Genericity of trivial Lyapunov spectrum for Lp-cocycles derived from second order linear homogeneous differential equations

TL;DR

This paper proves that for a broad class of linear cocycles derived from second order differential equations, the Lyapunov spectrum is generically trivial, indicating stability in most cases.

Contribution

It establishes that trivial Lyapunov spectra are generic among $L^p$-continuous linear cocycles from second order differential equations.

Findings

Trivial Lyapunov spectrum is generic in the studied class.

The result applies to a Baire second category subset.

The topology used is $L^p$-like on the infinitesimal generator.

Abstract

Given an ergodic flow $\varphi^t\colon M\rightarrow M$ defined on a probability space $M$ we study a family of continuous-time kinetic linear cocycles associated to the solutions of the second order linear homogeneous differential equations $\ddot x +\alpha(\varphi^t(\omega))\dot x+\beta(\varphi^t(\omega))x=0$, where the parameters $\alpha,\beta$ evolve along the $\varphi^t$-orbit of $\omega\in M$. Our main result states that for a generic subset of kinetic continuous-time linear cocycles, where generic means a Baire second category with respect to an $L^p$-like topology on the infinitesimal generator, the Lyapunov spectrum is trivial.