Approximation and the Spectral Multiplicity of Finite Skew Products

On Spectral Properties of Skew Products over Irrational Rotations

“…Then, if m is a positive integer such that 0 + 23 < 2m/(ra + l), the spectral multiplicity of the operator V T>(j) cannot exceed m. Remark. Goodson [3] proved this theorem for a more general method of approximation of T. We have restricted our attention to cyclic KS approximations purely for convenience.…”

“…THEOREM 2. [3,8,9]. Suppose that the measure-preserving automorphism T admits a cyclic KS approximation with speed 6/n and the function 0 : X -* Z N is such that the partition n = ($~1(z): zeZ N } is approximated by the sequence of partitions £(ri) of Definition 1 with speed d/n.…”

“…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 V at p we depend on results of Veech [10].…”