On irreducibility of Oseledets subspaces

Abstract

For a cocycle of invertible real $n$-by-$n$ matrices, the Multiplicative Ergodic Theorem gives an Oseledets subspace decomposition of ${\mathbb{R}}^n$; that is, above each point in the base space, ${\mathbb{R}}^n$ is written as a direct sum of equivariant subspaces, one for each Lyapunov exponent of the cocycle. It is natural to ask if these summands may be further decomposed into equivariant subspaces; that is, if the Oseledets subspaces are reducible. We prove a theorem yielding sufficient conditions for irreducibility of the trivial equivariant subspaces ${\mathbb{R}}^2$ and ${\mathbb{C}}^2$ for $O_2(\mathbb{R})$-valued cocycles and give explicit examples where the conditions are satisfied.

arXiv commentv.2: Modified emphasis/language, added in clarifying remarks/appendix