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Minimality and unique ergodicity for adic transformations

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Abstract

We study the relationship between minimality and unique ergodicity for adic transformations. We show that three is the smallest alphabet size for a unimodular “adic counterexample”, an adic transformation which is minimal but not uniquely ergodic. We construct a specific family of counterexamples built from (3 × 3) nonnegative integer matrix sequences, while showing that no such (2 × 2) sequence is possible. We also consider (2 × 2) counterexamples without the unimodular restriction, describing two families of such maps.

Though primitivity of the matrix sequence associated to the transformation implies minimality, the converse is false, as shown by a further example: an adic transformation with (2 × 2) stationary nonprimitive matrix, which is both minimal and uniquely ergodic.

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Authors and Affiliations

  1. Institut de Mathématiques de Luminy (UPR 9016), 163 Avenue de Luminy, case 930, 13288, Marseille Cedex 9, France

    Sebastien Ferenczi

  2. Dept Mat IME-USP, rua do Matão 1010, CEP 05508-090, São Paulo SP, Brazil

    Albert M. Fisher

  3. C.M.I. Université de Provence LATP, CNRSUMR 6632, F-13453, Marseille Cedex 13, France

    Marina Talet

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  1. Sebastien Ferenczi
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  2. Albert M. Fisher
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  3. Marina Talet
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Correspondence to Sebastien Ferenczi.

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Ferenczi, S., Fisher, A.M. & Talet, M. Minimality and unique ergodicity for adic transformations. JAMA 109, 1–31 (2009). https://doi.org/10.1007/s11854-009-0027-y

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  • DOI: https://doi.org/10.1007/s11854-009-0027-y

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