Ergodic family :

one computes:


As  is irrationnal, the translations  act as an ergodic group on the circle.

Consequences:
The spectrum of  is independent of . (Pastur)
The absolutely continuous spectrum of  is independent of . (Last-Simon)
No discrete spectrum. (Pastur)
Almost sure existence of the Lyapunov exponent


The absolutely continuous spectrum is the essential closure of the set of energies 
with vanishing Lyapunov exponent (Ishii - Pastur - Kotani).
Almost sure existence of the integrated density of states (the harmonic conjugate
of the Lyapunov exponent). 
The spectrum is the set of growth points of the IDS.