On irreducibility of Oseledets subspaces

2016/06/07 by Bose, Christopher, Horan, Joseph, Quas, Anthony
#37A05 (Secondary) #37H15 (Primary) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1606.02209

Abstract

For a cocycle of invertible real n-by-n matrices, the Multiplicative Ergodic Theorem gives an Oseledets subspace decomposition of ℝⁿ; that is, above each point in the base space, ℝⁿ 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 ℝ² and ℂ² for O₂(ℝ)-valued cocycles and give explicit examples where the conditions are satisfied.

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