Spectral Properties of Limit-periodic Schrödinger Operators

We investigate the spectral properties of Schrödinger operators in `2(Z) with limit-periodic potentials. The perspective we take was recently proposed by Avila and is based on regarding such potentials as generated by continuous sampling along the orbits of a minimal translation of a Cantor group. This point of view allows one to separate the base dynamics and the sampling function. We show that for any such base dynamics, the spectrum is of positive Lebesgue measure and purely absolutely continuous for a dense set of sampling functions, and it is of zero Lebesgue measure and purely singular continuous for a dense Gδ set of sampling functions.