E-resources
Peer reviewed
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Steele, T.H.
Topology and its applications, 08/2022, Volume: 318Journal Article
Let C1(M) be the set 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 τ(x,f) generated by Newton's method using f and x either diverges, or generates a Cantor set as its attractor. Whenever the Cantor is an attractor, the dynamics on the attractor are described by a single type of adding machine, so that the dynamics on all of these attractive Cantor sets are topologically conjugate.
