Automatic sequences are orthogonal to aperiodic multiplicative functions

Given a finite alphabet $\mathbb{A}$ and primitive substitution $\theta:\mathbb{A}\to\mathbb{A}^\lambda$ (of constant length $\lambda$), let $(X_\theta,S)$ denote the corresponding dynamical system, where $X_{\theta}$ is closure of orbit via left shift $S$ fixed point natural extension $\theta$ to self-map $\mathbb{A}^{\mathbb{Z}}$. The main result paper that all continuous observables in are orthogonal any bounded, aperiodic, multiplicative function $\mathbf{u}:\mathbb{N}\to\mathbb{C}$, i.e. \[ \lim_{N\to\infty}\frac1N\sum_{n\leq N}f(S^nx)\mathbf{u}(n)=0\] for $f\in C(X_{\theta})$ $x\in X_{\theta}$. In particular, each automatic sequence, is, sequence read by automaton, function.