Topics in Dynamical Systems: Attractors, Dimension, Lattice Models

提案中有几个主要主题：1)非均匀双曲动力系统的热力学形式化-这是为了建立具有非零李雅普诺夫系统的相变统计物理。2)非均匀双曲系统的SRB测量-这是为了建立耗散系统的“物理自然”类不变测量。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Lyapunov指数和零熵区域共存的系统的特定示例。4)阿诺索夫刚性-这是建立均匀型和非均匀型双曲之间的微妙关系。5)非保形驱避剂尺寸——研究通用非保形驱避剂的豪斯多夫尺寸。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的不规则混沌运动。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线流动的遍历性质的开创性著作。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。双曲性的系统研究是由Smale、Anosov和Sinai发起的，他们研究了具有足够强双曲性的系统。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。PI的理论对整个领域产生了广泛的影响。在此奖项期间，PI将与研究生一起工作。

There are several 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. 2) SRB measures for nonuniformly hyperbolic systems - this is to to build "physically natural" class of invariant measures for dissipative systems. 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. 4) Anosov rigidity - this is to establish a subtle relation between uniform and nonuniform types of hyperbolicity. 5) Dimension of non-conformal repellers - this is to study Hausdorff dimension for generic non-conformal repellers.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 hyperbolic dynamical systems that provide 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 system along its trajectories. The study of hyperbolic phenomena originated in seminal works of Artin, Morse, Hedlund, and Hopf on 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 hyperbolicity was initiated by Smale, Anosov and Sinai who studied 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. The PI's theory has a wide impact on the entire field. The PI will work with graduate students during the period of this award.