Kotani-Last problem and Hardy spaces on surfaces of Widom type
  It is a small theory of non almost periodic ergodic families of Jacobi matrices with pure (however) absolutely continuous spectrum. And the reason why this effect may happen: under our "axioms" we found an analytic condition on the resolvent set that is responsible for (exactly equivalent to) this effect. 
A. Volberg and P. Yuditskii
math-ph math.CV math.MP math.SP