Uniform Spectral Properties of One-dimensional Quasicrystals, Iv. Quasi-sturmian Potentials

We consider discrete one-dimensional Schrr odinger operators with quasi-Sturmian potentials. We present a new approach to the trace map dy-namical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, it is shown that the operators have purely singular continuous spectrum and their spectrum is a Cantor set of Lebesgue measure zero. We also exhibit a subclass having purely-continuous spectrum. All these results hold uniformly on the hull generated by a given potential.