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Value distribution of derivatives in polynomial dynamics

Part of: Arithmetic and non-Archimedean dynamical systems Holomorphic mappings and correspondences Complex dynamical systems

Published online by Cambridge University Press:  05 January 2021

YÛSUKE OKUYAMA  and
GABRIEL VIGNY
YÛSUKE OKUYAMA
Affiliation:
Division of Mathematics, Kyoto Institute of Technology, Sakyo-ku, Kyoto 606-8585, Japan (e-mail: okuyama@kit.ac.jp)
GABRIEL VIGNY*
Affiliation:
LAMFA, UPJV, 33 rue Saint-Leu, 80039 Amiens Cedex 1, France
*
e-mail: gabriel.vigny@u-picardie.fr
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Abstract

For every $m\in \mathbb {N}$, we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in $\mathbb {C}\setminus \{0\}$ under the $m$th order derivatives of the iterates of a polynomials $f\in \mathbb {C}[z]$ of degree $d>1$ towards the harmonic measure of the filled-in Julia set of f with pole at $\infty $. We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field k for a sequence of effective divisors on $\mathbb {P}^1(\overline {k})$ having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of $\mathbb {C}^2$ has a given eigenvalue.

Keywords

complex dynamicsHénon maphigher derivativenon-archimedean dynamicsvalue distribution

MSC classification

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions
Secondary: 37P30: Height functions; Green functions; invariant measures 32H50: Iteration problems

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 12 , December 2021 , pp. 3780 - 3806
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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