The Moebius function and continuous extensions of rotations

TL;DR

This paper proves that for most irrational rotations, the associated skew product systems with smooth functions satisfy Sarnak's conjecture, linking number theory and dynamical systems.

Contribution

It demonstrates that generic irrational rotations extended by smooth functions fulfill Sarnak's conjecture, advancing understanding of randomness in dynamical systems.

Findings

For generic irrational $oldsymbol{ heta}$, the skew product satisfies Sarnak's conjecture.

The extension involves smooth functions of class $C^{1+oldsymbol{ ext{delta}}}$.

The result connects number theory with dynamical systems and ergodic theory.

Abstract

Let $f\colon \mathbb{T}\to \mathbb{R}$ 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 $T_{c,f}\colon \mathbb{T}^2\to\mathbb{T}^2$ of the irrational rotation $Tx=x+\alpha$, given by $T_{c,f}(x,u)=(x+\alpha, u+cx+f(x))$ ($\bmod\ 1$) satisfies Sarnak's conjecture.