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Symbolic representations of nonexpansive group automorphisms

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Abstract

Ifα is an irreducible nonexpansive ergodic automorphism of a compact abelian groupX (such as an irreducible nonhyperbolic ergodic toral automorphism), thenα has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts inX. In spite of this we are able to construct a symbolic spaceV and a class of shift-invariant probability measures onV each of which corresponds to anα-invariant probability measure onX. Moreover, everyα-invariant probability measure onX arises essentially in this way.

The last part of the paper deals with the connection between the two-sided beta-shiftV β arising from a Salem numberβ and the nonhyperbolic ergodic toral automorphismα arising from the companion matrix of the minimal polynomial ofβ, and establishes an entropy-preserving correspondence between a class of shift-invariant probability measures onV β and certainα-invariant probability measures onX. 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.

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Author notes

  1. Elon Lindenstrauss

    Present address: Department of Mathematics, Princeton University, 08544, Princeton, NJ, USA

Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, 10012, New York, NY, USA

    Elon Lindenstrauss

  2. Mathematics Institute, University of Vienna, Nordbergstrasse 15, A-1090, Vienna, Austria

    Klaus Schmidt

  3. Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090, Vienna, Austria

    Klaus Schmidt

Authors
  1. Elon Lindenstrauss
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  2. Klaus Schmidt
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Correspondence to Elon Lindenstrauss.

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Lindenstrauss, E., Schmidt, K. Symbolic representations of nonexpansive group automorphisms. Isr. J. Math. 149, 227–266 (2005). https://doi.org/10.1007/BF02772542

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  • DOI: https://doi.org/10.1007/BF02772542

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