capable of producing emergent dynamics: dynamical quantities, such as escape of mass and heat transfer, that appear as a result of interaction among components in a large system. In this project we aim to achieve a new state-of-the art in smooth ergodic theory and hyperbolic dynamics by developing novel transfer operator techniques to understand emergent dynamical quantities and macroscopic statistical properties of 'large dynamical systems' whose microscopic dynamics are piecewise hyperbolic systems with singularities.