Symbolic representations of nonexpansive group automorphisms

Abstract

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.