Simple Lyapunov spectrum for linear homogeneous differential equations with Lp parameters

TL;DR

This paper demonstrates that for a broad class of linear differential equations with coefficients evolving along ergodic flows, the Lyapunov exponents are generically distinct, highlighting a form of spectral simplicity in these systems.

Contribution

It proves that, in an $L^p$-like topology, the Lyapunov exponents for certain linear cocycles are almost everywhere distinct, extending to various types of second order equations.

Findings

Lyapunov exponents are generically distinct for these systems

Results apply to both damped and frictionless equations

Includes cases for Schrödinger equations with potential functions

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 $\Phi:\mathbb{R}\times M\to \text{SL}(2,\mathbb{R})$.