Geodesic
Combinatorial Invariants of Metric Filtrations and Automorphisms; the Universal Adic Graph
Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 4, pp. 23-37.

Voir la notice de l'article provenant de la source Math-Net.Ru

We suggest a combinatorial classification of metric filtrations of measure spaces; a complete invariant of such a filtration is its combinatorial scheme, a measure on the space of hierarchies of the group $\mathbb Z$. In turn, the notion of a combinatorial scheme is a source of new metric invariants of automorphisms approximated by means of basic filtrations. We construct a universal graph with an adic structure such that every automorphism can be realized on its path space.
Keywords: uniform approximation, combinatorial definiteness, universal adic graph.
Mots-clés : filtrations 
@article{FAA_2018_52_4_a1,
     author = {A. M. Vershik and P. B. Zatitskii},
     title = {Combinatorial {Invariants} of {Metric} {Filtrations} and {Automorphisms;} the {Universal} {Adic} {Graph}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {23--37},
     publisher = {mathdoc},
     volume = {52},
     number = {4},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2018_52_4_a1/}
}
TY  - JOUR
AU  - A. M. Vershik
AU  - P. B. Zatitskii
TI  - Combinatorial Invariants of Metric Filtrations and Automorphisms; the Universal Adic Graph
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2018
SP  - 23
EP  - 37
VL  - 52
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2018_52_4_a1/
LA  - ru
ID  - FAA_2018_52_4_a1
ER  - 
%0 Journal Article
%A A. M. Vershik
%A P. B. Zatitskii
%T Combinatorial Invariants of Metric Filtrations and Automorphisms; the Universal Adic Graph
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2018
%P 23-37
%V 52
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2018_52_4_a1/
%G ru
%F FAA_2018_52_4_a1
A. M. Vershik; P. B. Zatitskii. Combinatorial Invariants of Metric Filtrations and Automorphisms; the Universal Adic Graph. Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 4, pp. 23-37. http://geodesic.mathdoc.fr/item/FAA_2018_52_4_a1/

[1] A. M. Vershik, “Teorema o lakunarnom izomorfizme monotonnykh posledovatelnostei razbienii”, Funkts. analiz i ego pril., 2:3 (1968), 17–21 | MR | Zbl

[2] A. M. Vershik, “Ubyvayuschie posledovatelnosti izmerimykh razbienii i ikh primeneniya”, Dokl. AN SSSR, 193:4 (1970), 748–751 | Zbl

[3] A. M. Vershik, “Kontinuum poparno neizomorfnykh diadicheskikh posledovatelnostei”, Funkts. analiz i ego pril., 5:3 (1971), 16–18 | DOI | MR

[4] A. M. Vershik, “Metricheskii invariant avtomorfizmov prostranstva s meroi, svyazannyi s ravnomernoi approksimatsiei i posledovatelnostyami razbienii”, Dokl. AN SSSR, 209:1 (1973), 15–18 | Zbl

[5] A. M. Vershik, “Chetyre opredeleniya shkaly avtomorfizma”, Funkts. analiz i ego pril., 7:3 (1973), 1–17

[6] A. M. Vershik, “Ravnomernaya algebraicheskaya approksimatsiya operatorov sdviga i umnozheniya”, Dokl. AN SSSR, 259:3 (1981), 526–529 | MR | Zbl

[7] A. M. Vershik, “Teorema o markovskoi periodicheskoi approksimatsii v ergodicheskoi teorii”, Kraevye zadachi matematicheskoi fiziki i smezhnye voprosy teorii funktsii, v. 14, Zap. nauchn. sem. LOMI, 115, 1982, 72–82 | Zbl

[8] A. M. Vershik, “Teoriya ubyvayuschikh posledovatelnostei izmerimykh razbienii”, Algebra i analiz, 6:4 (1994), 1–68 | MR | Zbl

[9] A. M. Vershik, A. D. Gorbulskii, “Masshtabirovannaya entropiya filtratsii $\sigma$-algebr”, Teoriya veroyatn. i ee primen., 52:3 (2007), 446–467 | DOI

[10] A. M. Vershik, “Teoriya filtratsii podalgebr, standartnost i nezavisimost”, UMN, 72:2(434) (2017), 67–146 | DOI | MR | Zbl

[11] A. M. Vershik, P. B. Zatitskii, “Universalnaya adicheskaya approksimatsiya, invariantnye mery i masshtabirovannaya entropiya”, Izv. RAN. Ser. matem., 81:4 (2017), 68–107 | DOI | MR | Zbl

[12] E. Glasner, B. Weiss, “On the interplay between measurable and topological dynamics”, Handbook of Dynamical Systems, v. 1B, Elsevier, Amsterdam, 2006, 597–648 | MR | Zbl

[13] R. Gjerde, Ø. Johansen, “Bratteli–Vershik models for Cantor minimal systems associated to interval exchange transformations”, Math. Scand., 90:1 (2002), 87–100 | DOI | MR | Zbl

[14] S. Bezuglyi, O. Karpel, “Bratteli diagrams: structure, measures, dynamics”, Dynamics and numbers, Contemporary Math., 669, Amer. Math. Soc., Providence, RI, 2016, 1–36 | DOI | MR | Zbl

[15] A. B. Katok, “Monotonnaya ekvivalentnost v ergodicheskoi teorii”, Izv. AN SSSR. Ser. matem., 41:1 (1977), 104–157 | MR | Zbl

[16] A. B. Katok, A. M. Stepin, “Approksimatsii v ergodicheskoi teorii”, UMN, 22:5(137) (1967), 81–106 | MR | Zbl

[17] B. Hasselblatt, A. Katok, Handbook of Dynamical Systems, Elsevier, Amsterdam, 2002 | MR

[18] A. M. Stepin, “Ob entropiinom invariante ubyvayuschikh posledovatelnostei izmerimykh razbienii”, Funkts. analiz i ego pril., 5:3 (1971), 80–84 | MR

[19] A. M. Vershik, P. B. Zatitskii, “Ob universalnom borelevskom adicheskom prostranstve”, Zap. nauchn. sem. POMI, 468 (2018), 24–38 | MR