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

Okuyama, Yûsuke

doi:10.1090/S1088-4173-2014-00271-9

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

Conform. Geom. Dyn. 18 (2014), 217-228

DOI: https://doi.org/10.1090/S1088-4173-2014-00271-9

Published electronically: November 12, 2014

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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.

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