Nonexistence of Lyapunov Exponents for Matrix Cocycles

TL;DR

This paper investigates the conditions under which the set of points with divergent Lyapunov averages in matrix cocycles is large, showing that under certain dynamical properties, this set is residual despite classical results indicating it has zero measure.

Contribution

It demonstrates that for systems with exponential specification and Hölder continuous cocycles, the Lyapunov-irregular set can be residual if multiple ergodic measures with different spectra exist.

Findings

Lyapunov-irregular set can be residual under specified conditions

Classical zero measure result does not imply small size in topological sense

Existence of multiple ergodic measures with different spectra is crucial

Abstract

It follows from Oseledec Multiplicative Ergodic Theorem that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any dynamical system $f:X\rightarrow X$ with exponential specification property and a H$\ddot{\text{o}}$lder continuous matrix cocycle $A:X\rightarrow G (m,\mathbb{R})$, we show here that if there exist ergodic measures with different Lyapunov spectrum, then the Lyapunov-irregular set of $A$ is residual (i.e., containing a dense $G_\delta$ set).