Newton's method applied to rational functions: Fixed points and Julia sets
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914304591134720 |
|---|---|
| author | Nayak, Tarakanta Pal, Soumen Phogat, Pooja |
| author_facet | Nayak, Tarakanta Pal, Soumen Phogat, Pooja |
| contents | For a rational function $R$, let $N_R(z)=z-\frac{R(z)}{R'(z)}.$ Any such $N_R$ is referred to as a Newton map. We determine all the rational functions $R$ for which $N_R$ has exactly two attracting fixed points, one of which is an exceptional point. Further, if all the repelling fixed points of any such Newton map are with multiplier $2$, or the multiplier of the non-exceptional attracting fixed point is at most $\frac{4}{5}$, then its Julia set is shown to be connected. If a polynomial $p$ has exactly two roots, is unicritical but not a monomial, or $p(z)=z(z^n+a)$ for some $a \in \mathbb{C}$ and $n \geq 1$, then we have proved that the Julia set of $N_{\frac{1}{p}}$ is totally disconnected. For the McMullen map $f_λ(z)=z^m - \fracλ{z^n}$, $λ\in \mathbb{C}\setminus \{0\}$ and $m,n \geq 1$, we have proved that the Julia set of $N_{f_λ}$ is connected and is invariant under rotations about the origin of order $m+n$. All the connected Julia sets mentioned above are found to be locally connected. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_08498 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Newton's method applied to rational functions: Fixed points and Julia sets Nayak, Tarakanta Pal, Soumen Phogat, Pooja Dynamical Systems 37F10, 65H05 For a rational function $R$, let $N_R(z)=z-\frac{R(z)}{R'(z)}.$ Any such $N_R$ is referred to as a Newton map. We determine all the rational functions $R$ for which $N_R$ has exactly two attracting fixed points, one of which is an exceptional point. Further, if all the repelling fixed points of any such Newton map are with multiplier $2$, or the multiplier of the non-exceptional attracting fixed point is at most $\frac{4}{5}$, then its Julia set is shown to be connected. If a polynomial $p$ has exactly two roots, is unicritical but not a monomial, or $p(z)=z(z^n+a)$ for some $a \in \mathbb{C}$ and $n \geq 1$, then we have proved that the Julia set of $N_{\frac{1}{p}}$ is totally disconnected. For the McMullen map $f_λ(z)=z^m - \fracλ{z^n}$, $λ\in \mathbb{C}\setminus \{0\}$ and $m,n \geq 1$, we have proved that the Julia set of $N_{f_λ}$ is connected and is invariant under rotations about the origin of order $m+n$. All the connected Julia sets mentioned above are found to be locally connected. |
| title | Newton's method applied to rational functions: Fixed points and Julia sets |
| topic | Dynamical Systems 37F10, 65H05 |
| url | https://arxiv.org/abs/2503.08498 |