Homogeneous Dynamics and Number Theory

所要研究的问题是齐次空间上的动力系统，它是由离散的子群作为李群的商得到的。这样的空间自然产生于数论中的问题。主要目的是继续在齐次空间上建立动力学作为处理数论中某些问题的有力工具。本文的主要研究对象是：(I)某些子群的轨道闭合和轨道均衡分布结果的“有效性”；(Ii)建立齐次空间上“某些子群”的不变测度和轨道的“刚性”结果；特别关注几何有限流形上的Radon测度和正特征背景下的Raghunathan猜想的情况。李群及其离散子群是数学的中心研究对象之一。在过去的40年里，齐次空间上的动力学技术和思想在回答数论中某些长期存在的问题方面取得了丰硕的成果，特别是在丢番图近似中。相关的结果大多在刚性理论中，即关于对象的相当弱的初始信息几乎完全地对对象进行分类。这项提议寻求这条道路的延续。

The problems to be investigated are in the area of dynamical systems on homogeneous spaces which are obtained as a quotient of a Lie group by a discrete subgroup. Such spaces arise naturally from problems in number theory. The main objective is to continue the path of establishing dynamics on homogeneous spaces as a powerful tool to approach certain problems in number theory. The following problems will be main objects of the study: (i) "effectivization" of orbit closure and equdistribution results for orbits of certain subgroups (ii) establishing "rigidity" results for invariant measures and orbits of ''certain subgroups" on homogeneous spaces; special attention will be given to the case of Radon measures on geometrically finite manifolds and Raghunathan's conjecture in positive characteristic setting. Lie groups and their discrete subgroups are one of the central objects of study in mathematics. During the past 40 years, techniques and ideas from dynamics on homogeneous spaces have been fruitful in answering certain long standing questions in number theory, in particular in Diophantine approximation. The relevant results are mostly in the rigidity theory of the kind that a rather weak initial information about an object almost fully classifies the object. This proposal seeks continuation of this path.