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On the Krieger-Araki-Woods ratio set. (English) Zbl 0836.28008

The ratio sets of the quasi-invariant ergodic measures with respect to the group of finite coordinate changes on the increasing products of finite alphabets are investigated. In more details so-called \(G\)-measures are treated, which are generalizations of well-known M. Keane’s \(g\)- measures. In the main result (Theorem 4.4) the necessary and (slightly different but close) sufficient conditions are given for a number \(r\) to belong to the ratio set of a \(G\)-measure.
The following elegant statement is worth to be pointed out.
Proposition 5.4. Let \(\nu\) be the Riesz product which is the \(\text{weak }^*\)-limit of the measures \(\prod^n_{k = 1} (1 + a_k 2 \pi \cos 3^kt) dt\). Suppose that \(a_k \searrow 0\) as \(k \to \infty\). Then the ratio set of \(\nu\) is contained in the set \(\{1,0, \infty\}\).

MSC:

28D15 General groups of measure-preserving transformations
47A35 Ergodic theory of linear operators
42A55 Lacunary series of trigonometric and other functions; Riesz products
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References:

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