Endomorphisms and automorphisms of minimal symbolic systems with sublinear complexity

We show that if the complexity difference function p(n+1)-p(n) of a infinite minimal shift is bounded, then the the automorphism group of the one-sided shift is finite, and the automorphism group of the corresponding two-sided shift “modulo the shift” is finite. For the minimal Sturmian and minimal substitution shifts, the bounds can be explicitly computed, and for linearly recurrent shifts, the bound can be expressed as a function of the linear recurrence constant. We also show that any endomorphism of a linearly recurrent shift is a root of a power of the shift map.