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Izv. RAN. Ser. Mat., 2017, Volume 81, Issue 4, Pages 68–107 (Mi izv8610)  

This article is cited in 3 scientific papers (total in 3 papers)

Universal adic approximation, invariant measures and scaled entropy

A. M. Vershikabc, P. B. Zatitskiidea

a St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
b St. Petersburg State University, Department of Mathematics and Mechanics
c Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow
d Ècole Normale Supérieure, Département de mathématiques et applications, Paris
e Chebyshev Laboratory, St. Petersburg State University, Department of Mathematics and Mechanics

Abstract: We define an infinite graded graph of ordered pairs and a canonical action of the group $\mathbb{Z}$ (the adic action) and of the infinite sum of groups of order two $\mathcal{D}=\sum_1^{\infty} \mathbb{Z}/2\mathbb{Z}$ on the path space of the graph. It is proved that these actions are universal for both groups in the following sense: every ergodic action of these groups with invariant measure and binomial generator, multiplied by a special action (the ‘odometer’), is metrically isomorphic to the canonical adic action on the path space of the graph with a central measure. We consider a series of related problems.

Keywords: graph of ordered pairs, universal action, adic transformation, scaled entropy.

Funding Agency Grant Number
Russian Science Foundation 14-11-00581
This research was financially supported by the Russian Science Foundation under grant no.~14-11-00581).


DOI: https://doi.org/10.4213/im8610

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English version:
Izvestiya: Mathematics, 2017, 81:4, 734–770

Bibliographic databases:

UDC: 517.518
MSC: Primary 37A35; Secondary 28D05, 37A05, 37A50, 60G99
Received: 02.10.2016

Citation: A. M. Vershik, P. B. Zatitskii, “Universal adic approximation, invariant measures and scaled entropy”, Izv. RAN. Ser. Mat., 81:4 (2017), 68–107; Izv. Math., 81:4 (2017), 734–770

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, “The theory of filtrations of subalgebras, standardness, and independence”, Russian Math. Surveys, 72:2 (2017), 257–333  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. A. M. Vershik, P. B. Zatitskii, “Combinatorial Invariants of Metric Filtrations and Automorphisms; the Universal Adic Graph”, Funct. Anal. Appl., 52:4 (2018), 258–269  mathnet  crossref  crossref  isi  elib
    3. A. M. Vershik, P. B. Zatitskii, “On a universal Borel adic space”, J. Math. Sci. (N. Y.), 240:5 (2019), 515–524  mathnet  crossref
  • Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya Izvestiya: Mathematics
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