Dynamical Systems, Rigidity and Related Topics

该计划的中心主题是研究动力学中刚性的各个方面。 PI及其合作者最近引入了新的方法和见解， 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在数论中，并提供了第一个例子存在不变的几何结构， 大类行动。主要研究方向有：1.高阶阿贝尔群的部分双曲和抛物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.在代数动力系统的刚度测度理论中寻找新的方法.研究了高阶阿贝尔群的一般作用类的不变几何结构的存在性，这些作用类由全局拓扑、同伦或动力学条件决定。4. Anosov系统的分类，直至维斯各类模的可微共轭。5. Gauss动力系统的光滑实现的构造。6.酉群表示理论在动力学上同调问题中的应用数学上的“刚性”概念有很多方面。它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更多的方程。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），则这些数字都相等（n-1个等式）。PI早期研究中的一个例子在概念上类似，但技术上要复杂得多：负曲率的紧凑表面，即有界几何形状，其中任何测地线三角形的角度之和小于180度，其中两个数字表征全局和统计体积增长（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的和由面积唯一地确定。 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多个领域引人注目的应用的扩展和发展。 下面是对区域3的一些结果的解释。以上确定性动力系统中混沌行为的研究对于现代动力系统理论在自然科学和社会科学各个领域的应用起着核心作用。 这是阻碍更全面理解重要模型的关键困难：虽然建立产生混沌行为的某些初始条件的存在通常相对容易，但证明大多数或许多（在相空间中的体积意义上）的混沌行为超出了目前甚至预期的数学方法。PI和他的合作者发现，对于具有多维时间的系统，在某些非常一般的条件下，这个困难被克服了：拓扑或动力学性质的全局条件先验地保证只存在一些混沌轨道，实际上意味着存在这样的轨道，这些轨道填充相空间中的正体积。

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 provided first examples of existence of invariant geometric structure for large classes of actions. There are several major directions in the proposed program:1.Local differentiable rigidity for partially hyperbolic and parabolic actions of higher rank abelian groups with the emphasis on the combination of the dynamical systems, harmonic analysis/group representation and geometric methods.2. Finding new methods in the theory of measure rigidity for algebraic dynamical systems.3. Existence of invariant geometric structures for general classes of actions of higher rank abelian groups determined by global topological, homotopical or dynamical conditions.4.Classification of Anosov systems up to differentiable conjugacy vis various classes of moduli.5.Construction of smooth realization of Gaussian dynamical systems. 6.Application of the theory of unitary group representations to the cohomological problems in dynamics.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 a special type may imply much larger number of equation. For example, if the 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 conceptually similar albeit technically much more sophisticated: a compact surface of negative curvature, i.e. a bounded geometric shape where any geodesic triangle has the sum of its angles less than 180 degrees, for which two numbers characterizing global and statistical volume growth (topological and metric entropy) coincide has constant negative curvature, i.e. the sum of the angles of a geodesic triangle is uniquely determined by the area. The research under the present grant involves both deeper investigation 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. Here is an interpretation of some results from the area 3. above. Investigation of chaotic behavior in deterministic dynamical systems plays a central role in application of the modern theory of dynamical systems to various areas on natural and social sciences. Here is the crucial difficulty which impedes efforts in more comprehensive understanding of important models: while it is often relatively easy to establish existence of some initial conditions which produce chaotic behavior, proving chaotic behavior of most or many (in the sense of volume in the phase space) is beyond the present on even anticipated mathematical methods. PI and his collaborators discovered that for systems with multidimensional time under certain very general conditions this difficulty is overcome: global conditions of topological or dynamical nature which a priori guarantee only existence of some chaotic orbits in fact imply existence of such orbits which fill positive volume in the phase space.