Computing automorphism groups of shifts using atypical equivalence classes

We study the automorphism group of an infinite minimal shift (X,σ) such that the complexity difference function, p(n+1)− p(n), is bounded. We give some new bounds on Aut(X,σ)/⟨σ⟩ and also study the one-sided case. For a class of Toeplitz shifts, including the class of shifts defined by constant-length primitive substitutions with a coincidence, and with height one, we show that the two-sided automorphism group is a cyclic group. We next focus on shifts generated by primitive constant-length substitutions. For these shifts, we give an algorithm that computes their two-sided automorphism group. Finally we show that with the same techniques, we are able to compute the set of conjugacies between two such shifts