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Attracting Orbits in Newton's Method

Mike Hurley
Transactions of the American Mathematical Society
Vol. 297, No. 1 (Sep., 1986), pp. 143-158
DOI: 10.2307/2000461
Stable URL: http://www.jstor.org/stable/2000461
Page Count: 16
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Attracting Orbits in Newton's Method
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Abstract

It is well known that the dynamical system generated by Newton's Method applied to a real polynomial with all of its roots real has no periodic attractors other than the fixed points at the roots of the polynomial. This paper studies the effect on Newton's Method of roots of a polynomial "going complex". More generally, we consider Newton's Method for smooth real-valued functions of the form fμ(x) = g(x) + μ, μ a parameter. If μ0 is a point of discontinuity of the map μ → (the number of roots of fμ), then, in the presence of certain nondegeneracy conditions, we show that there are values of μ near μ0 for which the Newton function of fμ has nontrivial periodic attractors.

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