H\"older continuity of Oseledets splittings for semi-invertible operator cocycles

TL;DR

This paper proves the H"older continuity of Oseledets splittings for semi-invertible cocycles, extending previous results to noninvertible cases and demonstrating nonuniform hyperbolicity on large measure sets.

Contribution

It extends H"older continuity results of Oseledets splittings to semi-invertible, possibly noninvertible cocycles, including those with compact operator values.

Findings

H"older continuity of Oseledets subspaces established on large measure sets

Noninvertible cocycles with nonzero Lyapunov exponents show Pesin-type hyperbolic behavior

Extension of previous invertible cocycle results to noninvertible and compact operator cases

Abstract

For H\"older continuous cocycles over an invertible, Lipschitz base, we establish the H\"older continuity of Oseledets subspaces on compact sets of arbitrarily large measure. This extends a result of Ara\'{u}jo, Bufetov, and Filip by considering possibly noninvertible cocycles, which in addition may take values in the space of compact operators on a Hilbert space. As a by-product of our work, we also show that a noninvertible cocycle with nonvanishing Lyapunov exponents exhibits nonuniformly hyperbolic behaviour (in the sense of Pesin) on a set of full measure.