Symbolic representations of nonexpansive group automorphisms

Elon Lindenstrauss*, Klaus Schmidt

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

Abstract

If α is an irreducible nonexpansive ergodic automorphism of a compact abelian group X (such as an irreducible nonhyperbolic ergodic toral automorphism), then a 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.

Original languageAmerican English
Pages (from-to)227-266
Number of pages40
JournalIsrael Journal of Mathematics
Volume149
DOIs
StatePublished - 2005
Externally publishedYes

Bibliographical note

Funding Information:
ACKNOWLEDGEMENT: This research has been supported in part by NSF grants DMS 0140497 and DMS 0434403 (E.L.) and FWF Project P16004-N05 (K.S.). During part of this work, both authors received support from the American Institute of Mathematics and NSF grant DMS 0222452. We would furthermore like to express our gratitude to the Mathematics Departments of the University of Washington, Stanford University, the Newton Institute, Cambridge and the ETH Ziirich for hospitality during parts of this work. E.L. is a Clay Research Fellow and is grateful for this generous support from the Clay Mathematics Institute. E. L. would also like to thank Rick Kenyon for an interesting and helpful discussion on these and related topics.

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