Dynamics of translations on maximal compact subgroups

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.