Rigidity in Homogeneous Dynamics with Connections to Number Theory

本课题涉及高阶部分双曲阿贝尔作用量的刚性测度及其应用。证明了在某些情况下，在这种作用下，很少有概率测度是不变的和遍历的（Furstenberg，Margulis）。到目前为止，所有目前的工作依赖于额外的假设，正熵（约翰逊，鲁道夫），有时在额外的遍历沿着临界方向（加里宁，Katok，Spatzier）。最近在避免不必要的遍历型假设以及测度刚性的应用（Einsiedler，Katok，Lindenstrauss）方面取得了实质性的进展。 艾因西德勒将继续他的研究invariantmeasures避免额外的遍历型假设，工作disjointness性质的各种高阶动力系统，进一步应用的刚性，和相关的代数topics.The理论的动力系统是一个相对较新的，但importantmathematical理论与许多连接到其他部分ofmathematics以及其他科学，如物理学，气象学，或计算机科学。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（例如天体力学）随时间的演化发展而来的。特别是当这个过程过于复杂而无法精确预测长期结果时，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有效和安全的编码算法的关键。利用动力学解决其他数学领域的问题也有着悠久的传统。高阶动力系统（时间不止一维）的研究在过去的几年里受到了广泛的关注，部分原因是它与物理学（特别是统计力学）和计算机科学（高维数据存储方法）的联系。拟议的研究链接三个独立的数学领域，即遍历理论，李理论和代数。这些领域之间丰富的相互作用有助于通过使用来自不同领域的工具来解决问题。最突出的联系是齐性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.