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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2016
- DOI:
- 10.48550/arXiv.1606.02209
- arXiv:
- arXiv:1606.02209
- Bibcode:
- 2016arXiv160602209B
- Keywords:
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- Mathematics - Dynamical Systems;
- 37H15 (Primary);
- 37A05 (Secondary)
- E-Print:
- v.2: Modified emphasis/language, added in clarifying remarks/appendix