Topics in Dynamical Systems: Attractors, Dimension, Lattice Models

拟议的工作涉及动力系统理论(特别是双曲理论)的广泛问题，以及它在维度理论、数学物理(包括耦合映射格子)、遍历理论和统计力学中的应用及其关系。动力系统维理论的工作包括：描述各种双曲型动力系统的多重分形谱(包括维谱、熵谱和Lyapunov指数谱)以及相应的多重分形分解(非共形扩展映射、公理A微分同胚等)；建立多维共形扩展映射的多重分形刚性现象(直到多重分形谱的动力系统的分类)。在耦合映象格子(CML)理论中，主要研究：CML的无限维SRB度量，包括建立特征性质和热力学极限，描述Lyapunov谱，证明熵公式；CML行波解的稳定性以及CML行波解与相应的偏微分方程组(如Ginzburg-Landau方程、Kolmogorov-Petrovski-Peskunov方程、Huxley方程等)之间的关系。对称和自相似原理是自然界最美丽的创造。例如，它们用分形图表示，分形图以其美丽但复杂的几何形状而闻名。分形图的例子多种多样，从广为人知的成本线或山脉，到不太为人所知的恒星在星系和宇宙中的星系或植物根系中的分布。维度理论是一种用来解释分形结构的数学理论。而在动力学中，不变分形的存在往往会导致不稳定的“类湍流”运动，并与“混沌”行为相联系。因此，对分形学的研究有助于理解最复杂的现象，如海洋或大气中的湍流。这项拟议的工作涉及最近发展起来的一个领域的研究，该领域位于维理论和动力系统理论之间的界面上。主要研究不变分形学及其对系统随机性质的影响，旨在对动力系统中的现代维度理论进行全面和系统的研究。预计结果不仅对高级数学家，而且对依赖动态过程数学建模的广泛范围的科学家，包括物理学家、数值建模专家、工程师、分子生物学家等都具有重要意义。

The proposed work involves a broad range of problems in the theory of dynamical systems (in particular, hyperbolic theory) and its applications to and relations with dimension theory, mathematical physics (including coupled map lattices), ergodic theory, and statistical mechanics. Projects in the dimension theory of dynamical systems include: the description of several multifractal spectra (including dimension spectra, entropy spectra, and spectra for Lyapunov exponents) and the corresponding multifractal decompositions for various classes of dynamical systems of hyperbolic type (non-conformal expanding maps, Axiom A diffeomorphisms, etc.); and the establishment of the multifractal rigidity phenomenon for multidimensional conformal expanding maps (the classification of dynamical systems up to multifractal spectra). In the theory of coupled map lattices (CML), the principal investigator will study: infinite-dimensional SRB measures for CML, which includes establishing the characteristic property and thermodynamical limit, describing the Lyapunov spectrum, and proving the entropy formula; and the stability of traveling wave solutions of CML and relations between traveling wave solutions for CML and the corresponding PDE (such as Ginzburg-Landau equation, Kolmogorov-Petrovski-Peskunov equation, Huxley equation, etc.). The principles of symmetry and self-similarity are nature's most beautiful creations. For example, they are expressed in fractals which are famous for their beautiful but complicated geometric shapes. Examples of fractals vary from well-known ones-cost lines or mountain ranges-to less known-distribution of stars in galaxies and galaxies in the universe or root systems of plants. Dimension theory is a mathematical theory which is designed to explain fractals' structure. And in dynamics the presence of invariant fractals often results in unstable ``turbulent-like'' motions and is associated with ``chaotic'' behavior. Thus the study of fractals can help understand the most complicat ed phenomena such as turbulence in the ocean or atmosphere. The proposed work involves research in a recently developing area which lies in the interface between dimension theory and the theory of dynamical systems. Focusing on invariant fractals and their influence on stochastic properties of systems, the principal investigator intends to provide a comprehensive and systematic treatment of modern dimension theory in dynamical systems. Results are expected to be of great importance not only to advanced mathematicians but to a wide range of scientists who depend upon mathematical modeling of dynamical processes, including physicists, specialists in numerical modeling, engineers, molecular biologists, etc.