  Avila recently introduced a new method for the study of the discrete
Schr\"odinger Operator with limit periodic potential. I adapt this method to
the context of orthogonal polynomials in the unit circle with limit periodic
Verblunsky Coefficients. Specifically, I represent these Verblunsky
Coefficients as a continuous sampling of the orbits of a Cantor group by a
minimal translation. I then investigate the measures that arise on the unit
circle as I vary the sampling function.
