2018/11/01 by Lemańczyk, Mariusz, Müllner, Clemens · 1 citation
#11B85 #37A05 #37B10 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1811.00594
Given a finite alphabet \mathbbA and a primitive substitution θ:\mathbbA→\mathbbAλ (of constant length λ), let (Xθ,S) denote the corresponding dynamical system, where Xθ 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 \mathbbAℤ. The main result of the paper is that all continuous observables in Xθ are orthogonal to any bounded, aperiodic, multiplicative function u:ℕ→ℂ, i.e. limN→∞\frac1N∑n≤ Nf(Snx)u(n)=0 for all f∈ C(Xθ) and x∈ Xθ. In particular, each primitive automatic sequence, that is, a sequence read by a primitive finite automaton, is orthogonal to any bounded, aperiodic, multiplicative function.