Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrödinger equations
- 1 February 2005
- journal article
- research article
- Published by Cambridge University Press (CUP) in Ergodic Theory and Dynamical Systems
- Vol. 25 (4), 1015-1045
- https://doi.org/10.1017/s0143385704000999
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 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.