Anzai skew products with Lebesgue component of infinite multiplicity

摘要

Let f be a 1-periodic C-1-function whose Fourier coefficients satisfy the condition Sigma(n)(3) (f) over cap(n)(2) < infinity. For every alpha is an element of RQ and m is an element of Z{0}, we consider the Anzai skew product T(x, y) = (x + alpha, y + mx + f(x)) acting on the 2-torus. It is shown that T has infinite Lebesgue spectrum on the orthocomplement L(2)(dw)(perpendicular to) of the space of functions depending only on the first variable. This extends some earlier results of Kushnirenko, Choe, Lemanczyk, Rudolph, and the author.