Möbius disjointness along ergodic sequences for uniquely ergodic actions

Abstract

We show that there are an irrational rotation $Tx=x+\alpha$ on the circle $\mathbb{T}$ and a continuous $\phi :\mathbb{T}\to \mathbb{R}$ such that for each (continuous) uniquely ergodic flow $\mathcal{S}=(S_t{)}_{t\in \mathbb{R}}$ acting on a compact metric space $Y$, the automorphism $T_{\phi ,\mathcal{S}}$ acting on $(X\times Y,\mu \otimes \nu )$ by the formula $T_{\phi ,\mathcal{S}}(x,y)=(Tx,S_{\phi (x)}(y))$, where $\mu$ stands for Lebesgue measure on $\mathbb{T}$ and $\nu$ denotes the unique $\mathcal{S}$-invariant measure, has the property of asymptotically orthogonal powers. This gives a class of relatively weakly mixing extensions of irrational rotations for which Sarnak's conjecture on Möbius disjointness holds for all uniquely ergodic models of $T_{\phi ,\mathcal{S}}$. Moreover, we obtain a class of "random" ergodic sequences $(c_n)\subset \mathbb{Z}$ such that if $\mathbf{\mu }$ denotes the Möbius function, then $$\underset{N\to \infty }{\lim }\frac{1}{N}\sum _{n\le N}g(S_{c_n}y)\mathbf{\mu }(n)=0$$ for all (continuous) uniquely ergodic flows $\mathcal{S}$, all $g\in C(Y)$ and $y\in Y$.

38 pages