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

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.