A Comprehensive Program in Modern Dynamics with Emphasis on Rigidity

该计划旨在推进动力系统的现代结构理论的主要领域的几个研究方向：双曲（主要和非均匀），部分双曲，抛物和椭圆，特别强调 对各种刚性现象和跨领域的几个强大的方法的适用性。PI及其合作者在过去几年中引入了新的方法和见解。 最近的进展允许获得算术结构几乎无处不在光滑的意义上的惠特尼一组任意大的措施一类最大秩行动（排名的行动组是一个小于相空间的维度）下一个非常一般的假设足够的复杂性。确定了一个二分法：对于此类行动：要么根本没有指数复杂性，要么它的度量（熵的一种形式）从下到上被一个常数统一限制，这个常数随着维数的增加而增加到无穷大。该程序的发展方向包括将算术性程序推广到秩与维数无关的更广泛的作用类，测度刚性在Zimmer程序中的应用，进一步发展高阶阿贝尔作用的熵理论和熵刚性，完成高阶阿贝尔群的部分双曲代数作用的可微刚性程序，研究完全非双曲的幂幺作用的刚性现象，这与经典情况形成鲜明对比，非标准KAM型不变曲线定理和进一步研究光滑实现问题。动力系统作为时间演化的数学模型， 自然科学和社会科学领域的各种进程。它也有一个令人惊讶的广泛的应用范围内的核心数学学科，最特别的是各种领域的几何和数论。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，甚至是更一般的性质，由群的关键数学概念描述。 一维时间的经典情形与多维时间的经典情形有着本质的区别：在一维时间的经典情形中，各种复杂性和混沌现象会逐渐地、混沌地出现 在同一个系统中，行为可以而且通常确实与有序行为共存（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），在后者中，正如PI及其合作者所建立的那样）复杂性通常是全局性的，事实上，混沌以一种高度结构化的方式出现，并且具有高的复杂性水平，尽管可以通过适当的版本来测量熵的基本概念。

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.