Measure rigidity and smooth rigidity of abelian actions

测量阿贝尔动作的刚度和平滑刚度

• 批准号: 0701292
• 负责人: Boris Kalinin
• 金额: $ 8.71万
• 依托单位: University of South Alabama
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701292&HistoricalAwards=false
• 关键词: Measure; rigidity; smooth; abelian; actions

The proposed research is in the realm of dynamical systems and smooth ergodic theory. The main goal of the project is to study smooth actions of (discrete and continuous) higher rank Abelian groups, actions that exhibit some degree of hyperbolicity. They represent natural generalizations of diffeomorphisms and flows on smooth manifolds. Such higher rank dynamical systems appear in various geometric and algebraic situations. The main examples include actions by commuting hyperbolic and partially hyperbolic automorphisms of tori and by commuting translations on cosets of Lie groups. In contrast to the properties of a single hyperbolic diffeomorphism or flow, many remarkable rigidity phenomena have been discovered for certain classes of higher rank Abelian actions. These phenomena include smooth rigidity, cocycle rigidity, and rigidity of invariant measures. Using dynamical, analytic, and group theoretic methods the principal investigator will further the study of such rigidity properties. In particular, he will consider broader classes of these actions than have been investigated hitherto, including nonalgebraic and nonuniformly hyperbolic actions. Another goal of the project is to study certain questions in smooth dynamics of a single Anosov or partially hyperbolic system.Dynamical systems and ergodic theory are relatively new fields of mathematics that study the evolution of physical and mathematical systems over time (e.g., planetary motion, flow of air or other fluids). With origins in differential equations and celestial mechanics, these modern branches of mathematics provide numerous applications not only to other areas of mathematics but also to such natural sciences as physics, biology, meteorology, sociology, and computer science. Dynamical systems and ergodic theory have introduced new mathematical ideas into these sciences, including the study of long-term qualitative behavior, along with various analytic and probabilistic methods. One of the main goals of the proposed research is to investigate complex dynamical systems consisting of several different systems that ?commute? with each other, meaning that the evolution of any of the systems is not affected by the changes brought about by the others. Complex systems of this kind appear naturally in algebra, geometry, and physics. The recent study of such systems has produced exciting applications to number theory and quantum mechanics.

建议的研究领域的动力系统和光滑遍历理论。该项目的主要目标是研究（离散和连续）高阶阿贝尔群的光滑行为，表现出一定程度的双曲性的行为。它们代表了光滑流形上的自同态和流的自然推广。这样的高阶动力系统出现在各种几何和代数的情况。主要的例子包括交换双曲和部分双曲自同构环面和交换平移李群陪集的行动。与单个双曲型同态或流的性质相反，对于某些高秩阿贝尔作用类，已经发现了许多显著的刚性现象。这些现象包括光滑刚性、上循环刚性和不变测度刚性。使用动力学，分析和群论方法的主要研究人员将进一步研究这种刚性属性。特别是，他将考虑更广泛的类，这些行动比迄今已调查，包括非代数和非一致双曲行动。该项目的另一个目标是研究单个Anosov或部分双曲系统的光滑动力学中的某些问题。动力系统和遍历理论是相对较新的数学领域，研究物理和数学系统随时间的演化（例如，行星运动， 空气或其它流体的流动）。这些数学的现代分支起源于微分方程和天体力学，不仅在数学的其他领域，而且在物理学、生物学、气象学、社会学和计算机科学等自然科学中也有许多应用。动力系统和遍历理论引入了新的数学思想到这些科学中，包括长期定性行为的研究，沿着各种分析和概率方法。所提出的研究的主要目标之一是调查复杂的动力系统组成的几个不同的系统，？通勤？这意味着任何系统的进化都不会受到其他系统所带来的变化的影响。这类复杂系统自然地出现在代数、几何和物理学中。最近对这类系统的研究在数论和量子力学中产生了令人兴奋的应用。