A Note on the Continuity of Oseledets Subspaces for Fiber-Bunched Cocycles

TL;DR

This paper proves that for fiber-bunched $GL(2,\mathbb{R})$ cocycles, the Oseledets subspaces change continuously in measure, providing insight into their stability under perturbations.

Contribution

It establishes the measure-continuity of Oseledets subspaces specifically for fiber-bunched cocycles, a significant step in understanding their stability.

Findings

Oseledets subspaces vary continuously in measure for fiber-bunched cocycles.

Continuity holds within the subset of fiber-bunched elements.

Provides a foundation for further stability analysis of cocycles.

Abstract

We prove that restricted to the subset of fiber-bunched elements of the space of $GL(2,\mathbb{R})$-valued cocycles Oseledets subspaces vary continuously, in measure, with respect to the cocycle.