Finite rank bratteli diagrams: Structure of invariant measures

S. Bezuglyi, J. Kwiatkowski, K. Medynets, B. Solomyak

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)2637-2679
Number of pages43
JournalTransactions of the American Mathematical Society
Volume365
Issue number5
DOIs
StatePublished - 2013
Externally publishedYes

Keywords

  • Bratteli diagrams
  • Ergodicity
  • Invariant measures
  • Mixing
  • Vershik maps

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • General Mathematics

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