A Comprehensive Program in Modern Dynamics with Emphasis on Rigidity

该计划旨在推进现代动力系统结构理论主要领域的几个研究方向：双曲（大部分和非均匀），部分双曲，抛物线和椭圆，特别强调各种刚性现象和几种跨领域的强大方法的适用性。在过去的几年里，PI和他的合作者引入了新的方法和见解。最近的进展允许在具有足够复杂性的非常一般的假设下，在任意大测度集上获得一类最大秩作用（作用群的秩比相空间的维数小1）的几乎处处光滑的惠特尼意义上的算术结构。对于这样的行为，建立了一个二分法：要么根本不存在指数复杂性，要么其度量（熵的一种形式）从下到下被一个随维度增长到无穷大的常数统一限定。规划的方向包括：将算术规划推广到更广泛的秩与维数无关的作用类；测度刚性在Zimmer规划中的应用；进一步发展熵理论和高阶阿贝尔作用的熵刚性；完成高阶阿贝尔群的部分双曲代数作用的可微刚性规划；单能作用对经典情况的鲜明对比，非标准的kam型不变曲线定理和光滑实现问题的进一步研究。动力系统是自然科学和社会科学领域中各种过程的时间演化的数学模型。它在核心数学学科中也有令人惊讶的广泛应用，特别是在几何和数论的各个领域。在许多情况下，“时间”不一定是通常的一维时间，但它可以是多维的，甚至更一般的性质，这是由关键的数学概念群描述的。一维时间的经典情况和多维时间的经典情况之间有一个关键的区别：而在前一种情况下，各种复杂性和混沌现象可能逐渐出现，混沌行为可以并且通常在同一系统中与有序行为共存（例如，有序的行星运动与太阳系中一些小行星和较小物体的混沌行为）。在后者（由PI和他的合作者建立）中，复杂性通常是全球性的，事实上，混沌以一种高度结构化的方式出现，并且通过熵的基本概念的适当版本来衡量，具有高的（尽管可计算的）复杂性水平。

The Proposed program aims at advancing of several directions of research across the principal areas of the modern structural theory of dynamical systems: hyperbolic (mostly and non-uniform), partially hyperbolic, parabolic and elliptic with a particular emphasis on various kinds of rigidity phenomena and on applicability of several powerful methods across areas. New methods and insights have been introduced by the PI and his collaborators over the past years. The most recent advances allowed to obtain arithmetic structure almost everywhere smooth in the sense of Whitney on a set of arbitrarily large measure for a class of maximal rank actions (the rank of the acting group is one less than the dimension of phase space) under a very general assumption of sufficient complexity. A dichotomy was established: for such actions: either there is not exponential complexity at all, or its measure ( a version of entropy) is uniformly bounded from below by a constant that grows to infinity with dimension. Directions in the proposed program include extension of the arithmeticity program to broader classes of actions with the rank not related to dimension, applications of measure rigidity to Zimmer program, further development of the entropy theory and entropy rigidity for higher rank abelian actions, completing the program of differentiable rigidity of partially hyperbolic algebraic actions of higher rank abelian groups, the study of rigidity phenomena for totally non- hyperbolic, unipotent actions that present a striking contrast for the classical case, non-standard KAM-types invariant curve theorem and further investigation of the smooth realization problem.Dynamical systems serve as mathematical models of time evolution of various process across the areas of natural and social sciences. It also has a surprisingly broad range of applications within core mathematical disciplines, most particularly to various areas of geometry and number theory. Within many of these contexts the "time" is not necessarily the usual one-dimensional time but it can be multi-dimensional or, of even more general nature that is described by the key mathematical concept of group. There is a crucial difference between the classical case of one-dimensional time and that of multi-dimensional time: while in the former case various complexity and chaotic phenomena may appear gradually and chaotic behavior can and usually does coexists with ordered one within the same systems (e.g. ordered planetary motions vs. chaotic behavior of some asteroids and smaller objects in the solar system), in the latter as was established by the PI and his collaborators) complexity is often global and in fact chaos appears in a highly structured way and with high albeit calculable levels of complexity as measured by a proper version of the fundamental notion of entropy.