Symbolic Representations of Nonexpansive Group Automorphisms

If α is an irreducible nonexpansive ergodic automorphism of a compact abelian group X (such as an irreducible nonhyperbolic ergodic toral automorphism), then α 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 α-invariant probability measure on X. Moreover, every α-invariant probability measure on X arises essentially in this way. The last part of the paper deals with the connection between the twosided beta-shift Vβ arising from a Salem number β and the nonhyperbolic ergodic toral automorphism α arising from the companion matrix of the minimal polynomial of β, and establishes an entropy-preserving correspondence between a class of shift-invariant probability measures on Vβ and certain α-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.