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We consider discrete one-dimensional Schr\"odinger operators withquasi-Sturmian potentials. We present a new approach to the trace map dynamicalsystem which is independent of the initial conditions and establish acharacterization of the spectrum in terms of bounded trace map orbits. Usingthis, it is shown that the operators have purely singular continuous spectrumand their spectrum is a Cantor set of Lebesgue measure zero. We also exhibit asubclass having purely $\alpha$-continuous spectrum. All these results holduniformly on the hull generated by a given potential.

Uniform spectral properties of one-dimensional quasicrystals, iv. quasi-sturmian potentials