On the distribution of the angle between Oseledets spaces

TL;DR

This study analyzes when the angle between Oseledets subspaces is log-integrable for GL_2-cocycles. In the i.i.d. setting, a finite first moment can yield a non-log-integrable angle, while a finite second moment ensures log-integrability; in contrast, the general measurable cocycle framework exhibits complete flexibility, allowing arbitrary joint distributions of the Oseledets spaces (subject to a bounded-gap condition in the bounded case). The results reveal a sharp distinction between random matrix products and broader cocycle dynamics, and point to extensions to higher dimensions and connections with stationary measures. Overall, the work clarifies when angle regularity is guaranteed and demonstrates broad generative capacity for Oseledets data under measurable dynamics.

Abstract

We study the distribution of the angles between Oseledets subspaces and their log-integrability, focusing on dimension $2$. For random i.i.d. products of matrices, we construct examples of probability measures on $\mathrm{GL}_2(\mathbb{R})$ with finite first moment where the Oseledets angle is not log-integrable. We also show that for probability measures with finite second moment the angle is always log-integrable. We then consider general measurable $\mathrm{GL}_2(\mathbb{R})$-cocycles over an arbitrary ergodic automorphism of a non-atomic Lebesgue space, proving that no integrability condition on the matrix distribution ensures log-integrability of the angle. In fact, the joint distribution of the Oseledets spaces can be chosen arbitrarily. A similar flexibility result for bounded cocycles holds under an unavoidable technical restriction.

Figures (1)

• Figure 1: The skyscraper with base $B$, labeled according to the function $\ell(\mathord{\cdot})$.

Theorems & Definitions (22)

• theorem 1
• theorem 2
• theorem 3
• theorem 4
• lemma 5: Triangular cocycles
• lemma 6: Weierstrass product inequalities
• lemma 7
• proof
• proof : Proof of Theorem \ref{['t.nonintegrable']}
• proof : Proof of Theorem \ref{['t.integrable']}