Dynamical Systems, Rigidity and Related Topics

摘要：该计划的中心主题是研究动力学中刚性的各个方面。PI和他的合作者最近提出了新的方法和见解，导致在高阶阿贝尔群作用的不变测度的刚性和轨道结构的可微刚性问题上取得了重大进展。基于这些方法取得的进展在数论中的丢番图近似问题和算术量子混沌问题上产生了卓有成效的应用。建议的方案有几个主要方向：1。高阶阿贝尔群部分双曲作用的局部可微刚性，重点讨论了动力系统、调和分析/群表示和几何方法的结合。基于不变刚性几何结构的各种方法的Anosov动作的全局刚性。3 .利用创新的高熵和低熵方法研究正熵情况下不变测度的刚性以及零熵情况下的新方法。5.多边形中Finsler测地流和台球的量子唯一遍历性和伤痕的存在性问题。具有非均匀双曲性质的广义动力系统周期轨道数目增长的精确渐近和乘法下界。可测动力系统的平滑实现问题。数学中的“刚性”概念有许多方面。它最简单和最基本的表现可以从下面的基本例子中看到：少数特定类型的方程或不等式可能隐含着大量的方程。例如，如果n个数字的算术平均值与几何平均值一致（一个方程），那么这些数字都相等（n-1个方程）。PI早期研究的一个例子在概念上是相似的，尽管在技术上要复杂得多：一个负曲率的紧曲面，即一个有界的几何形状，其中任何测地三角形的角度之和小于180度，其中两个表征全局和统计体积增长的数字（拓扑和度量熵）重合，具有恒定的负曲率，即测地三角形的角度之和唯一地由面积决定。本基金的研究包括对多维时间动力系统的刚性现象的深入研究，以及在数学和数学物理的几个领域的显著应用的扩展和发展。后者包括：1)用有理数同时逼近几个无理数的问题；2)当普朗克常数趋于零时，某类量子力学系统的行为与其经典极限之间的联系。中心思想是经典极限的某些性质（如双曲性或“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，如“量子态的均匀分布”和“伤痕”。

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".