How extensions impact the factor complexity of morphic images

A classical result states that, when applying a non-erasing morphism to a word, the factor complexity of the image is bounded by the initial factor complexity multiplied by a constant depending on the morphism. However, when looking at the proof, one easily sees that this can be vastly improved. This is the object of this talk in which we exhibit a large family of words for which, when applying a morphism, the factor complexity grows at most by an additive constant.