The Schwartzman group of an affine transformation

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.