Equidistribution of the zeros of higher order derivatives in polynomial dynamics
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909066783096832 |
|---|---|
| author | Okuyama, Yûsuke |
| author_facet | Okuyama, Yûsuke |
| contents | For every $m\in\mathbb{N}$, we establish the convergence of the averaged distributions of the zeros of the $m$-th order derivatives $(f^n)^{(m)}$ of the iterated polynomials $f^n$ of a polynomial $f\in\mathbb{C}[z]$ of degree $>1$ towards the harmonic measure of the filled-in Julia set of $f$ with pole at $\infty$ as $n\to+\infty$, when $f$ has no exceptional points in $\mathbb{C}$. This complements our former study on the zeros of $(f^n)^{(m)}-a$ for any value $a\in\mathbb{C}\setminus\{0\}$. The key in the proof is an approximation of the higher order derivatives of a solution of the Schröder or Abel functional equations for a meromorphic function on $\mathbb{C}$ with a locally uniform non-trivial error estimate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_03296 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Equidistribution of the zeros of higher order derivatives in polynomial dynamics Okuyama, Yûsuke Dynamical Systems Complex Variables For every $m\in\mathbb{N}$, we establish the convergence of the averaged distributions of the zeros of the $m$-th order derivatives $(f^n)^{(m)}$ of the iterated polynomials $f^n$ of a polynomial $f\in\mathbb{C}[z]$ of degree $>1$ towards the harmonic measure of the filled-in Julia set of $f$ with pole at $\infty$ as $n\to+\infty$, when $f$ has no exceptional points in $\mathbb{C}$. This complements our former study on the zeros of $(f^n)^{(m)}-a$ for any value $a\in\mathbb{C}\setminus\{0\}$. The key in the proof is an approximation of the higher order derivatives of a solution of the Schröder or Abel functional equations for a meromorphic function on $\mathbb{C}$ with a locally uniform non-trivial error estimate. |
| title | Equidistribution of the zeros of higher order derivatives in polynomial dynamics |
| topic | Dynamical Systems Complex Variables |
| url | https://arxiv.org/abs/2309.03296 |