  Let $\g$ be a semisimple Lie algebra and $G = \Int(\g)$. In this article, we
relate the Jordan decomposition of $X \in \g$ (or $g \in G$) with the dynamics
induced on generalized flag manifolds by the right invariant continuous-time
flow generated by $X$ (or the discrete-time flow generated by $g$). We
characterize the recurrent set and the finest Morse decomposition (including
its stable sets) of these flows and show that its entropy always vanishes. We
characterize the structurally stable ones and compute the Conley index of the
attractor Morse component. When the nilpotent part of $X$ is trivial, we
compute the Conley indexes of all Morse components. Finally, we consider the
dynamical aspects of linear differential equations with periodic coefficients
in $\g$, which can be regarded as an extension of the dynamics generated by an
element $X \in \g$. In this context, we generalize Floquet theory and extend
the previous results to this case.
