Spectral types of skewed Bernoulli shift

Abstract

For the transformation T : x bar right arrow kx (mod 1) for k greater than or equal to 2, it is proved that a real-valued function f(x) of modulus 1 is not a multiplicative coboundary if the discontinuities 0 < x(1) < ... < x(n) less than or equal to 1 of f(x) are k-adic points and x(1) greater than or equal to 1/k. It is also proved that the weakly mixing skew product transformations arising from Bernoulli shifts have Lebesgue spectrum.