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Conformal Geometry and Dynamics

ISSN 1088-4173

 

 

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


Author: Yûsuke Okuyama
Journal: Conform. Geom. Dyn. 18 (2014), 217-228
MSC (2010): Primary 37F45
DOI: https://doi.org/10.1090/S1088-4173-2014-00271-9
Published electronically: November 12, 2014
MathSciNet review: 3276585
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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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Additional Information

Yûsuke Okuyama
Affiliation: Division of Mathematics, Kyoto Institute of Technology, Sakyo-ku, Kyoto 606-8585 Japan
Email: okuyama@kit.ac.jp

DOI: https://doi.org/10.1090/S1088-4173-2014-00271-9
Keywords: Holomorphic family, marked critical point, superattracting periodic point, equidistribution, activity current, bifurcation current
Received by editor(s): February 24, 2014
Received by editor(s) in revised form: July 11, 2014, and August 12, 2014
Published electronically: November 12, 2014
Article copyright: © Copyright 2014 American Mathematical Society