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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.14383 |
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Table of 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.