Value distribution of derivatives in polynomial dynamics
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 ha...
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| Published in: | Ergodic theory and dynamical systems Vol. 41; no. 12; pp. 3780 - 3806 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Cambridge, UK
Cambridge University Press
01.12.2021
Cambridge University Press (CUP) |
| Subjects: | |
| ISSN: | 0143-3857, 1469-4417 |
| Online Access: | Get full text |
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| Summary: | 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. |
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| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0143-3857 1469-4417 |
| DOI: | 10.1017/etds.2020.125 |