Ratner's work on unipotent flows and its impact.
Ratner's work on unipotent flows and its impact.
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拉特纳关于单能流及其影响的工作。
DOI：
10.1090/noti/1829
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发表时间：
2019
影响因子：
--
通讯作者：
Wilkinson, Amie
中科院分区：
文献类型：
--
作者：
Lindenstrauss, Elon;Sarnak, Peter;Wilkinson, Amie
达尼。顾名思义，这些定理断言这些流的轨道的闭包以及相关特征是非常受限的（刚性）。因此，它们为解决与这些流有关的问题提供了一个基本而有力的工具。拉特纳在建立这种刚性时引入和发展的出色技术，已经成为最近在其他情况下证明的类似刚性定理的蓝图。我们首先描述具有实数项且行列式等于1的𝑑×𝑑矩阵群的建立，即SL（𝑑，∈）。如果𝑔−1是幂零矩阵（我们用1表示𝐺中的单位元），则元素𝑔∈SL（𝑑，∈）是幂零的，如果𝑈的每个元素都是幂零的，则我们说一个群𝑈<𝐺是幂零的。SL（𝑑，l）的连通幂偶子群，特别是单参数幂偶子群，是Ratner著作中的基本对象。如果实数加性群在群上存在一个由多项式定义的满射同态，则称该群为单参数单幂群；例如𝑢（𝑡）=（1𝑡）
Dani above. As the name suggests, these theorems assert that the closures, as well as related features, of the orbits of such flows are very restricted (rigid). As such they provide a fundamental and powerful tool for problems connected with these flows. The brilliant techniques that Ratner introduced and developed in establishing this rigidity have been the blueprint for similar rigidity theorems that have been proved more recently in other contexts. We begin by describing the setup for the group of 𝑑× 𝑑 matrices with real entries and determinant equal to 1—that is, SL (𝑑, ℝ). An element 𝑔∈ SL (𝑑, ℝ) is unipotent if 𝑔− 1 is a nilpotent matrix (we use 1 to denote the identity element in 𝐺), and we will say a group 𝑈< 𝐺 is unipotent if every element of 𝑈 is unipotent. Connected unipotent subgroups of SL (𝑑, ℝ), in particular one-parameter unipotent subgroups, are basic objects in Ratner’s work. A unipotent group is said to be a one-parameter unipotent group if there is a surjective homomorphism defined by polynomials from the additive group of real numbers onto the group; for instance 𝑢 (𝑡)=(1 𝑡