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的几个主要领域的研究方向：双曲线(主要是非均匀的)、部分双曲线、抛物线和椭圆，特别强调各种刚性现象和几种强大的方法在各个领域的适用性。在过去的几年里，PI和他的合作者引入了新的方法和见解。最近的进展使得在一个非常一般的足够复杂性的假设下，对于一类最大秩动(作用群的秩比相空间的维度小一)，在一组任意大的测度上，几乎处处都可以得到Whitney意义下的光滑算术结构。建立了一个二分法：对于这样的行为：要么根本不存在指数复杂性，要么它的度量(熵的一个版本)从下到下一致地被一个随维度增长到无穷大的常数所限定。所提出的程序的方向包括：将算术程序推广到更广泛的与维度无关的作用类，将度量刚性应用到Zimmer程序，进一步发展了熵理论和高阶阿贝尔作用的熵刚性，完成了高阶阿贝尔群的部分双曲代数作用的可微刚性程序，研究了与经典情况截然不同的完全非双曲、幂等作用的刚性现象，非标准Kam型不变曲线定理，以及进一步研究了光滑实现问题。动力系统是自然科学和社会科学领域各种过程的时间演化的数学模型。它在核心数学学科中的应用范围也令人惊讶地广泛，尤其是在几何和数论的各个领域。在许多这样的背景下，“时间”不一定是通常的一维时间，但它可以是多维的，或者是由关键的数学概念群所描述的更一般的性质。一维时间的经典情况与多维时间的经典情况之间有一个至关重要的区别：虽然在前一种情况下，各种复杂和混沌现象可能会逐渐出现，并且混沌行为可以而且通常确实与同一系统内的有序行星共存(例如，有序的行星运动与太阳系中一些小行星和较小天体的混沌行为)，在后者，正如PI及其合作者所建立的那样，复杂性往往是全球性的，而且事实上混沌以一种高度结构化的方式出现，并且通过熵基本概念的适当版本来衡量，具有很高的复杂性，尽管可以计算。

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.