Topics in Dynamical Systems, Ergodic Theory and Geometry

项目内容包括紧流形上高阶交换群（离散和连续）的Anosov（通常双曲）和若干类部分双曲作用的分类，高阶半简单李群中若干类格的光滑作用的分类，系统地研究了高阶离散和连续阿贝尔群的Anosov和部分双曲代数作用的不变测度，高阶阿贝尔群作用的双曲测度的一般理论的发展，以及各种动力系统的不变分布的统一理论的发展。其中的主要工具是光滑遍历理论（李雅普诺夫特征指数和非一致双曲），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，它为这些领域提供了主要的范式和严格的分析工具。这些范式反过来又在自然科学、社会科学和工程领域众多问题的数学模型的发展和分析中发挥了关键作用。动力学中的标准设置认为时间是一维的，要么是离散的，要么是连续的。其中一个中心结论是，在各种情况下，轨道结构丰富，其鲁棒性可以用易于理解的符号模型来描述。本基金的研究着眼于时间是多维的，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用有限的数值参数集来描述。事实证明，这些案例中的主要范例截然不同。一方面，符号模式不再有效。另一方面，轨道结构被证明是“刚性的”，即不仅它的鲁棒特征，而且更微妙的结构，包括渐近行为的精细统计，在摄动下不会改变。本项目旨在识别和系统地研究一套“通用模型”，以取代一维时间下的符号模型。

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.