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Typical dynamics of Newton's method

Jan Dudák, T. H. Steele

Abstract

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.

Typical dynamics of Newton's method

Abstract

This article consists of two papers: by Steele and by Dudák and Steele. Let be the space of continuously differentiable real-valued functions defined on . We show that for the typical element in , there exists a set , both residual and of full measure in , such that for any , the trajectory generated by Newton's method using and either diverges, converges to a root of , 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.
Paper Structure (3 sections, 11 theorems, 34 equations)

This paper contains 3 sections, 11 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. The results

Key Result

Theorem 2.2

For every $\alpha \in (\mathbb{N} \setminus \{ 1 \})^\mathbb{N}$ and every prime number $p$, let $M_\alpha (p)$ be the element of $\mathbb{N}_0 \cup \{ \infty \}$ defined by Then for any $\alpha , \beta \in (\mathbb{N} \setminus \{ 1 \})^\mathbb{N}$, the adding machines $(\Delta_\alpha, \varphi_\alpha)$, $(\Delta_\beta, \varphi_\beta)$ are topologically conjugate if and only if $M_\alpha (p) = M_

Theorems & Definitions (23)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 3.2
  • proof
  • Claim 3.2.1
  • Proof
  • Proposition 3.3
  • ...and 13 more