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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, “Ob universalnom borelevskom adicheskom prostranstve”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye metody. XXIX, Zap. nauchn. sem. POMI, 468, POMI, SPb., 2018, 24–38  mathnet
  • Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya Izvestiya: Mathematics
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