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

上个世纪后半叶最伟大的发现之一，影响了许多科学分支，是被称为“确定性混沌”的现象--在纯粹确定性系统中出现不规则的混沌运动。具有这种类型的运动的模型可以广泛地在物理学、生物学、化学以及工程和经济学中找到。双曲性理论是动力系统一般理论的重要组成部分，它为研究人员提供了一种工具，使他们能够利用非线性动力系统轨迹的无穷小双曲行为的信息来描述系统的全局性质，从而为确定性混沌现象提供了数学基础。双曲现象的研究起源于Artin，莫尔斯，Hedlund和Hopf的开创性著作，但双曲动力系统的系统研究是由Smale，Anosov和Sinai发起的，他们研究具有强双曲行为的动力系统，这些动力系统具有高水平的不可预测性并表现出强混沌行为。在这个项目中，PI考虑了双曲性的最弱（因此也是最普遍的）形式，称为非均匀双曲性。后者起源于PI的工作。李雅普诺夫指数理论是研究非一致双曲系统的基础，它为检测和描述非一致双曲系统的双曲性提供了“实用”工具。现代非均匀双曲性理论在遍历理论、数学和统计物理、黎曼几何和其他数学领域以及更远的领域有着众多的应用。该项目为研究生提供了研究培训的机会。PI将在过去成果的基础上开展广泛的研究计划，其中包括以下主题：（1）非一致双曲动力系统的热力学形式。利用几何测度理论中的一些思想来构造双曲动力学的平衡测度。新方法是基于动力学推进与势产生的Caratheodory维数结构相关联的Caratheodory测度;（2）双曲与非双曲行为的本质共存。这是为了理解两种不同类型的动力学行为-全双曲（正熵）和非双曲（零熵）-如何以一种基本的方式共存。（3）研究动力学中的两个重要问题：（1）Katok熵猜想，即保体积的唯一遍历同态的Kolmogorov-Sinai熵为零;（ii）Baire范畴猜想，认为光滑动力系统的不规则集和其上的连续上圈都有第二Baire范畴;（4）紧流形上的关联指数衰减和多项式衰减的映射。其目标是通过证明任何光滑流形都允许具有多项式或指数衰减的相关性的保体积双曲型同构，并满足中心极限定理，从而大大推进对光滑实现问题的理解。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.