2024/02/22 by Jan Dudák, Dudák, Jan, Timothy Steele +1
Decision Sciences · Engineering · Mathematics · #37B20 #Advanced Measurement and Metrology Techniques #Dynamical Systems (math.DS) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Scientific Measurement and Uncertainty Evaluation
paper · pdf · doi:10.48550/arxiv.2402.14383
openalex publication_date 2024/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This article consists of two papers: Typical dynamics of Newton's method by Steele and Erratum to "Typical dynamics of Newton's method" by Dudák and Steele. Let C1(M) be the space of continuously differentiable real-valued functions defined on [-M,M]. We show that for the typical element f in C1(M), there exists a set S ⊆ [-M,M], both residual and of full measure in [-M,M], such that for any x ∈ 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.