, Volume 109, Issue 1, pp 1-31
Date: 19 Jan 2010

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.