Typical dynamics of Newton's method

This article consists of two papers: Typical dynamics of Newton's method by Steele and Erratum to "Typical dynamics of Newton's method" by Dud\'ak 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.