Control Sets on Flag Manifolds and Ideal Boundaries of Symmetric Spaces

Given a semigroup $S$ with non-empty interior, contained in a semisimple real Lie group of non-compact type $G$, the effective control sets of $S$ in the flag manifolds are well known. In this work we consider the orbits of $S$ in a symmetric space and its images by the Weyl group and describe the effective control sets in the flag manifolds as images of those orbits.