Topics in Dynamical Systems: Attractors, Dimension, Lattice Model

拟开展的研究涉及光滑动力系统中的问题及其在遍历理论、数学统计物理和几何中的应用。研究的主要对象是具有足够强轨迹不稳定性的动力系统--即所谓的双曲动力系统。双曲理论为广为人知的“确定性混沌”范式提供了数学基础--在纯粹的确定性动力系统中出现不规则的混沌运动。这一范式认为，一个具有足够强双曲性行为的非线性动力系统的全局性质可以通过研究沿其轨迹线性化的系统来得出。双曲现象的研究起源于Artin、Morse、Hedlund和Hopf关于紧曲面上测地线流的不稳定性和遍历性质的开创性工作。后来，在其他情况下也观察到了双曲线行为(例如，斯梅尔马蹄形和双曲线自同构)。双曲动力系统的系统研究是由斯梅尔、阿诺索夫和西奈开创的，他们研究了具有足够强双曲行为的动力系统。这类系统具有高度的不可预测性，并表现出强烈的混沌行为。在该建议中，PI考虑最弱(因此，最普遍)的双曲性形式，称为非均匀双曲性。非一致双曲动力系统理论起源于PI的工作(有时被称为‘’Pesin理论‘’)，对非一致双曲动力系统的研究是基于Lyapunov指数理论。提案中有三个主要主题。1.非一致双曲系统的热力学--这是在马尔可夫扩张和塔结构的最新工作的基础上，建立非零李雅普诺夫指数系统相变的统计物理。2.具有混合双曲性和稳定遍历性的动力系统--这是为了研究具有非一致双曲性行为的系统有多“典型”。Dolgopyat和PI最近的一个结果表明，这种系统存在于任何相空间上。3.双曲和非双曲行为的共存--这是对著名的Kolmogorov-Arnold-Moser(KAM)理论的补充，构造了非零Lyapunov指数和零熵区域共存的系统的特例。PI还建议将他的工作应用于Fitzhugh-Nagumo方程--描述电压脉冲通过神经轴突传播的著名模型。这提供了有趣的新的和“自然”出现的非一致双曲“奇怪”吸引子的例子，并演示了从相对简单的莫尔斯-斯梅尔系统到“奇怪”吸引子到斯梅尔马蹄形吸引子的转变。

The proposed research deals with problems in Smooth Dynamical Systems and their applications to Ergodic Theory, Mathematical and Statistical Physics and Geometry. The main subject of study is dynamical systems with sufficiently strong instability of trajectories -- the so-called hyperbolic dynamical systems. Hyperbolic theory provides a mathematical foundation for the paradigm that is widely known as ``deterministic chaos'' -- the appearance of irregular chaotic motions in purely deterministic dynamical systems. This paradigm asserts that conclusions about global properties of a nonlinear dynamical system with sufficiently strong hyperbolic behavior can be deduced from studying the linearized systems along its trajectories. The study of hyperbolic phenomena originated in seminal works of Artin, Morse, Hedlund, and Hopf on the instability and ergodic properties of geodesic flows on compact surfaces. Later, hyperbolic behavior was observed in other situations (for example, Smale horseshoes and hyperbolic toral automorphism). The systematic study of hyperbolic dynamical systems was initiated by Smale, Anosov and Sinai who studied dynamical systems with sufficiently strong hyperbolic behavior. Such systems possess high level of unpredictability and exhibit strong chaotic behavior. In the proposal the PI considers the weakest (hence, most general) form of hyperbolicity known as nonuniform hyperbolicity. The theory of nonuniformly hyperbolic dynamical systems originated in the work of the PI (sometimes this theory is referred to as ``Pesin theory'') and the study of these systems is based upon the theory of Lyapunov exponents. There are three main topics in the proposal. 1. Thermodynamics of nonuniformly hyperbolic systems -- this is to build statistical physics of phase transitions for systems with nonzero Lyapunov exponents based on recent works on Markov extensions and tower constructions. 2. Dynamical systems with mixed hyperbolicity and stable ergodicity -- this is to study how "typical" the systems with nonuniform hyperbolic behavior are. A recent result by Dolgopyat and the PI shows that such systems exist on any phase space. 3. Coexistence of hyperbolic and non-hyperbolic behavior -- this is to complement the famous Kolmogorov-Arnold-Moser (KAM) theory by constructing particular examples of systems with coexistence of nonzero Lyapunov exponents and areas with zero entropy. The PI also proposes to apply his work to the FitzHugh-Nagumo equation -- the famous model that describes the propagation of voltage impulse through a nerve axon. This provides interesting new and "naturally" appearing examples of nonuniformly hyperbolic "strange" attractors as well as demonstrates transition from relatively simple Morse-Smale systems to "strange" attractors to Smale horseshoes.