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

TL;DR

This paper studies the set of Lyapunov-Perron regular points in linear cocycles over compact spaces, showing it is typically meager unless a rigid structure causes complete regularity with uniform Lyapunov exponents.

Contribution

It establishes conditions under which the set of regular points is meager or when the cocycle exhibits complete regularity with uniform exponents.

Findings

The set of Lyapunov-Perron regular points is meager unless a rigid structure exists.

Rigid structures lead to complete regularity with uniform Lyapunov exponents.

In some cases, the exponents are defined everywhere and are independent of the point.

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.