TITLE: Cantor spectrum and KDS eigenstates

AUTHOR: Joaquim Puig 

INSTITUTION: Departament de Matemātica Aplicada I, Universitat Polit\`ecnica de Catalunya.
Av. Diagonal 647, 08028 Barcelona, Spain 

ABSTRACT: 
  
In this note we consider KDS  eigenstates of one-dimensional Schrödinger operators with ergodic potential, which are a class of
generalized eigenfunctions including Bloch eigenstates. We show that if the spectrum, restricted to an interval, has zero Lyapunov
exponents and is a Cantor set,  then for a residual subset of energies, KDS eigenstates do not exist. In particular, we show that the
quasi-periodic Schrödinger operators whose  quasi-periodic cocycles are reducible for all energies have a limit
band-type spectrum.