Generalized Perturbed Symmetry

LetwRdenote the reversal of the wordw, and letwE = w. We examine a generalization of Mendès Franceʹs ‘perturbed symmetry’. Specifically, we study a map of the form w∏1 ≤ i ≤ k − 1xiwei, i ∈ {E,R}, and the infinite word arising from iteration of this map. In particular, if thexiare fixed strings of identical length, the resulting infinite word isk-automatic, and we characterize precisely when such a word can be ultimately periodic. Finally, we apply our technique to prove a 1994 conjecture by Blanchard and Fabre.