Automatic sequences are orthogonal to aperiodic multiplicative functions

Given a finite alphabet $ \mathbb{A} $ and a 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 the closure of the orbit via the left shift $ S $ of a fixed point of the natural extension of $ \theta $ to a self-map of $ \mathbb{A}^{ {\mathbb{Z}}} $. The main result of the paper is that all continuous observables in $ X_\theta $ are orthogonal to any bounded, aperiodic, multiplicative function $ \boldsymbol{u}: {\mathbb{N}}\to {\mathbb{C}} $, i.e.

$ \lim\limits_{N\to\infty}\frac1N\sum\limits_{n\leq N}f(S^nx) \boldsymbol{u}(n) = 0 $

for all $ f\in C(X_\theta) $ and $ x\in X_\theta $. In particular, each primitive automatic sequence, that is, a sequence read by a primitive finite automaton, is orthogonal to any bounded, aperiodic, multiplicative function.