On irreducibility of Oseledets subspaces

TL;DR

This paper investigates the conditions under which Oseledets subspaces for matrix cocycles are irreducible, providing theoretical criteria and explicit examples for certain low-dimensional cases.

Contribution

The authors establish sufficient conditions for the irreducibility of Oseledets subspaces in specific low-dimensional settings and construct explicit examples satisfying these conditions.

Findings

Sufficient conditions for irreducibility of Oseledets subspaces in low dimensions.

Explicit examples demonstrating these conditions.

Analysis focused on $O_2( )$-valued cocycles.

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.