Topics in Dynamical Systems: Attractors, Dimension, Lattice Model

提出的研究涉及光滑动力系统理论中的问题及其在数学、统计物理和几何中的应用。研究的主要主题是所谓的双曲动力系统，它为被广泛称为“确定性混沌”的范式提供了数学基础-在纯确定性动力系统中出现不规则混沌运动。这一范式表明，对于具有足够强双曲行为的非线性动力系统，可以通过沿其轨迹研究线性化系统而得出关于其全局性质的结论。双曲现象的研究起源于Artin、Morse、Hedlund和Hopf关于紧致表面上测地流的不稳定性和遍历性的开创性著作。后来，在其他情况下（例如，小马蹄铁和双曲整体自同构）也观察到双曲行为。双曲动力系统的系统研究是由Smale、Anosov和Sinai发起的，他们研究了具有足够强双曲行为的动力系统。这种系统具有高度的不可预测性，并表现出强烈的混沌行为。在这个提议中，PI考虑了最弱的（因此也是最一般的）双曲形式，即非均匀双曲。非均匀双曲动力系统的理论起源于PI的工作（有时这个理论被称为“Pesin理论”），这些系统的研究是基于李雅普诺夫指数理论。提案中有三个主要主题。1. 非均匀双曲动力系统的热力学形式——这是基于最近关于马尔可夫扩展和塔结构的工作，为具有非零李雅普诺夫指数的系统建立相变的统计物理。2. 混合双曲性和稳定遍历性——这是为了研究具有非均匀双曲性的系统有多“典型”。Dolgopyat和PI最近的一个结果表明，这样的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，通过构建具有非零李雅普诺夫指数和零熵区域共存的系统的特定示例。PI还建议将他的工作应用于FitzHugh-Nagumo方程和Brusselator模型——神经生物学和化学中的著名模型。他们提供了有趣的新的和“自然”出现的非均匀双曲系统的例子，并展示了从相对简单的莫尔斯-小系统到“奇怪的”吸引子和小马蹄铁的转变。

The proposed research deals with problems in the theory of smooth dynamical systems and their applications to mathematical and statistical physics and geometry. The main subject of study is the so- called hyperbolic dynamical systems that 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 (e,g, 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. Thermodynamic formalism for nonuniformly hyperbolic dynamical 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. 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 and the Brusselator model -- the famous models in neurobiology and chemistry. They provide interesting new and "naturally" appearing examples of nonuniformly hyperbolic systems as well as demonstrate transitions from relatively simple Morse-Smale systems to "strange" attractors and to Smale horseshoes.