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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Finite rank Bratteli diagrams: Structure of invariant measures


Authors: S. Bezuglyi, J. Kwiatkowski, K. Medynets and B. Solomyak
Journal: Trans. Amer. Math. Soc. 365 (2013), 2637-2679
MSC (2010): Primary 37B05, 37A25, 37A20
Published electronically: November 7, 2012
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Abstract: We consider Bratteli diagrams of finite rank (not necessarily simple) and ergodic invariant measures with respect to the cofinal equivalence relation on their path spaces. It is shown that every ergodic invariant measure (finite or ``regular'' infinite) is obtained by an extension from a simple subdiagram. We further investigate quantitative properties of these measures, which are mainly determined by the asymptotic behavior of products of incidence matrices. A number of sufficient conditions for unique ergodicity are obtained. One of these is a condition of exact finite rank, which parallels a similar notion in measurable dynamics. Several examples illustrate the broad range of possible behavior of finite rank diagrams and invariant measures on them. We then prove that the Vershik map on the path space of an exact finite rank diagram cannot be strongly mixing, independent of the ordering. On the other hand, for the so-called ``consecutive'' ordering, the Vershik map is not strongly mixing on all finite rank diagrams.


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Additional Information

S. Bezuglyi
Affiliation: Institute for Low Temperature Physics, National Academy of Sciences of Ukraine, Kharkov, Ukraine
Email: bezuglyi@ilt.kharkov.ua

J. Kwiatkowski
Affiliation: Department of Mathematics, University of Warmia and Mazury, 10-719 Olsztyn, Poland
Email: jkwiat@mat.uni.torun.pl

K. Medynets
Affiliation: Department of Mathematics, United States Naval Academy, Annapolis, Maryland 21402
Email: medynets@usna.edu

B. Solomyak
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
Email: solomyak@math.washington.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05744-8
PII: S 0002-9947(2012)05744-8
Keywords: Bratteli diagrams, Vershik maps, mixing, ergodicity, invariant measures
Received by editor(s): December 23, 2010
Received by editor(s) in revised form: September 19, 2011
Published electronically: November 7, 2012
Additional Notes: The research of the second author was supported by grant MNiSzW N N201384834.
The fourth author was supported in part by NSF grants DMS-0654408 and DMS-0968879.
Article copyright: © Copyright 2012 American Mathematical Society