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的测量刚度及其应用。据推测，在某些情况下，在这种行为下，很少有不变的和遍历的概率度量（Furstenberg, Margulis）。到目前为止，目前所有的工作都依赖于额外的正熵假设（Johnson, Rudolph），有时也依赖于沿临界方向的额外遍历性（Kalinin, Katok, Spatzier）。最近在避免不必要的遍历型假设以及测量刚度的应用方面取得了实质性进展（einsedler, Katok,Lindenstrauss）。eintredler将继续研究避免额外遍历型假设的不变测度，各种高阶动力系统的工作条件不相交性质，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，但重要的数学理论，与数学的其他部分以及其他科学，如物理学，气象学或计算机科学有许多联系。从历史上看，它是从对确定性但复杂的物理过程（例如天体力学）随时间演变的研究中发展起来的。特别是如果这个过程太复杂而无法精确地预测长期结果，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有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着悠久的传统。高阶动力系统（时间不止一个维度）的研究在过去几年受到了很多关注，部分原因是它与物理（特别是统计力学）和计算机科学（高维数据存储方法）的联系。提议的研究将数学的三个独立领域联系起来，即遍历论、李论和代数。这些领域之间丰富的相互作用有助于使用来自不同领域的工具来解决问题。其中最突出的联系是动力学非齐次空间与数论之间的联系，这是丢番图近似理论中一些问题的关键。然而，动力系统的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通常更适合于对其动力特性的详细研究，而遇到的现象在更大的一般动力系统背景下是有趣的。

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.