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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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