Superattracting cycles of the relaxed Newton’s method for entire functions

Abstract

In this paper, we study certain dynamical properties of the relaxed Newton’s method for entire functions. We present a procedure for constructing entire functions so that superattracting cycles of any prescribed period occur when the relaxed Newton’s method is applied. Based on the above conclusion, we give a model of any prescribed type cycles for four disjoint subclasses of meromorphic functions, including polynomials, transcendental entire functions, rational functions but not polynomials, and transcendental meromorphic functions but not entire functions.