  This survey gives an account of an algebraic construction of symbolic covers
and representations of ergodic automorphisms of compact abelian groups,
initiated by A.M. Vershik around 1992 for hyperbolic automorphisms of
finite-dimensional tori. The key ingredient in this approach, which was
subsequently extended to arbitrary expansive automorphisms of compact abelian
groups, is the use of homoclinic points of the automorphism.
  Although existence and abundance of homoclinic points is intimately connected
to expansiveness of the automorphism, it is nevertheless possible to extend
certain aspects of this construction to nonexpansive irreducible automorphisms
of compact abelian groups (like irreducible toral automorphisms whose dominant
eigenvalue is a Salem number). The later sections of this survey discuss the
phenomena and problems arising in this extension.
