Dynamics of translations on maximal compact subgroups

Mauro Patrão, Ricardo Sandoval
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Abstract

In this article, we study the dynamics of translations of an element of a semisimple Lie group $G$ acting on its maximal compact subgroup $K$. First, we extend to our context some classical results in the context of general flag manifolds, showing that when the element is hyperbolic its dynamics is gradient and its fixed points components are given by some suitable right cosets of the centralizer of the element in $K$. Second, we consider the dynamics of a general element and characterizes its recurrent set, its minimal Morse components and their stable and unstable manifolds in terms of the Jordan decomposition of the element, and we show that each minimal Morse component is normally hyperbolic.
最大紧凑子群上的平移动力学
在这篇文章中,我们研究了作用于其最大紧凑子群 $K$ 的半不简单李群 $G$ 的元素平移的动力学。首先,我们将一般旗状manifolds 的一些经典结果延伸到我们的语境中,证明当元素是双曲的时候,它的动力学是梯度的,它的定点分量是由元素在 $K$ 中的中心化的一些合适的右余弦给出的。其次,我们考虑了一般元素的动力学,并根据元素的乔丹分解描述了其循环集、最小莫尔斯分量及其稳定和不稳定流形,并证明了每个最小莫尔斯分量都是正常双曲的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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