Exceptional and Generic Orbits in Homogeneous Dynamics and Number Theory

齐次动力学和数论中的例外和一般轨道

• 批准号: 0801064
• 负责人: Dmitry Kleinbock
• 金额: $ 23.07万
• 依托单位: Brandeis University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801064&HistoricalAwards=false
• 关键词: Exceptional; Generic; Orbits; Homogeneous; Dynamics

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.

这个项目涉及代数动力系统及其在数论中的应用。长期以来，人们一直认为，丢番图近似的许多问题都可以归结为适当的齐次流的轨道的性质。特别是，与整数方程或不等式理论相关的各种现象可以以一种有用的方式解释为某些轨迹相对于齐次空间上的某些自然测度是通用的，而另一些则需要构造具有特殊行为的轨迹。首席研究员计划继续研究这些轨道，旨在数论的新应用。使用的动力学工具有：测度刚性、遍历定理、混合和等分布、约化理论和定量不发散性。动力系统在这里代表一个抽象的点集以及一个控制点随时间移动的演化律。事实证明，许多关于用有理数同时逼近真实的数的问题，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，这使得有可能使用各种复杂的工具进行调查。