Dynamics of the Newton maps of rational functions

Roger W. Barnard, Jerry Dwyer, Erin Williams, G. Brock Williams

Research output: Contribution to journalArticlepeer-review

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 (z-r1)(z-r2)z-p. We prove that these Newton maps are all conjugate to z2, allowing us to completely describe their Julia sets.

Original languageEnglish
JournalJournal of Analysis
DOIs
StateAccepted/In press - 2021

Keywords

  • Complex dynamics
  • Newton map
  • Rational functions

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