On automorphism groups of Toeplitz subshifts

TL;DR

This paper investigates the structure of automorphism groups of Toeplitz subshifts, revealing their abelian nature, conditions for finite generation, and constructing examples with complex automorphism groups across different complexity regimes.

Contribution

It characterizes automorphism groups of Toeplitz subshifts under various complexity conditions and constructs examples illustrating the limits of these characterizations.

Findings

Automorphism groups are abelian and finitely generated torsion subgroups are finite and cyclic.

In non-superlinear complexity, automorphism groups are generated by roots of the shift, modulo a finite cyclic group.

Examples show automorphism groups can be arbitrarily complex even with low complexity.

Abstract

In this article we study automorphisms of Toeplitz subshifts. Such groups are abelian and any finitely generated torsion subgroup is finite and cyclic. When the complexity is non superlinear, we prove that the automorphism group is, modulo a finite cyclic group, generated by a unique root of the shift. In the subquadratic complexity case, we show that the automorphism group modulo the torsion is generated by the roots of the shift map and that the result of the non superlinear case is optimal. Namely, for any $\varepsilon > 0$ we construct examples of minimal Toeplitz subshifts with complexity bounded by $C n^{1+\epsilon}$ whose automorphism groups are not finitely generated. Finally, we observe the coalescence and the automorphism group give no restriction on the complexity since we provide a family of coalescent Toeplitz subshifts with positive entropy such that their automorphism groups are arbitrary finitely generated infinite abelian groups with cyclic torsion subgroup (eventually restricted to powers of the shift).