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Some examples of adic transformations and automorphisms of substitutions. (English. Russian original) Zbl 1033.28502

Sel. Math. Sov. 11, No. 1, 83-104 (1992); translation from Dep. # 7384-88 VINITI (1988).
Adic transformations on the space of infinite paths of a Bratteli diagram were introduced by A. M. Vershik [J. Sov. Math. 28, 667–674 (1985); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 115, 72–82 (1982; Zbl 0505.47006)]. He proved that any ergodic automorphism on a Lebesgue space is metrically isomorphic to an adic transformation. For stationary adic transformations there is a connection with substitutions [N. Livshits, Math. Notes 44, 920–925 (1988); translation from Mat. Zametki 44, No. 6, 785–793 (1988; Zbl 0668.28005)]. B. Host [Ergodic Theory Dyn. Syst. 6, 529–540 (1986; Zbl 0625.28011)] has given some sufficient conditions for a substitution to have discrete spectrum and to be weakly mixing. In this paper the author proves an analogous theorem for adic transformations and studies various examples illustrating different situations which may arise.

MSC:

28D05 Measure-preserving transformations
37A05 Dynamical aspects of measure-preserving transformations
37A25 Ergodicity, mixing, rates of mixing
54H20 Topological dynamics (MSC2010)