Automatic sequences are orthogonal to aperiodic multiplicative functions

Abstract

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 $\mathbf{u}:\mathbb{N}\to\mathbb{C}$, i.e. \[ \lim_{N\to\infty}\frac1N\sum_{n\leq N}f(S^nx)\mathbf{u}(n)=0\] for 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.