Complete regularity of linear cocycles and the Baire category of the set of Lyapunov-Perron regular points

TL;DR

The paper studies the topological size of the LP-regular set $\mathcal{R}$ for linear cocycles and shows it is meager in broad settings unless a rigid structure—complete regularity—holds. It establishes a sharp all-or-meager dichotomy under minimal dynamics, linking complete regularity to a constant Lyapunov spectrum and globally defined Oseledets data, and connects these notions to the Sacker–Sell spectrum and amenable reductions. The authors develop a robust Baire-category framework, a criterion for generic irregularity, and a detailed analysis via singular values and exterior powers to detect dominated splittings. They provide concrete examples illustrating the independence of LP-regularity and spectrum constancy, and they extend results to Hölder cocycles over hyperbolic dynamics, deriving equivalences among complete regularity, amenable reduction, and periodic-spectrum uniformity. Overall, the work clarifies when regularity phenomena are topologically prevalent versus exceptional, with implications for the structure of Lyapunov splittings and spectral theory of cocycles.

Abstract

Given a continuous linear cocycle A over a homeomorphism f of a compact metric space X, we investigate its set R of Lyapunov-Perron regular points, that is, the collection of trajectories of f that obey the conclusions of the Multiplicative Ergodic Theorem. We obtain results roughly saying that the set R is of first Baire category (i.e., meager) in X, unless some rigid structure is present. In some settings, this rigid structure forces the Lyapunov exponents to be defined everywhere and to be independent of the point; that is what we call complete regularity.

Theorems & Definitions (57)

• proposition 2.1
• proposition 2.2
• proposition 2.3: Semi-uniform subadditive ergodic theorem Sch, Stu
• corollary 2.4
• theorem 3.1
• proposition 3.2
• proof
• proof : Proof of Theorem \ref{['t.uniformity']}
• corollary 3.3
• proof