On a degenerating limit theorem of DeMarco--Faber

One of our aims is to complement the proof of DeMarco--Faber's degenerating limit theorem for the family of the unique maximal entropy measures parametrized by a punctured open disk associated to a meromorphic family of rational functions on the complex projective line degenerating at the puncture. This complementation is done by our main result, which rectifies a key computation in their argument. We also establish and use a direct and explicit translation from degenerating complex dynamics into quantized Berkovich dynamics, instead of using DeMarco--Faber's more conceptual transfer principle between those dynamics.