Transfer operators and emergent dynamics in hyperbolic systems

动力系统是一个数学领域，涉及研究随时间演化的现象。它与分析、数论、概率和几何等其他数学领域有着深厚的联系。许多有趣的动力系统本质上都是混沌的。这意味着它们表现出对初始条件的敏感依赖性，并且无法通过跟踪它们的轨道来预测它们的长期行为。因此，从概率的角度研究此类系统的行为是很自然的。推断混沌动力系统统计方面的工具称为传递算子。这样的算子描述了在动态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有关潜在动态统计数据的显着信息，例如相关衰减率、极端事件和罕见事件。一类基本的混沌平滑双曲动力系统称为阿诺索夫系统。在过去的十五年中，转移算子技术在平滑、“理想化”（单点）、阿诺索夫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（即数学分析中的现代主题半经典分析）建立了显着的新联系。然而，传递算子技术尚未首创用于耦合阿诺索夫系统，因此显然不适用于更一般的具有奇点的耦合分段双曲系统。这种耦合系统自然地以网络模型的形式出现在工程、物理和生物科学中，并且在研究非平衡热力学中至关重要。这使得传递算子和遍历理论的卓有成效的方法无法对这种能够产生紧急动态的复杂系统的行为提供统计见解：动态量，例如质量和传热的逃逸，是大型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颖的传递算子技术来理解“大型动力系统”的涌现动力量和宏观统计特性，其微观动力学是具有奇点的分段双曲系统，从而在平滑遍历理论和双曲动力学方面实现新的最先进技术。

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.