Abstract
Newton’s method is a well-known iterative method to find roots of a function. The related Newton maps have primarily been studied for polynomials, but recently extended to rational and transcendental functions. We describe how a rational function r influences the degree and fixed points of its Newton map R. We then analyze the Julia sets of the Newton maps of Möbius transformations. In doing so, we verify a conjecture of Corte and expand on that result. We also consider Newton maps of rational functions of the form \(\displaystyle \frac{(z-r_1)(z-r_2)}{z-p}\). We prove that these Newton maps are all conjugate to \(z^2\), allowing us to completely describe their Julia sets.
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Roger W. Barnard declares that he has no conflict of interest. Jerry Dwyer declares that he has no conflict of interest. Erin Williams declares that she has no conflict of interest. G. Brock Williams declares that he has no conflict of interest.
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Communicated by Samy Ponnusamy.
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Barnard, R.W., Dwyer, J., Williams, E. et al. Dynamics of the Newton maps of rational functions. J Anal 29, 1055–1070 (2021). https://doi.org/10.1007/s41478-021-00304-x
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DOI: https://doi.org/10.1007/s41478-021-00304-x