Conditions for Equality between Lyapunov and Morse Decompositions

Let \(Q\rightarrow X\) be a continuous principal bundle whose group \(G\) is reductive. A flow \(\phi \) 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 pertubations leaving unchanged the flow on the base space.