Singular analytic linear cocycles with negative infinite Lyapunov exponents

TL;DR

This paper characterizes linear analytic cocycles with all Lyapunov exponents at negative infinity as nilpotent and explores their conjugation properties, perturbations, and spectral splittings in the one-frequency case.

Contribution

It provides a classification of such cocycles as nilpotent and describes their conjugation to upper triangular or Jordan forms, including criteria for spectral domination.

Findings

Cocycles with all Lyapunov exponents at negative infinity are nilpotent.

Such cocycles can be analytically conjugated to upper triangular or Jordan normal forms.

Small perturbations can produce distinct Lyapunov exponents.

Abstract

We show that linear analytic cocycles where all Lyapunov exponents are negative infinite are nilpotent. For such one-frequency cocycles we show that they can be analytically conjugated to an upper triangular cocycle or a Jordan normal form. As a consequence, an arbitrarily small analytic perturbation leads to distinct Lyapunov exponents. Moreover, in the one-frequency case where the $k$-th Lyapunov exponent is finite and the $k+1$st negative infinite, we obtain a simple criterion for domination in which case there is a splitting into a nilpotent part and an invertible part.