An ergodic family:

one computes:

As  is irrational, the action of the translations  on the unit circle is ergodic.
Consequences:
The spectrum of  does not depend on . (Pastur)
The absolutely continuous spectrum  does not depend on . (Last-Simon)
There is no discrete spectrum. (Pastur)
The Lyapunov exponent exists almost surely


The absolutely continuous spectrum is the essential closure of the set of energies
where the Lyapunov exponent vanishes (Ishii - Pastur - Kotani).
Almost sure existence of the density of states (it is the harmonic conjugate of the
Lyapunov exponent - Thouless formula). 
The spectrum is the set of growth point of the integrated density of states.