M\"obius 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 $\varphi\colon\mathbb{T}\to\mathbb{R}$ 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 $T_{\varphi,\mathcal{S}}$ acting on $(X\times Y,\mu\otimes\nu)$ by the formula $T_{\varphi,\mathcal{S}}(x,y)=(Tx,S_{\varphi(x)}(y))$, 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 M\"obius disjointness holds for all uniquely ergodic models of $T_{\varphi,\mathcal{S}}$. Moreover, we obtain a class of "random" ergodic sequences $(c_n)\subset\mathbb{Z}$ such that if $\boldsymbol{\mu}$ denotes the M\"obius function, then $$ \lim_{N\to\infty}\frac1N\sum_{n\leq N}g(S_{c_n}y)\boldsymbol{\mu}(n)=0 $$ for all (continuous) uniquely ergodic flows $\mathcal{S}$, all $g\in C(Y)$ and $y\in Y$.