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Funktsional. Anal. i Prilozhen., 2014, Volume 48, Issue 4, Pages 26–46 (Mi faa3166)  

This article is cited in 16 scientific papers (total in 17 papers)

The Problem of Describing Central Measures on the Path Spaces of Graded Graphs

A. M. Vershikabc

a Saint Petersburg State University
b St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
c Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow

Abstract: We suggest a new method for describing invariant measures on Markov compacta and on path spaces of graphs and, thereby, for describing characters of certain groups and traces of $AF$-algebras. The method relies on properties of filtrations associated with a graph and, in particular, on the notion of a standard filtration. The main tool is an intrinsic metric introduced on simplices of measures; this is an iterated Kantorovich metric, and the central result is that the relative compactness in this metric guarantees the possibility of a constructive enumeration of ergodic invariant measures. Applications include a number of classical theorems on invariant measures.

Keywords: invariant and central measures, projective limit of simplices, filtrations, intrinsic metric, uniform compactness

Funding Agency Grant Number
Russian Science Foundation 14-11-00581


DOI: https://doi.org/10.4213/faa3166

Full text: PDF file (433 kB)
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English version:
Functional Analysis and Its Applications, 2014, 48:4, 256–271

Bibliographic databases:

UDC: 517.9
Received: 08.08.2014

Citation: A. M. Vershik, “The Problem of Describing Central Measures on the Path Spaces of Graded Graphs”, Funktsional. Anal. i Prilozhen., 48:4 (2014), 26–46; Funct. Anal. Appl., 48:4 (2014), 256–271

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. M. Vershik, “Equipped graded graphs, projective limits of simplices, and their boundaries”, J. Math. Sci. (N. Y.), 209:6 (2015), 860–873  mathnet  crossref
    2. A. M. Vershik, A. V. Malyutin, “Phase transition in the exit boundary problem for random walks on groups”, Funct. Anal. Appl., 49:2 (2015), 86–96  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. A. M. Vershik, “Standardness as an Invariant Formulation of Independence”, Funct. Anal. Appl., 49:4 (2015), 253–263  mathnet  crossref  crossref  isi  elib
    4. Alexander V. Kolesnikov, Danila A. Zaev, “Exchangeable optimal transportation and log-concavity”, Theory Stoch. Process., 20(36):2 (2015), 54–62  mathnet  mathscinet  zmath
    5. È. B. Vinberg, S. E. Kuznetsov, “Evgenii (Eugene) Borisovich Dynkin (obituary)”, Russian Math. Surveys, 71:2 (2016), 345–371  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. A. R. Minabutdinov, “Limiting curves for polynomial adic systems”, J. Math. Sci. (N. Y.), 224:2 (2017), 286–303  mathnet  crossref  mathscinet
    7. Vershik A.M., “Asymptotic theory of path spaces of graded graphs and its applications”, Jap. J. Math., 11:2 (2016), 151–218  crossref  mathscinet  zmath  isi  scopus
    8. A. M. Vershik, “The theory of filtrations of subalgebras, standardness, and independence”, Russian Math. Surveys, 72:2 (2017), 257–333  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    9. A. M. Vershik, P. B. Zatitskii, “Universal adic approximation, invariant measures and scaled entropy”, Izv. Math., 81:4 (2017), 734–770  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    10. A. V. Kolesnikov, D. A. Zaev, “Optimal transportation of processes with infinite Kantorovich distance: independence and symmetry”, Kyoto J. Math., 57:2 (2017), 293–324  crossref  mathscinet  zmath  isi  scopus
    11. A. M. Vershik, A. V. Malyutin, “Infinite geodesics in the discrete Heisenberg group”, J. Math. Sci. (N. Y.), 232:2 (2018), 121–128  mathnet  crossref
    12. A. M. Vershik, A. V. Malyutin, “The Absolute of Finitely Generated Groups: II. The Laplacian and Degenerate Parts”, Funct. Anal. Appl., 52:3 (2018), 163–177  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    13. A. M. Vershik, A. V. Malyutin, “The absolute of finitely generated groups: I. Commutative (semi)groups”, Eur. J. Math., 4:4 (2018), 1476–1490  crossref  mathscinet  isi  scopus
    14. J. Math. Sci. (N. Y.), 240:5 (2019), 539–550  mathnet  crossref
    15. A. M. Vershik, A. V. Malyutin, “Asymptotic behavior of the number of geodesics in the discrete Heisenberg group”, J. Math. Sci. (N. Y.), 240:5 (2019), 525–534  mathnet  crossref
    16. Matveev K., “Macdonald-Positive Specializations of the Algebra of Symmetric Functions: Proof of the Kerov Conjecture”, Ann. Math., 189:1 (2019), 277–316  crossref  mathscinet  zmath  isi  scopus
    17. A. M. Vershik, “Three Theorems on the Uniqueness of the Plancherel Measure from Different Viewpoints”, Proc. Steklov Inst. Math., 305 (2019), 63–77  mathnet  crossref  crossref  isi  elib
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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