Uniform Convergence of Schrödinger Cocycles over Bounded Toeplitz Subshift

For locally constant cocycle defined on an aperiodic subshift, Damanik and Lenz proved that if the subshift satisfies a certain condition (B), then the cocycle is uniform. For any simple Toeplitz subshift, we proved that the corresponding Schrödinger cocycle is uniform, although it does not satisfy condition (B) in general. In this paper, we study bounded Toeplitz subshift. In general, it does not satisfy condition (B); and it contains non-simple case, which make us cannot use Chebishev polynomial. By a combination of trace formula and avalanche principle, we prove that for any bounded Toeplitz subshift, the corresponding Schrödinger cocycle is also uniform.