  Lind and Schmidt have shown that the homoclinic group of a cyclic $\Z^k$
algebraic dynamical system is isomorphic to the dual of the phase group. We
show that this duality result is part of an exact sequence if $k=1$. The exact
sequence is a well known algebraic object, which has been applied by Schmidt in
his work on rigidity. We show that it can be derived from dynamical
considerations only. The constructions naturally lead to an almost 1-1-coding
of certain Pisot automorphisms by their associated $\beta$-shift, generalizing
similar results for Pisot automorphisms of the torus.
