Exceptional and Generic Orbits in Homogeneous Dynamics and Number Theory

本课题研究代数动力系统及其在数论中的应用。很长一段时间以来，人们已经观察到许多关于丢番图近似的问题可以用适当的均匀流的轨道行为来表示。具体地说，与整数方程或不等式理论有关的各种现象可以以一种有用的方式解释为关于齐次空间上的某些自然度量的某些轨迹是通用的，而其他一些现象则要求构造具有特殊行为的轨迹。首席研究员计划继续他对这些轨道的研究，目标是在数论中有新的应用。可以使用的动力学工具有：测量刚性、遍历定理、混合和均匀分布、约化理论和定量无发散。在这里，动力系统代表一组抽象的点，以及控制点随时间移动的方式的演化规律。结果表明，许多关于有理数同时逼近实数的问题可以用某些轨道的行为来理解。此外，在这种情况下出现的系统是代数性质的，这使得使用各种各样的复杂工具来研究它们成为可能。

This project deals with algebraic dynamical systems and their applications to number theory. It has been observed for a long time that many problems concerning diophantine approximation an be cast in terms of the behavior of orbits of a suitable homogeneous flow. In particular, various phenomena related to the theory of integer equations or inequalities can be in a useful way interpreted as certain trajectories being generic with respect to some natural measures on homogeneous spaces, while other call for constructing trajectories with exceptional behavior. The principal investigator plans to continue his study of those orbits, aiming at new applications to number theory. Dynamical tools to be used are: measure rigidity, ergodic theorems, mixing and equidistribution, reduction theory, and quantitative nondivergence.A dynamical system here stands for an abstract set of points together with an evolution law which governs the way points move over time. It turns out that many problems concerning simultaneous approximation of real numbers by rational numbers can be understood in terms of the behavior of certain orbits. Furthermore, systems that arise in this context are of algebraic nature, which makes it possible to use a wide variety f sophisticated tools for their investigation.