Chain transitive sets for flows on flag bundles

We study the chain transitive sets and Morse decompositions of flows on fiber bundles whose fibers are compact homogeneous spaces of Lie groups. The emphasis is put on generalized flag manifolds of semi-simple (and reductive) Lie groups. In this case an algebraic description of the chain transitive sets is given. Our approach consists in shadowing the flow by semigroups of homeomorphisms to take advantage of the good properties of the semigroup actions on flag manifolds. The description of the chain components in the flag bundles generalizes the Theorem of Selgrade for projective bundles with an independent proof.