Dominated splittings for semi-invertible operator cocycles on Hilbert space
Abstract
A theorem of J. Bochi and N. Gourmelon states that an invertible linear cocycle admits a dominated splitting if and only if the singular values of its iterates become separated at a uniform exponential rate. It is not difficult to show that for cocycles of non-invertible linear maps over an invertible dynamical system -- which we refer to as semi-invertible cocycles -- this criterion fails to imply the existence of a dominated splitting. In this article we show that a simple modification of Bochi and Gourmelon's singular value criterion is equivalent to the existence of a dominated splitting in both the invertible and the semi-invertible cases. This result extends to the more general context of semi-invertible cocycles of bounded linear operators acting on a Hilbert space, and generalises previous results due to J.-C. Yoccoz, J. Bochi and N. Gourmelon, and the present author.
Keywords
Cite
@article{arxiv.1403.0824,
title = {Dominated splittings for semi-invertible operator cocycles on Hilbert space},
author = {Ian D. Morris},
journal= {arXiv preprint arXiv:1403.0824},
year = {2015}
}
Comments
This paper has been withdrawn by the author due to a critical error: step 8 is written as if the image of the operator P(x) were \mathcal{U}(x), but it is actually \mathcal{W}(x). This error invalidates the proof of the main theorem and the entire article should be treated as incorrect