Nonsingular $\alpha$-rigid maps: Short proof

Abstract

It is shown that for every $\alpha$, where $\alpha\in [0, 1/2]$, there exists an $\alpha$-rigid transformation whose spectrum has Lebesgue component. This answers the question posed by Klemes and Reinhold in [7]. We apply a certain correspondence between weak limits of powers of a transformation and its skew products.