Connected components of open semigroups in semi-simple Lie groups

This paper studies connected components of open subsemigroups of non-compact semi-simple Lie groups by using control sets in the flag manifolds and their coverings. The concept of recurrent component is introduced and a method is given by which their number can be computed. It is shown that the union of all recurrent components is a semigroup. The idea of mid-reversibility comes up to show that an open semigroup has just one semigroup component if the identity belongs to its closure. A necessary and sufficient condition for mid-reversibility is proved showing that e.g. in a complex group any open semigroup is mid-reversible.