Minimality and unique ergodicity for adic transformations
- First Online:
- 95 Downloads
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.
Unable to display preview. Download preview PDF.
- [BKMS09]S. Bezuglyi, J. Kwiatkowski, K. Medynets and B. Solomyak, Invariant measures on stationary Bratteli diagrams, Ergodic Theory Dynam. Systems (2009), to appear.Google Scholar
- [Fer02]S. Ferenczi, Substitutions and symbolic dynamical systems, in Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Math. 1794, Springer, Berlin, 2002, pp. 101–142.Google Scholar
- [FZ06]S. Ferenczi and L. Zamboni, Eigenvalues and simplicity of 4-interval exchanges, preprint 2009, http://iml.univ-mrs.fr/~ferenczi/fz2.pdf.
- [Fis]A. M. Fisher, Abelian differentials, interval exchanges, and adic transformations, in preparation, 2008.Google Scholar
- [Fis92]A. M. Fisher, Integer Cantor sets and an order-two ergodic theorem, Ergodic Theory Dynam. Systems 13 (1992), 45–64.Google Scholar
- [Fis08]A. M. Fisher, Nonstationary mixing and the unique ergodicity of adic transformations, Stochastics and Dynamics, 2009, to appear.Google Scholar
- [Hos00]B. Host, Substitution subshifts and Bratteli diagrams, in Topics in Symbolic Dynamics and Applications, Cambridge Univ. Press, Cambridge, 2000, pp. 35–55.Google Scholar
- [Kri70]W. Krieger, On unique ergodicity, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. II: Probability Theory, Univ. California press, Berkeley, Calif., 1972, 327–346.Google Scholar
- [LM95]D. Lind and B. Marcus, Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.Google Scholar
- [Ver89]A.M. Vershik, A new model of the ergodic transformations, in Dynamical Systems and Ergodic Theory, Banach Center Publications 23, PWN-Polish Scientific Publishers, Warsaw, 1989, pp. 381–384.Google Scholar
- [Ver81]A. M. Vershik, Uniform algebraic approximation of shift and multiplication operators, Soviet Math. Dokl. 24 (1981), 101–103.Google Scholar
- [Ver95a]A. M. Vershik, The adic realizations of the ergodic actions with the homeomorphisms of the Markov compact and the ordered Bratteli diagrams, Teor. Predstav. Din. Systemy Kombin. i Algoritm Metody. I (1995), 120–126; translated in J. Math. Sci. (New York) 87 (1997), 4054–4058.Google Scholar
- [Via06]M. Viana, Dynamics of interval exchange transformations and Teichmüller flows, Lecture notes, July 6, 2008, http://w3.impa.br/~viana/out/ietf.pdf.