Equidistribution of rational functions having a superattracting periodic point towards the activity current and the bifurcation current

Abstract

We establish an approximation of the activity currentTcT_cin the parameter space of a holomorphic familyffof rational functions having a marked critical pointccby parameters for whichccis periodic underff, i.e., is a superattracting periodic point. This partly generalizes a Dujardin–Favre theorem for rational functions having preperiodic points, and refines a Bassanelli–Berteloot theorem on a similar approximation of the bifurcation currentTfT_fof the holomorphic familyff. The proof is based on a dynamical counterpart of this approximation.