M\"obius disjointness for non-uniquely ergodic skew products

TL;DR

This paper proves that certain smooth skew product maps on the torus, which preserve a measurable section, are disjoint from the Möbius sequence, extending to non-uniquely ergodic cases and revealing new disjointness properties.

Contribution

It establishes Möbius disjointness for a class of non-uniquely ergodic skew products on the torus, generalizing previous results to less restrictive ergodic conditions.

Findings

Skew product maps with a measurable section are Möbius disjoint.

Non-uniquely ergodic skew products have finite index factors disjoint from Möbius.

The result applies to $C^ au$ smooth maps with $ au > 2$.

Abstract

For $\tau>2$, let $T$ be a $C^\tau$ skew product map of the form $(x+\alpha,y+h(x))$ on $\mathbb T^2$ over a rotation of the circle. We show that if $T$ preserves a measurable section, then it is disjoint to the M\"{o}bius sequence. This in particular implies that any non-uniquely ergodic $C^\tau$ skew product map on $\mathbb T^2$ has a finite index factor that is disjoint to the M\"{o}bius sequence.