This paper studies reversibility and transitivity of semigroups acting on homogeneous spaces. Properties of the reversor set of a subsemigroup acting on homogeneous spaces are presented. Let G be a topological group and L a subgroup of G. Assume that S is a subsemigroup of G with nonempty interior. It is presented a study of the reversibility of the S-action on $G/L$ in terms of the actions of S and L on homogeneous spaces of G. The results relate the reversibility and the transitivity of S in $G/L$ with the minimality of the action of L on homogeneous spaces of G. In addition, sufficient conditions for S to be right reversible in G if S is reversible in $G/L$ are presented.

We give various realizations of the adjoint orbits of a semi-simple Lie group and describe their symplectic geometry. We then use these realizations to identify a family of Lagrangian submanifolds of the orbits.

Let $Q\rightarrow X$ be a continuous principal bundle whose group $G$ is reductive. A flow ${\it\phi}$ of automorphisms of $Q$ endowed with an ergodic probability measure on the compact base space $X$ induces two decompositions of the flag bundles associated to $Q$: a continuous one given by the finest Morse decomposition and a measurable one furnished by the multiplicative ergodic theorem. The second is contained in the first. In this paper we find necessary and sufficient conditions so that they coincide. The equality between the two decompositions implies continuity of the Lyapunov spectra under perturbations leaving unchanged the flow on the base space.

In this paper we study characteristic exponents of flows in relation with the dynamics of flows on flag bundles. The starting point is a flow on a principal bundle with semi-simple group G. Projection against the Iwasawa decomposition G=KAN defines an additive cocycle over the flow with values in 𝔞=log A. Its Lyapunov exponents (limits along trajectories) and Morse exponents (limits along chains) are studied. A symmetric property of these spectral sets is proved, namely invariance under the Weyl group. We also prove that these sets are located in certain Weyl chambers, defined from the dynamics on the associated flag bundles. As a special case linear flows on vector bundles are considered.

It has recently been shown by Mo and Negreiros that a necessary condition for an invariant almost complex structure on the complex full flag manifold ${\bb F}(n)$ to admit a (1, 2)-symplectic invariant metric is that its associated tournament be cone-free. In this paper, a canonical stair-shaped form is given for such tournaments, and this is applied to show that the condition is also sufficient; in the process, all the associated (1, 2)-symplectic metrics are described.