  This paper is concerned with uniform convergence in the multiplicative
ergodic theorem on aperiodic subshifts. If such a subshift satisfies a certain
condition, originally introduced by Boshernitzan, every locally constant
SL(2,R)-valued cocycle is uniform. As a consequence, the corresponding
Schr\"odinger operators exhibit Cantor spectrum of Lebesgue measure zero.
  An investigation of Boshernitzan's condition then shows that these results
cover all earlier results of this type and, moreover, provide various new ones.
In particular, Boshernitzan's condition is shown to hold for almost all circle
maps and almost all Arnoux-Rauzy subshifts.
