Symbolic representations of nonexpansive group automorphisms

Abstract

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 two-sided beta-shift $V_β$ 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 $V_β$ 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.