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
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.
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Sbornik: Mathematics, 2001, 192:12, 1799–1828
MSC: Primary 37A25, 28D05; Secondary 54H20, 47A35
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
\paper Stochastic constructions of flows of rank~1
\jour Mat. Sb.
\jour Sb. Math.
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Ryzhikov V.V., “On the spectral and mixing properties of rank-one constructions in ergodic theory”, Dokl. Math., 74:1 (2006), 545–547
Danilenko A.I., “Mixing rank-one actions for infinite sums of finite groups”, Israel J. Math., 156 (2006), 341–358
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
El Abdalaoui, El Houcein, “A new class of rank-one transformations with singular spectrum”, Ergodic Theory Dynam. Systems, 27:5 (2007), 1541–1555
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
Danilenko A.I., “(C, F)-actions in ergodic theory”, In Memory of Alexander Reznikov, Progress in Mathematics, 265, 2008, 325–351
A. I. Bashtanov, “Property of almost independent images for ergodic transformations without partial rigidity”, Proc. Steklov Inst. Math., 271 (2010), 23–33
V. V. Ryzhikov, “Simple spectrum of the tensor product of powers of a mixing automorphism”, Trans. Moscow Math. Soc., 73 (2012), 183–191
Danilenko A.I., “Mixing Actions of the Heisenberg Group”, Ergod. Theory Dyn. Syst., 34:4 (2014), 1142–1167
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