Topics in Dynamical Systems: Attractors, Dimension, Lattice Models

该提案有几个主要主题：1）非一致双曲动力系统的热力学形式主义-这是建立非零李雅普诺夫系统相变的统计物理。2)非一致双曲系统的SRB测度-这是为了建立耗散系统的“物理自然”类不变测度。3)双曲和非双曲行为的共存-这是为了补充著名的Kolmogorov-Arnold-Moser（KAM）理论，通过构建非零李雅普诺夫指数和零熵区域共存的系统的特定示例。4)Anosov刚性-这是建立一个微妙的关系之间的一致和非一致类型的双曲。5)非共形排斥子的维数-这是研究一般非共形排斥子的Hausdorff维数。拟议的研究涉及光滑动力系统理论及其在数学和统计物理和几何中的应用问题。研究的主要课题是双曲动力系统，它为广为人知的“确定性混沌”范式提供了数学基础-在纯粹确定性动力系统中出现不规则的混沌运动。这种范式认为，具有足够强双曲行为的非线性动力系统的全局性质的结论可以从研究线性化系统沿着其轨迹推导出来。对双曲现象的研究起源于Artin、莫尔斯、Hedlund和Hopf关于紧致曲面上测地线流的遍历性质的开创性著作。后来，在其他情况下观察到双曲行为（例如，Smale马蹄铁和双曲toral自同构）。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行为的系统。此类系统具有高度的不可预测性并表现出强烈的混乱行为。在该提案中，PI考虑了最弱（因此也是最普遍）的双曲形式，称为非均匀双曲。非一致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的理论对整个领域有着广泛的影响。PI将在此期间与研究生合作。

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.