This paper studies 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. It is proved a symmetric property of these spectral sets, namely invariance under the Weyl group. It is proved also 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.AMS 2000 subject classification: Primary: 37B05, 37B35. Secondary: 20M20, 22E46.

Consider a continuous flow of automorphisms of a G-principal bundle which is chain transitive on its compact Hausdorff base. Here G is a connected non-compact semi-simple Lie group with finite centre. The finest Morse decomposition of the induced flows on the associated flag bundles were obtained in previous articles. Here we describe the stable sets of these Morse components and, under an additional assumption, their Conley indexes.

Let g be a real semisimple Lie algebra and G = Int(g). In this article, we relate the Jordan decomposition of X ∈ g (or g ∈ 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 ∈ g. In this context, we generalize Floquet theory and extend our previous results to this case.

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