Automatic sequences are orthogonal to aperiodic multiplicative functions

Abstract

Given a finite alphabet A and a primitive substitution θ : A → A λ (of constant length λ), let (X θ , S) denote the corresponding dynamical system, where X θ is the closure of the orbit via the left shift S of a fixed point of the natural extension of θ to a self-map of A Z. The main result of the paper is that all continuous observables in X θ are orthogonal to any bounded, aperiodic, multiplicative function u : N → C, i.e. lim N →∞ 1 N n≤N f (S n x)u(n) = 0 for all f ∈ C(X θ) and x ∈ X θ. In particular, each primitive automatic sequence, that is, a sequence read by a primitive finite automaton, is orthogonal to any bounded, aperiodic, multiplicative function.

Automorphisms of automatic shifts

“…In particular, if θ has column number c and c = Id with c minimal, then the factor defined above is an almost everywhere one-to-one extension of its maximal equicontinuous factor, and this is a convenient topological factor for a dynamical system to have; see, for example, [1], [17], or [22] for uses. It is not always the case that such a factor exists [12].…”

“…Otherwise, we would have a contradiction to the minimality of the column number c. This implies that θ is well defined on X and we denote it byθ. It follows quite easily thatθ is primitive (see [17,Proposition 6.3]).…”

“…Proof. This is [17,Equation (37)], but we include the verification here for completeness. We need to show that for any b ∈ A there exists M ∈ X with b ∈ M .…”

“…However, there is still one more condition that τ has to satisfy. For this remaining condition we use a generalization of Lemańczyk and Mentzen's idea for bijective substitutions to assign to a general θ a group of permutations called G. The following argument was presented by the first author and Lemańczyk in [17], and we follow their notation. Let us fix an arbitrary M ∈ X .…”

On multiplicative automatic sequences

“…In particular, the question of classifying automatic multiplicative sequences has been investigated by a number of authors, including . The interplay between multiplicative and automatic sequences is studied also in , among others.…”