Substitution Hamiltonians with Bounded Trace Map Orbits

We investigate discrete one-dimensional Schr¨odinger operators with aperiodic potentials generated by primitive invertible substitutions on a two-letter alphabet. We prove that the spectrum coincides with the set of energies having a bounded trace map orbit and show that it is a Cantor set of zero Lebesgue measure. This result confirms a suggestion arising from a study of Roberts and complements results obtained by Bovier and Ghez. As an application we present a class of models exhibiting a purely singular continuous spectrum with probability one