  If $\alpha $ is an irreducible nonexpansive ergodic automorphism of a compact
abelian group $X$ (such as an irreducible nonhyperbolic ergodic toral
automorphism), then $\alpha $ has no finite or infinite state Markov
partitions, and there are no nontrivial continuous embeddings of Markov shifts
in $X$. In spite of this we are able to construct a symbolic space $V$ and a
class of shift-invariant probability measures on $V$ each of which corresponds
to an $\alpha$-invariant probability measure on $X$. Moreover, every
$\alpha$-invariant probability measure on $X$ arises essentially in this way.
  The last part of the paper deals with the connection between the two-sided
beta-shift $V_\beta $ arising from a Salem number $\beta $ and the
nonhyperbolic ergodic toral automorphism $\alpha $ arising from the companion
matrix of the minimal polynomial of $\beta $, and establishes an
entropy-preserving correspondence between a class of shift-invariant
probability measures on $V_\beta $ and certain $\alpha $-invariant probability
measures on $X$. This correspondence is much weaker than, but still quite
closely modelled on, the connection between the two-sided beta-shifts defined
by Pisot numbers and the corresponding hyperbolic ergodic toral automorphisms.
