Corrigendum to “Typical dynamics of Newton's method” [Topol. Appl. 318 (2022) 108201]

Let C1(M) be the space of continuously differentiable real-valued functions defined on [−M,M]. Here, we address an irremediable flaw found in [4], and 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 tr...

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Bibliographic Details
Published inTopology and its applications Vol. 354; p. 108986
Main Authors Dudák, Jan, Steele, T.H.
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.09.2024
Subjects
Adding machine
Newton's method
Typical behavior
37B20
Adding machine
Typical behavior
Newton's method
26A18
54H25
65P40
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Summary:Let C1(M) be the space of continuously differentiable real-valued functions defined on [−M,M]. Here, we address an irremediable flaw found in [4], and 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.
Bibliography:erratum
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2024.108986