Dynamical Systems, Rigidity and Related Topics

翻译后摘要：拟议的计划的中心主题是刚性在动力学的各个方面的研究。新的方法和见解已被PI和他的合作者，导致最近推出， 在高阶阿贝尔群作用的不变测度刚性和轨道结构的可微刚性问题上取得了重要进展。基于这些方法取得的进展产生了富有成效的应用丢番图逼近问题的数论和算术量子混沌问题。该方案主要有以下几个方向：1.高阶阿贝尔群的部分双曲作用的局部可微刚性，重点是动力系统，调和分析/群表示和几何方法的结合.全局刚性Anosov行动，使用各种方法的基础上不变的刚性几何结构。在正熵情况下，利用创新的高熵和低熵方法以及零熵情况下的新方法，研究了不变测度的刚性.的量子唯一遍历性和疤痕存在性问题 Finsler测地线流和台球在1000。精确的渐近和乘法下界的增长的数量的周期轨道的广泛类动力系统的非一致双曲行为。可测动力系统的光滑实现问题。"刚性"的数学概念有很多方面。其最简单和最基本的表现可以从下面的初等例子中看出：少数几个特殊类型的方程或不等式可能意味着大量的方程。例如，如果n个数字的算术平均值与几何平均值一致，（一个等式）那么所有的数都相等（n-1个方程）。PI早期研究的一个例子在概念上相似，但技术上要复杂得多：负曲率的紧致曲面，即任何测地线三角形的角之和小于180度的有界几何形状，其中两个表征整体和统计体积增长的数字（拓扑熵和度量熵）重合，具有恒定的负曲率，即测地线三角形的角之和由面积不均匀地确定。 本基金的研究既涉及多维时间动力系统刚性现象的深入研究，也涉及数学和数学物理若干领域的突出应用的扩展和发展。后者包括：1）多个无理数同时用有理数逼近的问题; 2）当普朗克常数趋于零时，某些量子力学系统的行为与其经典极限之间的联系。其中心思想是，经典极限的某些性质（例如一方面是双曲性或“混沌”，另一方面是某些类型的周期轨道的存在）反映在量子系统的行为中，例如“量子态的均匀分布”和“疤痕”。

Abstract:The central theme of the proposed program is the study of various aspects of rigidity in dynamics.New methods and insights have been recently introduced by the PI and his collaborators which led to a significant progress in the problem of rigidity of invariant measures and the differentiable rigidity of orbit structure for actions of higher rank abelian groups. Advances achieved based on these methods engendered fruitful applications to Diophantine approximation problems in number theory and to the problem in arithmetic quantum chaos. There are several major directions in the proposed program:1. Local differentiable rigidity for partially hyperbolic actions of higher rank abelian groups with the emphasis on the combination of the dynamical systems, harmonic analysis/group representation and geometric methods.2. Global rigidity of Anosov actions, using various approaches based on invariant rigid geometric structures.3. Rigidity of invariant measures using the innovative high entropy and low entropy methods in the positive entropy case as well as new approaches to the zero entropy case.4. Problems of quantum unique ergodicity and existence of scars for Finsler geodesic flows and billiards in polygons.5. Precise asymptotic and multiplicative lower bounds for the growth of the number of periodic orbits for broad classes of dynamical systems with non-uniformly hyperbolic behavior.6. The problem of smooth realization of measurable dynamical systems.Mathematical concept of ``rigidity'' has many facets. Its simplest and most basic manifestations can be seen from the following elementary example: a small number of equations or inequalities of aspecial type may imply much larger number of equation. For example, ifthe arithmetic mean on n numbers coincides with the geometric mean(one equation) then the numbers are all equal ( n-1 equations).An example from the PI's earlier research is conceptuallysimilar albeit technically much more sophisticated: a compactsurface of negative curvature, i.e. a bounded geometric shape where anygeodesic triangle has the sum of its angles less than 180 degrees, forwhich two numbers characterizing global and statistical volume growth(topological and metric entropy) coincide has constant negativecurvature, i.e. the sum of the angles of a geodesic triangle isuniquely determined by the area. The research under the present grant involves both deeperinvestigation of rigidity phenomena for dynamical systems with multi-dimensional time,and expansion and development of striking application to several areas of mathematics and mathematical physics. Among the latter are: 1) problems of simultaneous approximation of several irrational numbers by rationals and 2) connection between the behavior of certain class of quantum mechanical systems and their classical limits when Plank constant goes to zero. The central idea is that certain properties of classical limits(such as hyperbolicity or ``chaos" on the one hand and presence of certain types of periodic orbits on the other) is reflected in the behavior of quantum systems such as``unifrom distibution of quantum states " and ``scars".