Random Iterations of Subhyperbolic Relaxed Newton's Methods

Abstract

The family of the natural iterations of a subhyperbolic relaxed Newton s method of a complex polynomial devides the Riemann sphere into two disjoint sets: the nonempty, perfect, nowhere dense, connected and locally connected Julia set with Lebesgue measure zero, and the open Fatou set with stable simply connected components. A special kind of perturbations of such families known as random iterations of the relaxed Newton s methods (or Euler s methods) of definite complex polynomials is investigated in this work. We study first the dynamics of a family of random iterations of the relaxed Newton s methods of the polynomials in a small neighborhood of a given fixed polynomial whose relaxed Newton s method is subhyperbolic. We show that the Julia set of such a family is still nonempty, perfect and nowhere dense, and there are no wandering domains among the connected components of its Fatou set.Then we consider families of random iterations of the relaxed Newton s methods of the polynomials convergent to a fixed polynomial with a subhyperbolic relaxed Newton s method. We prove that the Fatou set of these families consists of contracting simply connected components, and their Julia set is a nonempty, perfect, nowhere dense, connected and locally connected set of Lebesgue measure zero.Finally, we give an application of the methods discussed in the previous chapters to the case when the relaxed Newton s method of a given polynomial has at least one parabolic periodic point in its Julia set. Using quasiconformal surgery we show how the Leau domains related to the parabolic periodic point(s) can be approximated by means of families of random iterations of definite subhyperbolic relaxed Newton s methods which are convergent to the relaxed Newton s method of the given polynomial.