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

Abstract

Let $C^1\left(M\right)$ 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 $C^1\left(M\right)$, 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.