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Value distribution of derivatives in polynomial dynamics

Published online by Cambridge University Press:  05 January 2021

YÛSUKE OKUYAMA
Affiliation:
Division of Mathematics, Kyoto Institute of Technology, Sakyo-ku, Kyoto606-8585, Japan (e-mail: [email protected])
GABRIEL VIGNY*
Affiliation:
LAMFA, UPJV, 33 rue Saint-Leu, 80039Amiens Cedex 1, France

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.

Type
Original Article
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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