Cantor spectrum and Kotani eigenstates

In this note we consider Kotani eigenstates of one-dimensional Schrödinger operators with ergodic potential. 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, Kotani eigenstates do not exist. In particular, we show that the quasi-periodic Schrödinger operators whose Schrödinger quasi-periodic cocycles are reducible for all energies have a limit band-type spectrum.