Topics in Dynamical Systems, Ergodic Theory and Geometry

该项目涵盖紧流形上的阿诺索夫（通常是双曲）和高阶交换群（离散和连续）的某些类部分双曲作用的分类，高阶半简单李群中格子的几类光滑作用的分类，对阿诺索夫不变测度和离散和连续高阶阿贝尔群的部分双曲代数作用的系统研究，开发高阶阿贝尔群作用的双曲测度的一般理论，以及各类动力系统不变分布统一理论的发展。 主要工具包括平滑遍历理论（李雅普诺夫特征指数和非均匀双曲性）、非平稳范式理论、几何超刚性和群表示理论。 动力系统理论是快速发展的非线性动力学和混沌理论领域的数学基础，为这些领域提供了严格分析的主要范式和工具。这些范式反过来在自然科学、社会科学和工程学中众多问题的数学模型的开发和分析中发挥着关键作用。 动力学的标准设置将时间视为一维，离散或连续。中心结论之一是，在各种情况下，轨道结构是丰富的，其鲁棒特征可以通过易于理解的符号模型来描述。 目前资助的研究着眼于时间是多维而相空间仍然是有限维的情况，即所考虑的系统的状态可以通过一组有限的数值参数来描述。 这些案例中的主要范式被证明是截然不同的。一方面，符号模型不再有效。另一方面，轨道结构被证明是“刚性的”，即不仅其稳健的特征，而且更微妙的结构，包括渐近行为的精细统计，在扰动下都不会改变。该项目旨在识别并系统地研究一套“通用模型”，以取代一维时间情况下的符号模型。

The project covers a classification of Anosov (normally hyperbolic) and certain classes of partially hyperbolic actions of higher-rank commutative groups, both discrete and continuous, on compact manifolds, classification of several classes of smooth actions of lattices in higher-rank semi-simple Lie groups, a systematic study of invariant measures for Anosov and partially hyperbolic algebraic actions of discrete and continuous higher-rank abelian groups, development of a general theory of hyperbolic measures for actions of higher-rank abelian groups, and developments of a uniform theory of invariant distributions for various classes of dynamical systems. Among the principal tools are smooth ergodic theory (Lyapunov characteristic exponents and non-uniform hyperbolicity), the theory of non-stationary normal forms, geometric super-rigidity and the theory of group representations. The theory of dynamical systems is the mathematical foundation of the rapidly developing fields of non-linear dynamics and chaos theory which provides these fields with their principal paradigms and tools of rigorous analysis. Those paradigms in turn play a key role in the development and analysis of mathematical models for numerous problems within natural and social sciences and engineering. The standard setup in dynamics considers time as one-dimensional, either discrete or continuous. One of the central conclusions is that in a variety of situations, the orbit structure is rich and its robust features can be described by well-understood symbolic models. The research under the present grant looks at the situation when the time is multi-dimensional while the phase space is still finite-dinemsional, i.e. the state of a system under consideration can be described by a finite set of numerical parameters. The leading paradigms in these cases turn out to be strikingly different. On the one hand, the symbolic models are no longer valid. On the other, the orbit structure turn out to be "rigid", i.e. not only its robust f eatures, but much more subtle structure, including the fine statistics of asymptotic behavior, do not change under perturbations. The project intends to identify and systematically study a set of "universal models" which replace the symbolic models in the case of one-dimensional time.