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Spectral properties of skew-product diffeomorphisms of tori. (English) Zbl 0904.28014

The author studies generalizations of Anzai skew-products of the form: \[ T(x,y)= (x+ \alpha,y+ Mx+ F(x)),\quad T:\mathbb{T}^d\times \mathbb{T}^{d'}\to \mathbb{T}^d\times \mathbb{T}^{d'}, \] defined on the product of \(d\) and \(d'\)-dimensional tori, \(\alpha= (\alpha_1,\alpha_2,\dots, \alpha_d)\), \(M\) is a \(d'\times d\)-dimensional matrix with integer entries and \(F: \mathbb{R}^d\to \mathbb{R}^{d'}\) is a \(\mathbb{Z}^d\) measurable mapping. When product Lebesgue measure is used, \(T\) is an invertible measure-preserving transformation. The mapping \(\phi(x)= Mx+ F(x)\) is called a cocycle, and is said to be weakly mixing if there are no other eigenvalues besides those arising from the discrete component in the first coordinate.
The author [Stud. Math. 115, No. 3, 241-250 (1995; Zbl 0844.28007); Colloq. Math. 70, No. 1, 73-78 (1996; Zbl 0845.28009)] and others have studied \(T\) in the case \(d= d'=1\). Here, analogous results are established in the general case. These results also extend recent results of Fraczek (preprint) where different methods are used. Specifically, it is shown that weakly mixing cocycles form a \(G_\delta\) set in certain sets of cocycles endowed with a suitable topology. In addition, it is shown that under suitable conditions [similar to the authors one-dimensional result in Bull. Lond. Math. Soc. 29, No. 2, 195-199 (1997; Zbl 0865.28012)] \(T\) has a countable Lebesgue spectrum on the orthocomplement of the space \(L^2(dx)\).

MSC:

28D05 Measure-preserving transformations
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