Dynamics of the Newton maps of rational functions

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 Mobius 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.