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

Abstract

Given an ergodic flow ${\phi }^t:M\to M$ 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 $\ddot{x}+\alpha ({\phi }^t(\omega ))\dot{x}+\beta ({\phi }^t(\omega ))x=0$, where the parameters $\alpha ,\beta$ evolve along the ${\phi }^t$-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.