Möbius disjointness for non-uniquely ergodic skew products

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 Möbius 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 Möbius sequence.

The results in this manuscript are superseded by the more recent joint preprint arXiv:1707.06345 with Wen Huang and Xiangdong Ye. This version will be kept as a permanent preprint. 30 pages

References in corpus (1)

• An averaged form of Chowla's conjecture

Cited by in corpus (3)

• Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics
• Topological Wiener-Wintner ergodic theorem with polynomial weights
• Möbius disjointness for $C^{1+\epsilon }$ skew products