  We compute the Schwartzman group associated with an ergodic affine
automorphism of a compact connected abelian group given by the composition of
an automorphism of the group and a translation by an element in the path
component of the identity. We show that the Schwartzman group can be
characterized by evaluating the invariant characters of the automorphism at the
group element by which one translates. As a byproduct, we show that the set of
labels associated with the doubling map on the dyadic solenoid is trivial,
which in turn allows us to show that any ergodic family of Jacobi matrices
defined over the doubling map has connected almost-sure essential spectrum.
