The Moebius function and continuous extensions of rotations

Abstract

Let $f:\mathbb{T}\to \mathbb{R}$ be of class $C^{1+\delta }$ for some $\delta >0$ and let $c\in \mathbb{Z}$. We show that for a generic $\alpha \in \mathbb{R}$, the extension $T_{c,f}:{\mathbb{T}}^2\to {\mathbb{T}}^2$ of the irrational rotation $Tx=x+\alpha$, given by $T_{c,f}(x,u)=(x+\alpha ,u+cx+f(x))$ ($\mathrm{mod}1$) satisfies Sarnak's conjecture.

26 pages. The paper has been shortened. It covers now only compact group extensions of rotations, Rokhlin extensions have been removed. A remark on disjointness criterion for irrational rotation was added

References in corpus (2)

• On spectral disjointness of powers for rank-one transformations and Möbius orthogonality
• On the correlation of the Moebius function with random rank-one systems