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

TL;DR

This paper proves that the activity current of a critical point in a family of rational functions can be approximated by parameters where the critical point is superattracting, refining existing theorems on bifurcation currents.

Contribution

It generalizes previous results by approximating activity and bifurcation currents using superattracting periodic points in rational families.

Findings

Approximation of activity current by superattracting parameters

Refinement of bifurcation current approximation theorems

Extension of Dujardin--Favre and Bassanelli--Berteloot results

Abstract

We establish an approximation of the activity current $T_c$ 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 $T_f$ of the holomorphic family $f$. The proof is based on a dynamical counterpart of this approximation.