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Conformal Geometry and Dynamics

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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
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

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