Conditions for equality between Lyapunov and Morse decompositions (Q2813939)

The setting of the results obtained here is a continuous principal bundle \(Q\to X\) with group \(G\) assumed to be semi-simple or reductive (for example, the bundle of frames of a \(d\)-dimensional vector bundle \(V\to X\) over \(X\), in which case \(G\) is \(\mathrm{GL}_d(\mathbb{R})\)). An automorphism of \(Q\) induces a flow on the base space \(X\) and on bundles with typical fibres carrying actions of \(G\). In the reductive case with its flag manifold defining a flag bundle there is a finest Morse decomposition under a mild transitivity condition. On the other hand there is the Oseledets decomposition given by the multiplicative ergodic theorem. It is shown here that any component of the Oseledets decomposition is contained in a component of the Morse decomposition, and the main results here describe three conditions that together are necessary and sufficient for the decompositions to coincide, with the main work being involved in proving sufficiency of the conditions.