Control sets on maximal compact subgroups

Let G be a noncompact connected semisimple Lie group, with finite center. In this paper we study the action of semigroups S of G, with nonempty interior, acting on maximal compact connected subgroups K of G. When S is connected, it is well known that the invariant control sets on K are the connected components of \(\pi ^{-1}(C)\), where \(\pi \) is the canonical projection of K onto F, F is the flag type of S and C is the only invariant control set for S on F. One of the main results of the present paper describes the set of transitivity of a control set, not necessarily invariant, of a semigroup S, not necessarily connected, acting on K, as fixed points of regular elements in S. Furthermore, we show that the number of control sets on K is the product of the number of control sets on the maximal flag manifold of G by the number of invariant control sets on K.