Möbius disjointness along ergodic sequences for uniquely ergodic actions

Abstract

We show that there is an irrational rotation $Tx=x+\unicode[STIX]{x1D6FC}$ on the circle $\mathbb{T}$ and a continuous $\unicode[STIX]{x1D711}:\mathbb{T}\rightarrow \mathbb{R}$ 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 $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}$ acting on $(X\times Y,\unicode[STIX]{x1D707}\otimes \unicode[STIX]{x1D708})$ by the formula $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}(x,y)=(Tx,S_{\unicode[STIX]{x1D711}(x)}(y))$, where $\unicode[STIX]{x1D707}$ stands for the Lebesgue measure on $\mathbb{T}$ and $\unicode[STIX]{x1D708}$ 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 the Möbius disjointness holds for all uniquely ergodic models of $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}$. Moreover, we obtain a class of ‘random’ ergodic sequences $(c_{n})\subset \mathbb{Z}$ such that if $\boldsymbol{\unicode[STIX]{x1D707}}$ denotes the Möbius function, then $$\begin{eqnarray}\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n\leq N}g(S_{c_{n}}y)\boldsymbol{\unicode[STIX]{x1D707}}(n)=0\end{eqnarray}$$ for all (continuous) uniquely ergodic flows ${\mathcal{S}}$, all $g\in C(Y)$ and $y\in Y$.