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 $\ddot{x}+\alpha ({\phi }^t(\omega ))\dot{x}+\beta ({\phi }^t(\omega ))x=0$ are almost everywhere distinct. The coefficients $\alpha ,\beta$ evolve along the ${\phi }^t$-orbit for $\omega \in M$ and ${\phi }^t:M\to M$ is an ergodic flow defined on a probability space. We also obtain the corresponding version for the frictionless equation $\ddot{x}+\beta ({\phi }^t(\omega ))x=0$ and for a Schrödinger equation $\ddot{x}+(E-Q({\phi }^t(\omega )))x=0$, inducing a cocycle $\Phi :\mathbb{R}\times M\to \text{SL}(2,\mathbb{R})$.