On irreducibility of Oseledets subspaces

Christopher Bose, Joseph Horan, Anthony Quas

Published 2016-06-07Version 1

For a real invertible cocycle of $n$-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 equivariantly as a direct sum of subspaces, according to the Lyapunov exponents of the cocycle. It is natural to ask if these summands may be further decomposed; that is, if the Oseledets subspaces are irreducible. We reinterpret Oseledets's result as a statement about block diagonalizing the cocycle: the given cocycle is cohomologous to one which is block diagonal, and the blocks correspond to the Oseledets subspaces. It is shown that it is not always possible to block triangularize the cocycle (that is, block diagonalize with triangular blocks), even allowing the resulting cocycle to be over the complex numbers, therefore showing that Oseledets subspaces are not always reducible.