Topics in Dynamical Systems and Ergodic Theory

摘要：本项目主要研究动力系统中的刚性现象。刚性的一个方面涉及到当一个较弱的结构(例如：可测量的)决定了一个更强的（即可微的）在某些类别的系统。另一个方面涉及在小扰动（局部可微刚性）或在某些类型的系统（全局可微刚性）下可微轨道结构或其一些重要元素的保存。然而，另一种类型的刚性出现在某些属性（如不变量之间的关系，或不变量结构的规则）迫使系统属于专门化的狭窄类时。研究计划的一部分涉及经典动力系统中各种刚性现象的识别。提出的工作的一个基本特征是一个综合的方法，它同时观察动力学中出现的所有三个主要类别的行为：椭圆，抛物线和双曲线，探索这三种范式之间的相似性和对比。主要目标之一是确定可测量轨道结构决定可微分轨道结构的新情况。该计划的另一部分建立在PI早期在识别和分类高阶阿贝尔群行为的刚性现象方面的成功之上，即具有多维“时间”的动力系统，其显示的行为与经典情况有本质上的不同。该计划的其他目标是开发具有均匀遍历行为的真实解析动力系统的新技术，包括解决在一些简单的低维流形上周期性流动元素附近存在的长期存在的问题。数学上的“刚性”概念有许多方面。它最简单和最基本的表现可以在高中代数的水平上看到：少量的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重合（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。刚性的各个方面出现在几个主要数学学科的交汇处，包括微分几何、李群理论和动力系统理论。本资助下的研究项目旨在确定时间为一维的经典动力学和具有多维时间的动力系统中的各种刚性现象，其中这种现象更为明显和普遍。提出的研究的另一个中心主题是将动力现象分为双曲型和部分双曲型（通俗地说大致是“混沌”）椭圆型（稳定行为）和抛物型（伴随特殊特征的中等复杂性）。

ABSTRACT:The project primarily deals with rigidity phenomena in dynamical systems.One aspect of rigidity concerns situations when a weaker structure(i.e. measurable) determines a stronger one (i.e. differentiable)within certain classes of systems. Another aspect deals withpreservation of the differentiable orbit structure or some of itsimportant elements under small perturbations (local differentiablerigidity) or within certain classes of systems (global differentiablerigidity). Yet another type of rigidity appears when a certainproperty (such as a relation between invariants, or a regularity of aninvariant structure) forces the system to belong to a specializednarrow class. A part of the research program deals with theidentification of various rigidity phenomena in classical dynamicalsystems. An essential characteristic of proposed work is a syntheticapproach which looks simultaneously into all three principal classes ofbehavior which appear in dynamics: elliptic, parabolic and hyperbolicexploring both similarities and contrasts among these threeparadigms. Among the major goals is the identification of newsituations where measurable orbit structure determines differentiableorbit structure. Another part of the program builds upon PI's earlersuccesses in identifying and classifying rigidity phenomena for actionsof higher--rank abelian groups, i.e. dynamical systems withmultidimensional ``time'' which displays behavior essentially differentfrom the classical case. Among other goals of the program is thedevelopment of new techniques for construction of real--analyticdynamical systems with uniform ergodic behavior including solution ofthe long--standing problems of existence of such systems near elementsof periodic flows on some simple low--dimensional manifolds based onrecent advances in that direction . Mathematical concept of "rigidity" has many facets. Its simplest andmost basic manifestations can be seen at the level just abovehigh school algebra: 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) than 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. Various aspects of rigidity appear atthe junction of several major mathematical disciplines, includingdifferential geometry, the theory of Lie groups and the theory ofdynamical systems. The research program under the present grant aims atidentifying various rigidity phenomena both in classical dynamicswhen time is one--dimensional and for dynamical systems withmultidimensional time where such phenomena are more pronounced andprevalent. Another central theme of the proposed research is a generalclassification of dynamical phenomena into hyperbolic and partiallyhyperbolic (roughly "chaotic" in lay parlance) elliptic (stablebehavior) and parabolic (intermediate complexity accompanied bypeculiar special features).