Characterization of dominated splittings for operator cocycles acting on Banach spaces

TL;DR

This paper extends the theory of dominated splittings to operator cocycles on Banach spaces, providing geometric methods and explicit growth estimates, thus broadening the understanding of ergodic theorems in infinite-dimensional settings.

Contribution

It generalizes finite-dimensional dominated splitting results to Banach spaces using a geometric approach and offers explicit growth estimates in the finite-dimensional case.

Findings

Extended dominated splitting characterization to Banach spaces.

Developed a geometric approach based on approximate singular value decomposition.

Provided explicit growth estimates for finite-dimensional cases.

Abstract

Versions of the Oseledets multiplicative ergodic theorem for cocycles acting on infinite-dimensional Banach spaces have been investigated since the pioneering work of Ruelle in 1982 and are a topic of continuing research interest. For a cocycle to induce a continuous splitting in which the growth in one subbundle exponentially dominates the growth in another requires additional assumptions; a necessary and sufficient condition for the existence of such a dominated splitting was recently given by J. Bochi and N. Gourmelon for invertible finite-dimensional cocycles in discrete time. We extend this result to cocycles of injective bounded linear maps acting on Banach spaces (in both discrete and continuous time) using an essentially geometric approach based on a notion of approximate singular value decomposition in Banach spaces. Our method is constructive, and in the finite-dimensional case yields explicit growth estimates on the dominated splitting which may be of independent interest.