Rigidity in Homogeneous Dynamics with Connections to Number Theory

该项目涉及高阶部分双曲阿贝尔作用的刚性度量及其应用。有人推测，在某些情况下，在这种作用下，不变的和遍历的概率度量是非常少的(Furstenberg，Marguis)。到目前为止，目前的所有工作都依赖于正熵的额外假设(Johnson，Rudolph)，有时还依赖于关键方向上的额外遍历性(Kalinin，Katok，Spatzier)。最近，在避免不必要的遍历性类型假设以及在度量刚性的应用方面取得了实质性进展(Einsiedler，Katok，Lindenstrauss)。Einsiedler将继续他的不变度量的研究，避免额外的遍历型假设，致力于各种高阶动力系统的不交性质，刚性的进一步应用，以及相关的代数主题。动力系统理论是一个相对较新的重要数学理论，与数学的其他部分以及其他科学，如物理，气象学，或计算机科学有许多联系。从历史上看，动力系统理论是从研究确定性但复杂的物理过程(如天体力学)随时间的演化发展而来的。特别是如果这个过程太复杂，不能准确地预测长期结果，动力系统理论就很重要，因为它可以给出定性的预测。最近，符号动力学被证明是在计算机科学中找到高效和安全的编码算法的关键。使用动力学来解决其他数学领域的问题也有很长的传统。在过去的几年里，对高阶动力系统(时间具有不止一维的时间)的研究受到了极大的关注，部分原因是它将物理学(特别是统计力学)和计算机科学(高维数据存储方法)联系在一起。这项拟议的研究涉及三个不同的数学领域，即遍历理论、李理论和代数。这些领域之间的丰富相互作用有助于通过使用不同领域的工具来解决问题。最突出的联系是齐次空间上的动力学和数论之间的联系，这是丢番图逼近理论中问题的关键。然而，动力系统理论也可以从这种相互作用中受益，因为代数起源的动力系统往往更适合于详细研究它们的动力学性质，而所遇到的现象在一般动力系统的更大背景下是有趣的。

The project involves measure rigidity of higher rank partiallyhyperbolic abelian actions and applications thereof. It isconjectured that under certain circumstances there are very fewprobability measures both invariant and ergodic under such actions(Furstenberg, Margulis). So far all of the current work relies onthe additional assumption of positive entropy (Johnson, Rudolph),and sometimes on additional ergodicity along critical directions(Kalinin, Katok, Spatzier). Substantial progress has been maderecently in avoiding unnecessary ergodicity-type assumptions aswell as in applications of measure rigidity (Einsiedler, Katok,Lindenstrauss). Einsiedler will continue his study of invariantmeasures avoiding additional ergodicity-type assumptions, work ondisjointness properties of various higher rank dynamical systems,further applications of rigidity, and related algebraic topics.The theory of dynamical systems is a relatively new, but importantmathematical theory with many connections to other parts ofmathematics as well as other sciences such as physics,meteorology, or computer sciences. Historically it developed fromthe study of the evolution of a deterministic but complicatedphysical process (for instance in celestial mechanics) over time.Especially if this process is too complicated to predict long termoutcomes precisely, the theory of dynamical systems is importantbecause it can give qualitative predictions. More recentlysymbolic dynamics turned out to be crucial to find efficient andsafe coding algorithms in computer sciences. There is also a longtradition of using dynamics to solve problems in other areas ofmathematics. The study of higher rank dynamical systems (wheretime has more than one dimension) has received much attentionduring the last years, in part again because of its connections tophysics (in particular statistical mechanics) and computersciences (higher dimensional data storage methods). The proposedresearch links three separate areas of mathematics, namely ergodictheory, Lie theory, and algebra. The rich interplay among thesefields helps to attack problems by using tools from differentareas. The most prominent connection is between dynamics onhomogeneous spaces and number theory, which holds the key toproblems in the theory of Diophantine approximation. However, thetheory of dynamical systems can benefit from this interaction toosince dynamical systems of algebraic origin are often moreamenable to a detailed study of their dynamical properties whilethe phenomena encountered are interesting in the larger context ofgeneral dynamical systems.