Equidistribution of Rational Functions Having a Superattracting Periodic Point towards the Activity Current and the Bifurcation Current

We establish an approximation of the activity current Tc in the parameter space of a holomorphic family f of rational functions having a marked critical point c by parameters for which c is periodic under f , 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 current Tf of the holomorphic family f . The proof is based on a dynamical counterpart of this approximation.