Topics in Smooth Ergodic Theory: Stochastic Properties, Thermodynamic Formalism, Coexistence

上个世纪后半叶影响到许多科学分支的最伟大的发现之一是被称为“确定性混沌”的现象--在纯粹的确定性系统中出现不规则的混沌运动。具有这种运动类型的模型在物理、生物、化学以及工程和经济学中都有广泛的应用。双曲性理论是一般动力系统理论的重要组成部分，它为确定性混沌现象提供了数学基础，它为研究人员提供了一种工具，使他们能够利用有关轨道的无穷小双曲行为的信息来描述非线性动力系统的整体性质。双曲现象的研究起源于Artin、Morse、Hedlund和Hopf的开创性工作，而对双曲动力系统的系统研究则是由斯梅尔、阿诺索夫和西奈开创的，他们研究的是具有高度不可预测性和表现出强混沌行为的强双曲行为的动力系统。在这个项目中，PI考虑双曲性的最弱形式(因此是最普遍的)，称为非均匀双曲性。后者起源于国际和平协会的工作。非一致双曲系统的研究是基于Lyapunov指数理论的，它为检测和描述系统的双曲性质提供了一些“实用”的工具。现代非均匀双曲性理论在遍历理论、数学和统计物理、黎曼几何以及其他数学领域和更远的领域有许多应用。该项目为研究生提供了研究培训的机会。在以往成果的基础上，PI将开展广泛的研究计划，包括以下主题：(1)非一致双曲动力系统的热力学形式化。利用几何测度论中的一些思想来构造双曲动力学中的平衡测度。新方法的基础是通过动力学推进Caratheodory度量与由势产生的Caratheodory维度结构相关联；(2)双曲和非双曲行为本质共存。这是为了理解两种不同类型的动力学行为--完全双曲的(正熵)和非双曲的(零熵)--如何以一种基本的方式共存。该项目的目的是构造表现本质共存现象的哈密顿系统和测地流，从而为经典的Kolmogorov-Arnold-Moser理论提供新的见解；(3)研究动力学中的两个重要猜想：(I)Katok熵猜想，声称保持唯一遍历微分同构的体积有零Kolmogorov-Sinai熵；(Ii)Baire范畴猜想，声称光滑动力系统及其上的连续上循环的不规则集具有第二Baire范畴；(4)紧致流形上具有指数和多项式衰减相关关系的映射。其目的是通过证明任何光滑流形都允许具有多项式或指数相关衰减的保容双曲微分同态，并满足中心极限定理，从而大大促进对平滑实现问题的理解。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

One of the greatest discoveries of the second half of the last century, which impacted many branches of science, was the phenomenon known as ``deterministic chaos'' – the emergence of irregular chaotic motions in purely deterministic systems. Models with this type of motions can be widely found in physics, biology, chemistry, as well as in engineering and economics. Hyperbolicity theory, which is an important part of general theory of dynamical systems, provides a mathematical foundation for the deterministic chaos phenomenon by supplying researchers with tools that allow them to describe global properties of a nonlinear dynamical system using information about infinitesimal hyperbolic behavior of its trajectories. The study of hyperbolic phenomena originated in the seminal works of Artin, Morse, Hedlund and Hopf, but the systematic study of hyperbolic dynamical systems was initiated by Smale, Anosov and Sinai, who studied dynamical systems with strong hyperbolic behavior which possess high level of unpredictability and exhibit strong chaotic behavior. In this project, the PI considers the weakest (hence, most general) form of hyperbolicity, known as non-uniform hyperbolicity. The latter originated in the work of the PI. The study of non-uniformly hyperbolic systems is based upon the theory of Lyapunov exponents, which provides some “practical” tools to detect and describe hyperbolic properties of the systems. The modern non-uniform hyperbolicity theory has numerous applications to ergodic theory, mathematical and statistical physics, Riemannian geometry, and other areas of mathematics and beyond. The project provides research training opportunities for graduate students.Building upon past results the PI will carry out a broad research program which includes the following topics: (1) Thermodynamic formalism for non-uniformly hyperbolic dynamical systems. Some ideas from geometric measure theory are used to construct equilibrium measures in hyperbolic dynamics. The new method is based on pushing forward by the dynamics the Caratheodory measure associated with the Caratheodory dimension structure generated by the potential; (2) Essential coexistence of hyperbolic and non-hyperbolic behavior. This is to understand how two different types of dynamical behavior - fully hyperbolic (positive entropy) and non-hyperbolic (zero entropy) - can coexist in an essential way. The project is aimed at constructing Hamiltonian systems and geodesic flows which exhibit the essential coexistence phenomenon thus providing new insights in the classical Kolmogorov-Arnold-Moser theory; (3) The study of two important conjectures in dynamics: (i) Katok's entropy conjecture, claiming that a volume preserving uniquely ergodic diffeomorphism has zero Kolmogorov-Sinai entropy; (ii) Baire Category conjecture, claiming that irregular sets for smooth dynamical systems and for continuous cocycles over them have the 2nd Baire Category; (4) Maps with exponential and polynomial decay of correlations on compact manifolds. The goal is to substantially advance the understanding of smooth realization problem by showing that any smooth manifold admits a volume preserving hyperbolic diffeomorphism with polynomial or exponential decay of correlations and also satisfies the Central Limit Theorem.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.