ANZAI SKEW PRODUCTS WITH LEBESGUE COMPONENT OF INFINITE MULTIPLICITY

Abstract

Let f be a 1-periodic C¹-function whose Fourier coefficients satisfy the condition [sum ]_n[mid ]n[mid ]³[mid ] fˆ(n[mid ]²<∞. For every α∈R\Q and m∈Z\{0}, we consider the Anzai skew product T(x, y) = (x+α, y+mx+f(x)) acting on the 2-torus. It is shown that T has infinite Lebesgue spectrum on the orthocomplement L²(dx)^⊥ of the space of functions depending only on the first variable. This extends some earlier results of Kushnirenko, Choe, Lemańczyk, Rudolph, and the author.