Journal d'Analyse Mathématique

, Volume 109, Issue 1, pp 1–31

Minimality and unique ergodicity for adic transformations

Authors

    • Institut de Mathématiques de Luminy (UPR 9016)
  • Albert M. Fisher
    • Dept Mat IME-USP
  • Marina Talet
    • C.M.I. Université de Provence LATP
Article

DOI: 10.1007/s11854-009-0027-y

Cite this article as:
Ferenczi, S., Fisher, A.M. & Talet, M. JAMA (2009) 109: 1. doi:10.1007/s11854-009-0027-y

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.

Copyright information

© Hebrew University Magnes Press 2009