Invariant Control Sets on Flag Manifolds and Ideal Boundaries of Symmetric Spaces

Let G be a semisimple real Lie group of non-compact type, K a maximal compact subgroup and S ⊆ G a semigroup with nonempty interior. We consider the ideal boundary ∂∞(G/K) of the associated symmetric space and the flag manifolds G/PΘ . We prove that the asymptotic image ∂∞(Sx0) ⊆ ∂∞(G/K), where x0 ∈ G/K is any given point, is the maximal invariant control set of S in ∂∞(G/K). Moreover there is a surjective projection π : ∂∞(Sx0) → ⋃ Θ⊆Σ CΘ , where CΘ is the maximal invariant control set for the action of S in the flag manifold G/PΘ , with PΘ a parabolic subgroup. The points that project over CΘ are exactly the points of type Θ in ∂∞ (Sx0) (in the sense of the type of a cell in a Tits Building).