  The spectrum of one-dimensional discrete Schr\"odinger operators associated
to strictly ergodic dynamical systems is shown to coincide with the set of
zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists
uniformly. This is used to obtain Cantor spectrum of zero Lebesgue measure for
all aperiodic subshifts with uniform positive weights. This covers, in
particular, all aperiodic subshifts arising from primitive substitutions
including new examples as e.g. the Rudin-Shapiro substitution.
  Our investigation is not based on trace maps. Instead it relies on an
Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the
author.
