M\"{o}bius Disjointness for product flows of rigid dynamical systems and affine linear flows

TL;DR

This paper proves the M"obius Disjointness Conjecture for certain product flows involving affine linear flows on compact abelian groups and rigid dynamical systems, extending the conjecture's validity to new classes of systems.

Contribution

It establishes the M"obius Disjointness Conjecture for product flows of affine linear flows and rigid systems, including Lipschitz skew products on the torus, with new estimates for M"obius averages.

Findings

M"obius Disjointness holds for product flows of affine linear and rigid systems.

Logarithmic averages of M"obius are disjoint for these systems.

Application to Lipschitz skew products on the torus.

Abstract

We obtain that Sarnak's M\"{o}bius Disjointness Conjecture holds for product flows between affine linear flows on compact abelian groups of zero topological entropy and a class of rigid dynamical systems. To prove this, we show an estimate for the average value of the product of the M\"obius function and any polynomial phase over short intervals and arithmetic progressions simultaneously. In addition, we prove that the logarithmically averaged M\"obius Disjointness Conjecture holds for the product flow between any affine linear flow on a compact abelian group of zero entropy and any rigid dynamical system. As an application, we show that the logarithmically averaged M\"obius Disjointness Conjecture holds for every Lipschitz continuous skew product dynamical system on $\mathbb{T}^2$ over a rotation of the circle.