On the Julia set of König's root-finding algorithms (Q2845435)

It is well known that the Julia set of Newton's method applied to complex polynomials is connected. This paper deals with an extension of this problem. More precisely, the author considers König's root-finding algorithms, that is, a uni-parametric family of methods with arbitrarily high order of convergence \(\sigma\geq 2\). These methods include Newton's or Halley's methods as particular cases, for \(\sigma= 2\) and \(\sigma=3\), respectively. The main result of the paper establishes that for all order \(\sigma\geq 3\) there exists a complex polynomial such that the Julia set related to König's method applied to this polynomial is not connected. Consequently, the classical result for Newton's method does not possess a general extension to higher order root-finding methods.