  In the theory of ergodic one-dimensional Schrodinger operators, ac spectrum
has been traditionally expected to be very rigid. Two key conjectures in this
direction state, on one hand, that ac spectrum demands almost periodicity of
the potential, and, on the other hand, that the eigenfunctions are almost
surely bounded in the essential suport of the ac spectrum. We show how the
repeated slow deformation of periodic potentials can be used to break rigidity,
and disprove both conjectures.
