Topics in Dynamical Systems: Attractors, Dimension, Lattice Model

摘要。该项目概述了三个方向的科学活动。首先是耦合映射格（CML）理论，其中研究的重点是无限维SRB测量的热力学极限（在西奈的意义上）；行波解的稳定性，以及通过行波从cml到偏微分方程的过渡。它还包括一些著名的偏微分方程的研究（例如，神经生物学中非常感兴趣的Swift-Hohenberg方程）。第二个方向涉及动力系统的维数理论，包括Birkhoff平均的多重分形分析及其在数论中的应用，以及保角展开映射的多重分形刚性现象。第三个研究方向涉及光滑遍历理论的一些最新优势。特别地，提出了构造具有非零Lyapunov指数（在任意黎曼流形上且具有可数遍历分量）的保体积微分同态的一些新例子。本文还提出了二维双曲测度的多重分形形式。本研究的主要目的是进一步发展混沌的数学理论。它涉及该理论的各个方面，包括动力系统中混沌运动的出现（这是物理学，生物学，经济学等各种现象的数学模型）以及混沌制度与空间分形几何结构之间的相互作用。提出研究混沌行为的一些宏观特征（如熵和李雅普诺夫指数），并将它们与空间的分形维数联系起来。并提出了一些新的混沌系统的例子，包括保守和耗散，有限维和无限维。这些例子可以作为流体力学中的湍流、神经科学中的神经元和记忆活动、植物生物学中的植物生长等极其复杂现象的模型。

Abstract.The project outlines scientific activity in three directions. The first is the theory of coupled map lattices (CML) where research is focused on the thermodynamical limit for infinite-dimensional SRB measures (in the sense of Sinai); stability of traveling wave solutions, and transition from CMLs to PDEs via traveling waves.It also includes the study of some well-known PDEs (for example, Swift-Hohenberg equation which is of great interest in neurobiology). The second direction is concerned with the dimension theory of dynamical systems and includes the multifractal analysis of Birkhoff averages with applications to some problems in number theory as well asthe multifractal rigidity phenomenon for conformal expanding maps. The third direction of research deals with some recent advantages in smooth ergodic theory. In particular, it is proposed to construct some new examples of volume preserving diffeomorphismswith non-zero Lyapunov exponents (on any Riemannian manifold and with countable number of ergodic components). It is also proposed to effect a multifractal formalism for two-dimensional hyperbolic measures.The main goal of the proposed research is to develop further the mathematical theory of chaos. It deals with various aspects of this theory including the appearance of chaotic motions in dynamical systems (which are mathematical models of various phenomena in physics, biology, economics, etc. ) and the interplay between chaotic regimes and fractal geometric structure of the space. It is proposed to study some macro-characteristics of chaotic behavior (such as entropy and Lyapunov exponents) and relate them to fractal dimension of the space. It is also proposed to construct some new examples of chaotic systems both conservative and dissipative, finite-dimensional as well as infinite-dimensional. These examples may serve as models of such extremely complicated phenomena as turbulence in hydrodynamics, neuron and memory activity in neuroscience, plant growth in plant biology, etc.