Decomposition of quasi-all invariant bundles for the cocycle with a set of strongly invariant cones

Abstract

In this paper, we prove that a linear cocycle with a set of strongly invariant cones admits a decomposition of quasi-all invariant bundles. We further obtain the certain location-relations between the fibres of positive invariant bundles and these cones, as well as we get the order-relations among characteristic exponents of the linear cocycle on the positive invariant bundles. The key of proofs is to analyze relations among positive invariant bundles with respect to a linear cocycle that admits both of a $k$-exponential separation and a $k^{\prime}$-exponential separation. These results are applied to a class of semilinear equations on a Hilbert space.