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Most monothetic extensions are rank-1. (English) Zbl 0833.28009
Let \(T : (X, {\mathcal B}, \mu) \to (X, {\mathcal B}, \mu)\) be an ergodic automorphism defined on a standard Borel probability space. A measurable function \(\varphi : X \to G\), where \(G\) is a compact metrizable abelian group is called a cocyle. The automorphism \[ T_\varphi : X \times G \to X \times G; \quad T_\varphi (x,g) = \bigl( Tx, \varphi (x) + g\bigr) \] is a \(G\)-extension of \(T\).
The ergodic properties of such maps have been studied extensively since “skew products” were first introduced by H. Anzai [Osaka Math. J. 3, 83-89 (1951; Zbl 0043.112)]. For example, in Invent. Math. 72, 299-314 (1983; Zbl 0519.28008), E. A. Robinson jun. showed that “typically” \(G\)-extensions have simple spectrum.
The main result of this paper is to show that if \(G\) is a monothetic group and \(T\) admits a cyclic approximation with speed \(o(1/n)\) [in the sense of A. B. Katok and A. M. Stepin, Usp. Mat. Nauk 22, No. 5(137), 81-106 (1967) (Russian), engl. translation in Russ. Math. Surv. 22, No. 5, 77-102 (1967; Zbl 0172.072)] then most \(G\)-extensions are rank–1. Related results concerning “Anzai” cocycles are given, where \(X = G = S^1\), the unit circle, and \(Tz = e^{2 \pi i \alpha} z\), an irrational rotation.

MSC:
28D05 Measure-preserving transformations
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