Toeplitz subshift whose automorphism group is not finitely generated

Abstract

We compute an explicit presentation of the (topological) automorphism group of a particular Toeplitz subshift with subquadratic complexity. The automorphism group is a non-finitely generated subgroup of rational numbers, or alternatively the 5-adic integers, under addition, the shift map corresponding to the rational number 1. The group is ( (5/2) i | i ∈ N , +) ≤ (Q, +).

On automorphism groups of Toeplitz subshifts

“…These results follow from the study of subgroups of odometers and allow us to recover Theorem 12 from [7], where the authors consider a family of Toeplitz subshifts generated by substitutions. The condition on Theorem 3.2 is true for many Toeplitz in the family called (p, q)-Toeplitz [5], including the example in [19]. Then, using this class of Toeplitz subshifts we extend the example given by Salo in [19], proving that the condition of non-superlinear complexity of [9,13] cannot be relaxed.…”

“…The condition on Theorem 3.2 is true for many Toeplitz in the family called (p, q)-Toeplitz [5], including the example in [19]. Then, using this class of Toeplitz subshifts we extend the example given by Salo in [19], proving that the condition of non-superlinear complexity of [9,13] cannot be relaxed. More precisely, in Theorem 4.1 we prove that for any ε > 0 there exists a Toeplitz subshift in the aforementioned family such that the complexity verifies p X (n) ≤ Cn 1+ε for all n ∈ N and whose automorphism group is not finitely generated.…”

“…In this class one can get complexities that are arbitrarily close to non-superlinear (but always superlinear). This class was implicitly used by Salo [19] to give an example of a subshift of complexity p X (n) ≤ Cn 1.757 with a non-finitely generated automorphism group. We simplify and extend his result in the generality of Theorem 4.1.…”

“…These results showed that the automorphism group seems to gain in constraints when the complexity goes down but this is not always true. Interestingly, in [19] the author provided a Toeplitz subshift with complexity p X (n) ≤ Cn 1.757 , whose automorphism group is not finitely generated.…”

Cellular Automata and Discrete Complex Systems

“…An explicit example of such a subshift is given in [Sal14a]. Unlike in the case of linear complexity, there is no uniform bound on the k, that is, for a transitive subshift X of subquadratic complexity, if k(f ) is the least positive integer such that f k(f ) ∈ {σ n | n ∈ Z}, then k : End(X) → N may be unbounded.…”