  We consider discrete Schr\"odinger operators on the half line with potentials
generated by the doubling map and continuous sampling functions. We show that
the essential spectrum of these operators is always connected. This result is
obtained by computing the subgroup of the range of the Schwartzman homomorphism
associated with homotopy classes of continuous maps on the suspension of the
standard solenoid that factor through the suspension of the doubling map and
then showing that this subgroup characterizes the topological structure of the
spectrum.
