Hidden automatic sequences

Abstract

An automatic sequence is a letter-to-letter coding of a fixed point of a uniform morphism. More generally, morphic sequences are letter-to-letter codings of fixed points of arbitrary morphisms. There are many examples where an, a priori, morphic sequence with a non-uniform morphism happens to be an automatic sequence. An example is the Lysënok morphism a → aca, b → d, c → b, d → c, the fixed point of which is also a 2-automatic sequence. Such an identification is useful for describing the dynamical systems generated by the fixed point. We give several ways to uncover such hidden automatic sequences, and present many examples. We focus in particular on morphisms associated with Grigorchuk groups.

How to prove that a sequence is not automatic

“…For example, the fixed point of the morphism defined on {a, b, c, d} by a → aca, b → d, c → aba, d → c (see [24,Theorem 4.1]) is not automatic. Namely, as noted in [17], the matrix of this morphism is primitive and its characteristic polynomial, which is equal to x 4 − 2x 3 − 2x 2 − x + 2, clearly has no rational root.…”

“…The lexicographically least sequence in the orbit closure of a q-automatic sequence is also q-automatic. Thus we get the following result: Theorem 27 ( [17]) Let x = (x n ) n≥0 be a sequence over some alphabet A. Let A ′ be a proper subset of A.…”

“…Remark 29 One of the referees of the paper [17] has just noted that Theorem 27 can also be deduced from Theorem A in [38]. Furthermore this approach does not need that A ′ be a proper subset of A.…”

“…Example 13 Theorem 12 was used in [17] to prove the non-automaticity of fixed points of morphisms related to Grigorchuk-like groups. For example, the fixed point of the morphism defined on {a, b, c, d} by a → aca, b → d, c → aba, d → c (see [24,Theorem 4.1]) is not automatic.…”

“…Example 28 Theorem 27 was used in [17] to prove that the two fixed points of morphisms respectively given in [26, Theorem 2.9] and [23,Theorem 4.5] are not q-automatic for any q ≥ 2, namely…”