FINITE RANK BRATTELI DIAGRAMS: STRUCTURE OF INVARIANT MEASURES

S Bezuglyi; J Kwiatkowski; K Medynets; B Solomyak

doi:10.1090/S0002-9947-2012-05744-8

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FINITE RANK BRATTELI DIAGRAMS: STRUCTURE OF INVARIANT MEASURES

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FINITE RANK BRATTELI DIAGRAMS: STRUCTURE OF INVARIANT MEASURES

S Bezuglyi, J Kwiatkowski, K Medynets and B Solomyak

Transactions of the American Mathematical Society, Vol.365(5), pp.2637-2679

05/01/2013

DOI: 10.1090/S0002-9947-2012-05744-8

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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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Details

Title: Subtitle: FINITE RANK BRATTELI DIAGRAMS: STRUCTURE OF INVARIANT MEASURES
Creators: S Bezuglyi - National Academy of Sciences of Ukraine
J Kwiatkowski - University of Warmia and Mazury in Olsztyn
K Medynets - United States Naval Academy
B Solomyak - University of Washington
Resource Type: Journal article
Publication Details: Transactions of the American Mathematical Society, Vol.365(5), pp.2637-2679
DOI: 10.1090/S0002-9947-2012-05744-8
ISSN: 0002-9947
eISSN: 1088-6850
Publisher: American Mathematical Society
Number of pages: 43
Grant note: N201384834 / MNiSzW N DMS-0654408; DMS-0968879 / NSF Erwin Schrodinger International Institute for Mathematical Physics in Vienna
Language: English
Date published: 05/01/2013
Academic Unit: Mathematics
Record Identifier: 9984241056502771

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