Substitutive Systems and a Finitary Version of Cobham’s Theorem

Abstract

We study substitutive systems generated by nonprimitive substitutions and show that transitive subsystems of substitutive systems are substitutive. As an application we obtain a complete characterisation of the sets of words that can appear as common factors of two automatic sequences defined over multiplicatively independent bases. This generalises the famous theorem of Cobham.

(Logarithmic) densities for automatic sequences along primes and squares

“…A quite similar result can be found in[4]. They show that the minimal components of a subshift corresponding to a k-automatic sequence are given by primitive and prolongable k ℓ -automatic sequences.…”

“…Proof. This is in its essence only a reformulation of Proposition 2.2 in [4]. However, we provide nevertheless a proof as it highlights important ideas for the proof of Proposition A.5.…”

“…Example 3.1. We find (37) 2 = (1, 0, 0, 1, 0, 1), (37) 4 2 = (0, 1, 0, 1) and [(1, 0, 1, 1, 0)] 2 = 22. Definition 3.2.…”

Automaticity of uniformly recurrent substitutive sequences

“…k-automatic) (this follows e.g. from (the proof of) [6, Lemma 2.10]; note that in [6] a two-sided substitutive sequence is defined as a two-sided sequence x such that the one-sided sequences (x n ) n 0 and (x n ) n<0 are substitutive). We recall that for all n 1, a sequence x is k-automatic if and only if it is k n -automatic [3, Theorem 6.6.4], and that all periodic sequences are k-automatic with respect to any k 2 [3, Thm.…”

“…For each n 0, there exist y n ∈ X φ and c n ∈ Z such that y = T cn ( φfn (y n )) (see e.g. [6,Lem. 2.11]).…”