Thermodynamics and statistics of non-uniformly hyperbolic dynamical systems

人们很自然地会问，我们对现在的了解能在多大程度上告诉我们对未来的了解。如果我们研究一个根据已知规则随时间进化的系统，我们可以根据观察系统的当前状态做出什么预测？现在人们已经很清楚，即使是相对简单的系统也会表现出混乱的行为，在这种情况下，我们做出准确预测的能力随着我们展望未来而迅速衰减。在这种情况下，人们可能希望将系统视为随机过程，并使用概率论的工具进行统计预测。该方法已成功地应用于最强混沌的均匀双曲系统。然而，大多数物理上现实的例子都不在这一类中，包括气象学（洛伦兹系统）、人口动力学（逻辑图）等重要模型。这激发了对非均匀双曲系统的研究，在那里，目前的知识状态远不完整。在理解某些类型的非一致双曲系统方面已经取得了进展，但是对于已经研究的系统和更广泛类型的系统，仍然存在许多悬而未决的问题。该研究项目将为几种重要的非均匀双曲系统提供新的结果，并且是发展对显示混沌行为的物理相关系统更全面理解的重要一步。一致双曲理论的关键要素包括平衡状态和SRB测量的存在性和唯一性结果，以及这些测量的统计特性，如控制观测值在预期平均值附近长期波动的中心极限定理，以及描述远离该平均值的结果概率的大偏差原理。这些结果中的一些，但不是全部，是已知的一类一维图及其扰动（强耗散图），部分双曲系统和测地线流。PI将把这些结果扩展到弱耗散图和更一般的测地线流，并将加强所有这三个类别的现有结果。使这些扩展成为可能的关键创新是PI和他的合著者引入了非均匀双曲性的新工具：用于构造SRB测量的“有效双曲性”概念，以及用于平衡态的唯一性和统计性质的“热力学规范”概念。这些工具已经产生了许多新的结果，并且明确地适用于更广泛的系统类别。

It is natural to ask how much our knowledge of the present tells us about the future. If we study some system that evolves in time according to known rules, what predictions can we make based on observing the present state of the system? It is by now well-understood that even relatively simple systems can display chaotic behavior, in which our ability to make exact predictions decays quite quickly as we look further ahead. In this case one may hope to treat the system as a random process and make statistical predictions using tools from probability theory. This approach has been successfully carried out for uniformly hyperbolic systems, the most strongly chaotic. However, most physically realistic examples fall outside of this class, including important models from meteorology (the Lorenz system), population dynamics (the logistic map), and others. This motivates the study of non-uniformly hyperbolic systems, where the present state of knowledge is much less complete. There has been progress towards understanding certain classes of non-uniformly hyperbolic systems, but many open problems remain, both for systems that have been studied and for broader classes of systems. This research project will give new results for several important classes of non-uniformly hyperbolic systems, and is an important step in developing a more complete understanding of physically relevant systems displaying chaotic behavior.Key elements of the uniformly hyperbolic theory include existence and uniqueness results for equilibrium states and SRB measures, together with statistical properties for these measures, such as the central limit theorem governing long-term fluctuations of observations around an expected average, and large deviations principles describing the probability of outcomes far from that average. Some of these results, but not all, are known for classes of one-dimensional maps and their perturbations (strongly dissipative maps), partially hyperbolic systems, and geodesic flows. The PI will extend these results to weakly dissipative maps and more general geodesic flows, and will strengthen existing results in all three categories. The key innovation making these extensions possible is the introduction by the PI and his co-authors of new tools for non-uniform hyperbolicity: a notion of "effective hyperbolicity" for the construction of SRB measures, and a notion of "thermodynamic specification" for uniqueness and statistical properties of equilibrium states. These tools have already yielded a number of new results and have clear applicability to broader classes of systems.