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Zap. Nauchn. Sem. POMI, 2015, Volume 432, Pages 83–104 (Mi znsl6112)  

This article is cited in 15 scientific papers (total in 15 papers)

Equipped graded graphs, projective limits of simplices, and their boundaries

A. M. Vershikab

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

Abstract: In this paper, we develop a theory of equipped graded graphs (or Bratteli diagrams) and an alternative theory of projective limits of finite-dimensional simplices. An equipment is an additional structure on the graph, namely, a system of “cotransition” probabilities on the set of its paths. The main problem is to describe all probability measures on the path space of the graph with given cotransition probabilities; it goes back to the problem, posed by E. B. Dynkin in the 1960s, of describing exit and entrance boundaries for Markov chains. The most important example is the problem of describing all central measures; those of describing states on AF-algebras or characters on locally finite groups can be reduced to it. We suggest an unification of the whole theory, an interpretation of the notions of Martin, Choquet, and Dynkin boundaries in terms of equipped graded graphs and in terms of the theory of projective limits of simplices. In the last section, we study the new notion of “standardness" of projective limits of simplices and of equipped Bratteli diagrams, as well as the notion of "lacunarization.”

Key words and phrases: equipped Bratteli diagram, projective limits of simplices, ergodic measures, Martin boundary.

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English version:
Journal of Mathematical Sciences (New York), 2015, 209:6, 860–873

UDC: 519.173+519.217.72
Received: 17.01.2015

Citation: A. M. Vershik, “Equipped graded graphs, projective limits of simplices, and their boundaries”, Representation theory, dynamical systems, combinatorial methods. Part XXIV, Zap. Nauchn. Sem. POMI, 432, POMI, St. Petersburg, 2015, 83–104; J. Math. Sci. (N. Y.), 209:6 (2015), 860–873

Citation in format AMSBIB
\Bibitem{Ver15}
\by A.~M.~Vershik
\paper Equipped graded graphs, projective limits of simplices, and their boundaries
\inbook Representation theory, dynamical systems, combinatorial methods. Part~XXIV
\serial Zap. Nauchn. Sem. POMI
\yr 2015
\vol 432
\pages 83--104
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6112}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2015
\vol 209
\issue 6
\pages 860--873
\crossref{https://doi.org/10.1007/s10958-015-2533-z}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84939420890}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. 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
    2. A. M. Vershik, N. I. Nessonov, “Stable representations of the infinite symmetric group”, Izv. Math., 79:6 (2015), 1184–1214  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. P. B. Zatitskiy, “On the possible growth rate of a scaling entropy sequence”, J. Math. Sci. (N. Y.), 215:6 (2016), 715–733  mathnet  crossref  mathscinet
    4. D. A. Zaev, “On ergodic decompositions related to the Kantorovich problem”, J. Math. Sci. (N. Y.), 216:1 (2016), 65–83  mathnet  crossref  mathscinet
    5. J. Math. Sci. (N. Y.), 224:2 (2017), 199–213  mathnet  crossref  mathscinet
    6. 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
    7. Vershik A., “Smoothness and Standardness in the Theory of Af-Algebras and in the Problem on Invariant Measures”, Probability and Statistical Physics in St. Petersburg, Proceedings of Symposia in Pure Mathematics, 91, eds. Sidoravicius V., Smirnov S., Amer. Math. Soc., 2016, 423–436  crossref  mathscinet  zmath  isi
    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. V. Duzhin, N. Vasilyev, “Modeling of an asymptotically central Markov process on 3D Young graph”, Math. Comput. Sci., 11:3-4, SI (2017), 315–328  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
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