Elon Lindenstrauss, Klaus Schmidt
Symbolic Representations of Nonexpansive Group Automorphisms
Preprint series: ESI preprints

MSC:

28D05 Measure-preserving transformations

34C23 Bifurcation See mainly{58F14}

34C37 Homoclinic and heteroclinic solutions, See also {58F15}

58F14 Bifurcation theory and singularities

22D40 Ergodic theory on groups, See also {28Dxx, 43A60}

Abstract: If $\alpha $ is an irreducible nonexpansive ergodic automorphism of a compact abelian group $X$ (such as an irreducible nonhyperbolic toral automorphism), then
$\alpha $ has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of such shifts in $X$. In spite of this we construct a symbolic space $V$ and a class of shift-invariant probability measures on
$V$ such that every such measure $\nu $ on $V$ corresponds to an $\alpha $-invariant probability measure on $X$, and that \emph{every} $\alpha $-invariant probability measure on $X$ arises essentially in this manner.

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 automorphism.

Keywords: Partially hyperbolic group automorphisms, invariant measures, Markov partitions, Beta-shifts
Notes: second version