Topics in Dynamical Systems: Attractors, Dimension, Lattice Models

提出的研究涉及光滑动力系统理论中的问题及其在数学和统计物理以及几何领域的应用。研究的主要对象是双曲动力系统，系统表现出“确定性混沌”现象；也就是说，在纯确定性动力系统中出现不规则混沌运动。具有足够强双曲行为的动力系统的许多全局性质可以通过研究沿其轨道的线性化近似系统推导出来。对这一现象的研究起源于斯梅尔、阿诺索夫和西奈的开创性著作。这样的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。它是在PI的早期工作中引入和研究的，它基于近轨道随时间分离的理论，称为李亚普诺夫指数。本研究的目标是为该主题开发一套分析工具，并构造一些具有丰富统计性质的自然不变测度。本文主要研究了以下几个问题：1)非均匀双曲动力系统的热力学形式化——即建立平衡测度的存在唯一性，研究非零Lyapunov指数系统的相变；2)非均匀双曲系统的Sinai-Ruelle-Bowen （SRB）测度——即构造具有非零Lyapunov指数的耗散系统的“物理自然”类不变测度；3)通过不依赖于系统符号表示的新方法构建非均匀双曲系统的Gibbs和平衡测度；4)双曲和非双曲行为的本质共存——即通过构建具有非零Lyapunov指数区域和零拓扑熵区域共存的系统的特定示例来补充著名的Kolmogorov-Arnold-Moser （KAM）理论。

The proposed research deals with problems in the theory of smooth dynamical systems and their applications to mathematical and statistical physics as well as to areas in geometry. The main subject of study is hyperbolic dynamical systems, systems exhibiting "deterministic chaos" phenomena; that is, the appearance of irregular chaotic motions in purely deterministic dynamical systems. Many global properties of a dynamical system with sufficiently strong hyperbolic behavior can be deduced from studying the linearized approximation system along its orbits. The study of this phenomena originated in seminal works of Smale, Anosov and Sinai. Such systems possess a 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 non-uniform hyperbolicity. It was introduced and studied in earlier work of the PI and is based upon the theory of separation of close orbits over time, called Lyapunov exponents. The goal of the proposed research is to develop a set of analytic tools for the subject and to construct some natural invariant measures with rich statistical properties. There are several main topics in the proposal: 1) Thermodynamic formalism for non-uniformly hyperbolic dynamical systems -- that is to establish existence and uniqueness of equilibrium measures and study phase transitions for systems with nonzero Lyapunov exponents; 2) Sinai-Ruelle-Bowen (SRB) measures for non-uniformly hyperbolic systems - that is to construct "physically natural" classes of invariant measures for dissipative systems with nonzero Lyapunov exponents; 3) Gibbs and equilibrium measures for non-uniformly hyperbolic systems constructed via a new approach that does not rely on symbolic representation of the system; 4) Essential coexistence of hyperbolic and non-hyperbolic behavior -- that is to complement the famous Kolmogorov-Arnold-Moser (KAM) theory by constructing particular examples of systems with coexistence of areas with nonzero Lyapunov exponents and areas with zero topological entropy.