On the spectrum of $\alpha$-rigid maps

TL;DR

This paper constructs examples of $ ext{alpha}$-rigid transformations with spectrum containing Lebesgue components, answering a question about the spectral properties of such transformations and expanding understanding of their spectral types.

Contribution

It demonstrates the existence of $ ext{alpha}$-rigid maps with Lebesgue spectrum components and identifies a broad class of $ ext{alpha}$-rigid transformations with singular spectrum.

Findings

Existence of $ ext{alpha}$-rigid transformations with Lebesgue spectrum for $ ext{alpha} ext{ } extless= rac{1}{2}$

Construction of a large class of $ ext{alpha}$-rigid transformations with singular spectrum

Answer to a question posed by Klemes and Reinhold regarding spectral components

Abstract

It is shown that there exists an $\alpha$-rigid transformation with $\alpha$ less or equal to $\frac12$ whose spectrum has Lebesgue component. This answers the question raised by Klemes and Reinhold in \cite{Klemes-Reinhold}. We exhibit also a large class of $\alpha$-rigid transformations with singular spectrum.