Simple Lyapunov spectrum for linear homogeneous differential equations with Lp parameters

Abstract

In the present paper we prove that densely, with respect to an $L^p$-like topology, the Lyapunov exponents associated to linear continuous-time cocycles $\Phi:\mathbb{R}\times M\to \text{GL}(2,\mathbb{R})$ induced by second order linear homogeneous differential equations $\ddot x+\alpha(\varphi^t(\omega))\dot x+\beta(\varphi^t(\omega))x=0$ are almost everywhere distinct. The coefficients $\alpha,\beta$ evolve along the $\varphi^t$-orbit for $\omega\in M$ and $\varphi^t: M\to M$ is an ergodic flow defined on a probability space. We also obtain the corresponding version for the frictionless equation $\ddot x+\beta(\varphi^t(\omega))x=0$ and for a Schr\"odinger equation $\ddot x+(E-Q(\varphi^t(\omega)))x=0$, inducing a cocycle $\Phi:\mathbb{R}\times M\to \text{SL}(2,\mathbb{R})$.