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Let Q → X be a continuous principal bundle whose group G is reductive. A flow φ 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 pertubations leaving unchanged the flow on the base space.AMS 2010 subject classification: Primary: 37H15, 37B35, 37B05. Secondary: 22E46, 20M20.

It is well known that the Einstein equation on a Riemannian flag manifold (G/K, g) reduces to an algebraic system if g is a G-invariant metric. In this paper we obtain explicitly new invariant Einstein metrics on generalized flag manifolds of Sp(n) and SO(2n); and we compute the Einstein system for generalized flag manifolds of type Sp(n). We also consider the isometric problem for these Einstein metrics.

It is well known that the Einstein equation on a Riemannian flag manifold (G/K, g) reduces to a algebraic system, if g is a G-invariant metric. In this paper we described this system for all flag manifolds of a classical Lie group. We also determined the number of isotropy summands for all of these spaces and proved certain properties of the set of t-roots for flag manifolds of type Bn, Cn and Dn.

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