Nonsingular $\alpha$-rigid maps: Short proof

TL;DR

This paper proves that for any alpha in [0, 1/2], there exists an alpha-rigid transformation with Lebesgue spectrum, answering a question about the spectral properties of such transformations.

Contribution

It introduces a short proof demonstrating the existence of alpha-rigid maps with Lebesgue spectrum for all alpha in [0, 1/2], resolving an open question.

Findings

Existence of alpha-rigid transformations with Lebesgue spectrum for all alpha in [0, 1/2]

Application of correspondence between weak limits of powers and skew products

Provides a concise proof of a previously open problem

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.