Bratteli-Vershik adic representations of some one-sided substitution subshifts

We study one-sided substitution subshifts, and how they can be respresented using BratteliVershik systems. In particular we focus on minimal recognizable substitutions such that the generated one-sided substitution subshift contains only one non-shift-invertible element (branch point), and we call these substitutions quasi-invertible. We characterise these substitutions, and show that if the substitution is left proper, then the subshift is equal to another substitution subshift where the branch point is the substitution fixed point. We use these results to prove that any quasi-invertible substitution subshift has either a Bratteli-Vershik representation, or a ‘pinched’ such representation.