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On the disjointness problem for Gaussian automorphisms. (English) Zbl 0923.28007

Let \((T_\sigma,X,{\mathcal B},\mu_\sigma)\) be a Gaussian dynamical system with (symmetric) spectral measure \(\sigma=\sigma_{f_0}\), where \(f_0\) denotes the 0-coordinate projection. The authors show that any two Gaussian dynamical systems \((T_\sigma,X,{\mathcal B},\mu_\sigma)\) and \((T_\tau,X,{\mathcal B},\mu_\tau)\) for which \(\sigma\) and \(\tau\) are concentrated on independent sets are either spectrally disjoint or have a common factor. Thus in this case, disjointness as introduced by H. Furstenberg [Math. Systems Theory 1, 1-49 (1967; Zbl 0146.28502)] is equivalent to \(\sigma\) and \(\tau\) being mutually singular. As an application of the approach used in this paper, the authors were able to construct non-rigid Gaussian automorphisms which are spectrally determined: spectral isomorphism implies measure theoretic isomorphism.

MSC:

28D05 Measure-preserving transformations
43A05 Measures on groups and semigroups, etc.

Citations:

Zbl 0146.28502
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