arXiv subjects

Dinis Amaro

Publications and source records attributed to Dinis Amaro.

2 recordsLinked to original sources

arXiv2023

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

Given an ergodic flow $φ^t\colon M\rightarrow M$ 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 $\ddot x +α(φ^t(ω))\dot x+β(φ^t(ω))x=0$, where the parameters $α,β$ evolve along the $φ^t$-orbit of $ω\in M$. 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.

math.DS↗

arXiv2023

Simple Lyapunov spectrum for linear homogeneous differential equations with Lp parameters

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 $Φ:\mathbb{R}\times M\to \text{GL}(2,\mathbb{R})$ induced by second order linear homogeneous differential equations $\ddot x+α(φ^t(ω))\dot x+β(φ^t(ω))x=0$ are almost everywhere distinct. The coefficients $α,β$ evolve along the $φ^t$-orbit for $ω\in M$ and $φ^t: M\to M$ is an ergodic flow defined on a probability space. We also obtain the corresponding version for the frictionless equation $\ddot x+β(φ^t(ω))x=0$ and for a Schrödinger equation $\ddot x+(E-Q(φ^t(ω)))x=0$, inducing a cocycle $Φ:\mathbb{R}\times M\to \text{SL}(2,\mathbb{R})$.

math.DS↗