Topics in Dynamical Systems and Ergodic Theory

摘要：该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在某些系统类别中，较弱的结构（即可测）决定较强的结构（即可微）的情况。 另一个方面涉及到在小扰动（局部可微刚性）或某些系统类（全局可微刚性）下可微轨道结构或其重要元素的保持。还有一种类型的刚性出现在某种性质（如不变量之间的关系，或不变结构的规律性）迫使系统属于一个特定的狭义类。 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。建议工作的一个基本特征是一个综合的方法，同时着眼于所有三个主要类别的行为出现在动力学：椭圆，抛物线和双曲线探索这三个范例之间的相似性和对比。其中的主要目标是确定新的情况下，可测量的轨道结构决定微分轨道结构。 该计划的另一部分建立在PI的earlersuccessfully在识别和分类刚性现象的actionsof更高-秩阿贝尔群，即动态系统与多维“时间”，其中显示的行为本质上不同于经典的情况。 该计划的其他目标是开发新技术，用于构建具有均匀遍历行为的真实的分析动力系统，包括解决长期存在的问题，即基于该方向的最新进展，在一些简单的低维流形上周期流的元素附近存在这种系统。“刚性”的数学概念是多方面的。它最简单和最基本的表现形式可以在高中代数以上的水平上看到：少量的方程或特殊类型的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），则所有数字都相等（n-1个等式）。 PI早期研究的一个例子在概念上是相似的，尽管技术上要复杂得多：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，其中两个数字表征了整体和统计体积增长（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。刚性的各个方面出现在几个主要数学学科的交界处，包括微分几何，李群理论和动力系统理论。本研究计划的目的是在一维时间的经典动力学和多维时间的动力学系统中识别各种刚性现象，这些现象更加明显和普遍。 拟议研究的另一个中心主题是将动力学现象一般分类为双曲和部分双曲（通俗地说就是“混沌”）、椭圆（稳定行为）和抛物线（伴随着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).