Invariant Control Sets on Flag Manifolds and Ideal Boundary

Let $G$ be a semisimple real Lie group of non-compact type, $K$ a maximal compact subgroup and $S\subseteq G$ a semigroup with nonempty interior. We consider the ideal boundary $\partial_{\infty }\left( G/K\right) $ of the associated symmetric space and the flag manifolds $G/P_{\Theta }$. We prove that the asymptotic image $\partial _{\infty }\left( Sx_{0}\right) \subseteq \partial _{\infty }\left( G/K\right) $, where $x_{0}\in G/K$ is any given point, is the maximal invariant control set of $S$ in $\partial_{\infty }\left( G/K\right) $. Moreover there is a surjective projection $\pi :\partial _{\infty }\left( Sx_{0}\right) \rightarrow \bigcup\limits_{\Theta \subseteq \Sigma }C_{\Theta }$, where $C_{\Theta }$ is the maximal invariant control set for the action of $S$ in the flag manifold $G/P_{\Theta }$, with $P_{\Theta }$ a parabolic subgroup. The points that project over $C_{\Theta } $ are exactly the points of type $\Theta $ in $\partial _{\infty }\left( Sx_{0}\right) $ (in the sense of the type of a cell in a Tits Building).