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Mat. Sb., 2001, Volume 192, Number 12, Pages 61–92 (Mi msb616)  

This article is cited in 9 scientific papers (total in 9 papers)

Stochastic constructions of flows of rank 1

A. A. Prikhod'ko

M. V. Lomonosov Moscow State University

Abstract: Automorphisms of rank 1 appeared in the well-known papers of Chacon (1965), who constructed an example of a weakly mixing automorphism not having the strong mixing property, and Ornstein (1970), who proved the existence of mixing automorphisms without a square root. Ornstein's construction is essentially stochastic, since its parameters are chosen in a “sufficiently random manner” according to a certain random law.
In the present article it is shown that mixing flows of rank 1 exist. The construction given is also stochastic and is based to a large extent on ideas in Ornstein's paper. At the same time it complements Ornstein's paper and makes it more transparent. The construction can be used also to obtain automorphisms with various approximation and statistical properties. It is established that the new examples of dynamical systems are not isomorphic to Ornstein automorphisms, that is, they are qualitatively new.

DOI: https://doi.org/10.4213/sm616

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English version:
Sbornik: Mathematics, 2001, 192:12, 1799–1828

Bibliographic databases:

UDC: 517.9
MSC: Primary 37A25, 28D05; Secondary 54H20, 47A35
Received: 14.06.2001

Citation: A. A. Prikhod'ko, “Stochastic constructions of flows of rank 1”, Mat. Sb., 192:12 (2001), 61–92; Sb. Math., 192:12 (2001), 1799–1828

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. Ryzhikov V.V., “On the spectral and mixing properties of rank-one constructions in ergodic theory”, Dokl. Math., 74:1 (2006), 545–547  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    2. Danilenko A.I., “Mixing rank-one actions for infinite sums of finite groups”, Israel J. Math., 156 (2006), 341–358  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    3. El Abdalaoui E.H., Parreau F., Prikhod'ko A.A., “A new class of Ornstein transformations with singular spectrum”, Ann. Inst. H. Poincaré Probab. Statist., 42:6 (2006), 671–681  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    4. El Abdalaoui, El Houcein, “A new class of rank-one transformations with singular spectrum”, Ergodic Theory Dynam. Systems, 27:5 (2007), 1541–1555  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    5. Danilenko A.I., Silva C.E., “Mixing rank-one actions of locally compact Abelian groups”, Ann. Inst. H. Poincaré Probab. Statist., 43:4 (2007), 375–398  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    6. Danilenko A.I., “(C, F)-actions in ergodic theory”, In Memory of Alexander Reznikov, Progress in Mathematics, 265, 2008, 325–351  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    7. A. I. Bashtanov, “Property of almost independent images for ergodic transformations without partial rigidity”, Proc. Steklov Inst. Math., 271 (2010), 23–33  mathnet  crossref  mathscinet  isi  elib
    8. V. V. Ryzhikov, “Simple spectrum of the tensor product of powers of a mixing automorphism”, Trans. Moscow Math. Soc., 73 (2012), 183–191  mathnet  crossref  mathscinet  zmath  elib
    9. Danilenko A.I., “Mixing Actions of the Heisenberg Group”, Ergod. Theory Dyn. Syst., 34:4 (2014), 1142–1167  crossref  mathscinet  zmath  isi  scopus  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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