On the measure of gaps and spectra for discrete 1D Schrödinger operators

We study the lebesgue measure of gaps and spectra, of ergodic Jacobi matrices. We show that: |σ/A|+|G|≥v, where: σ is the spectrum, G is the union of the gaps, A is the set of energies where the Lyaponov exponent vanishes and v is an appropriate seminorm of the potential. We also study in more detail periodic Jacobi matrices, and obtain a lower bound and large coupling asymptotics for the measure of the spectrum. We apply the results of the periodic case, to limit periodic Jacobi matrices, and obtain sufficient conditions for |G|≥v and for |σ|>0.