Ergodic Theory and Dynamical Systems



Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrödinger equations


KRISTIAN BJERKLÖV a1
a1 Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden (e-mail: bjerklov@math.kth.se) Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 3G3 (e-mail: bjerklov@math.utoronto.ca)

Article author query
bjerklov k   [Google Scholar] 
 

Abstract

We study the discrete quasi-periodic Schrödinger equation \[-(u_{n+1}+u_{n-1})+\lambda V(\theta+n\omega)u_n=Eu_n\] with a non-constant C1 potential function $V:\mathbb{T}\to\mathbb{R}$. We prove that for sufficiently large $\lambda$ there is a set $\Omega\subset\mathbb{T}$ of frequencies $\omega$, whose measure tends to 1 as $\lambda\to\infty$, with the following property. For each $\omega\in\Omega$ there is a ‘large’ (in measure) set of energies E, all lying in the spectrum of the associated Schrödinger operator (and hence giving a lower estimate on the measure of the spectrum), such that the Lyapunov exponent is positive and, moreover, the projective dynamical system induced by the Schrödinger cocycle is minimal but not ergodic.

(Received January 23 2004)
(Revised October 30 2004)