Value distribution of derivatives in polynomial dynamics
| Autor: | Yûsuke Okuyama, Gabriel Vigny |
|---|---|
| Přispěvatelé: | Kyoto Institute of Technology |
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Polynomial
Pure mathematics Dirac measure Mathematics::Dynamical Systems Distribution (number theory) General Mathematics Mathematics::Number Theory 01 natural sciences symbols.namesake 0502 economics and business 0101 mathematics [MATH]Mathematics [math] Mathematics - Dynamical Systems Mathematics Sequence Mathematics - Number Theory Mathematics - Complex Variables Applied Mathematics 010102 general mathematics 05 social sciences Algebraic number field Harmonic measure Julia set Projective line symbols 050203 business & management |
| Zdroj: | Ergodic Theory and Dynamical Systems Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2021, 41 (12), pp.3780-3806. ⟨10.1017/etds.2020.125⟩ |
| ISSN: | 0143-3857 1469-4417 |
| DOI: | 10.1017/etds.2020.125⟩ |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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