On Spectral Properties of Skew Products over Irrational Rotations

In the process of obtaining these results we make some observations, which may be of more general interest, concerning the problems of singularity and continuity of the spectral types of skew products. See §2 and §4. Our approach is based on the method of approximations as formulated by Katok and Stepin [4, 5] and later developed and modified by a number of authors [1, 3, 8, 9]. For the continuity of the spectrum of Vat p we depend on results of Veech [10]. We present these results, in adapted form, in §4 and §5. In Veech's paper the reader may find a detailed study of the problem of ergodicity of Ta> p and related number theoretic questions. Note that Katok and Stepin [4, 5] have already obtained the results of Theorem 1 and its corollary under certain more restrictive conditions on a and ft. Further, in [4], Katok and Stepin state that when a and 0 are rationally independent, the spectral multiplicity of Ta> fi does not exceed two. See also Oseledec [7] for the result that, in the special case when /? is £, the operator Va> fi has continuous spectrum with spectral multiplicity not exceeding two, for all irrational a.