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 ; that is, above each point in the base space, 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 and for -valued cocycles and give explicit examples where the conditions are satisfied.

Keywords:

• Oseledets subspaces
• multiplicative ergodic theorem
• reducible matrix cocycles

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

All three authors are supported by grants from the Natural Sciences and Engineering Research Council of Canada.