Title: On automorphism groups of Toeplitz subshifts,
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).