Measure rigidity and smooth rigidity of abelian actions

所提出的研究是在动力系统和光滑遍历理论领域。该项目的主要目标是研究(离散和连续的)高阶Abel群的光滑行为，即表现出某种程度的双曲性的行为。它们表示光滑流形上微分同胚和流的自然推广。这样的高阶动力系统出现在各种几何和代数情况下。主要的例子包括通过交换环面的双曲和部分双曲自同构以及通过交换李群陪集上的平移所产生的作用。与单个双曲微分同胚或流的性质相反，对于某些高阶Abel作用量，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的刚性。主要研究人员将利用动力学、解析和群论方法进一步研究这种刚性性质。特别是，他将考虑比迄今所研究的更广泛的这类作用，包括非代数和非一致双曲作用。该项目的另一个目标是研究单个Anosov系统或部分双曲系统的光滑动力学中的某些问题。动力学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变(例如，行星运动、空气流动或其他流体)。这些现代数学分支起源于微分方程式和天体力学，不仅为数学的其他领域提供了大量的应用，也为物理、生物学、气象学、社会学和计算机科学等自然科学提供了许多应用。动力系统和遍历理论为这些科学引入了新的数学思想，包括对长期定性行为的研究，以及各种分析和概率方法。拟议研究的主要目标之一是研究由几个不同系统组成的复杂动力系统。这意味着任何一个系统的演变都不会受到其他系统带来的变化的影响。这类复杂系统在代数、几何和物理中自然出现。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。

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.