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In this note, we show that if all Lyapunov exponents of a matrix cocyclevanish, then it can be perturbed to become cohomologous to a cocycle takingvalues in the orthogonal group. This extends a result of Avila, Bochi andDamanik to general base dynamics and arbitrary dimension. We actually prove afibered version of this result, and apply it to study the existence ofdominated splittings into conformal subbundles of general matrix cocycles.

Almost reduction and perturbation of matrix cocycles