Typical dynamics of Newton's method
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909116733063168 |
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| author | Dudák, Jan Steele, T. H. |
| author_facet | Dudák, Jan Steele, T. H. |
| contents | This article consists of two papers: $\textit{Typical dynamics of Newton's method}$ by Steele and $\textit{Erratum to "Typical dynamics of Newton's method"}$ by Dudák and Steele.
Let $C^1(M)$ be the space of continuously differentiable real-valued functions defined on $[-M,M]$. We show that for the typical element $f$ in $C^1(M)$, there exists a set $S \subseteq [-M,M]$, both residual and of full measure in $[-M,M]$, such that for any $x \in S$, the trajectory generated by Newton's method using $f$ and $x$ either diverges, converges to a root of $f$, or generates a Cantor set as its attractor. Whenever the Cantor set is the attractor, the dynamics on the attractor are described by a single type of adding machine, so that the dynamics on all of these attracting Cantor sets are topologically equivalent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14383 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Typical dynamics of Newton's method Dudák, Jan Steele, T. H. Dynamical Systems 37B20 This article consists of two papers: $\textit{Typical dynamics of Newton's method}$ by Steele and $\textit{Erratum to "Typical dynamics of Newton's method"}$ by Dudák and Steele. Let $C^1(M)$ be the space of continuously differentiable real-valued functions defined on $[-M,M]$. We show that for the typical element $f$ in $C^1(M)$, there exists a set $S \subseteq [-M,M]$, both residual and of full measure in $[-M,M]$, such that for any $x \in S$, the trajectory generated by Newton's method using $f$ and $x$ either diverges, converges to a root of $f$, or generates a Cantor set as its attractor. Whenever the Cantor set is the attractor, the dynamics on the attractor are described by a single type of adding machine, so that the dynamics on all of these attracting Cantor sets are topologically equivalent. |
| title | Typical dynamics of Newton's method |
| topic | Dynamical Systems 37B20 |
| url | https://arxiv.org/abs/2402.14383 |