Abstract:

Newton's root finding algorithm, called Newton's method, applied to entire or meromorphic functions, has been widely studied in complex dynamics. In 1964 Traub introduced another root finding algorithm, somehow related to Newton's method, known as Traub's method. Motivated by the very interesting dynamical properties of Newton's map we consider a generalised family of maps, which includes Traub's map, and we look at some of its dynamical properties when applied to a polynomial $p(z)$. We give some of the basic properties of this family of maps and we focus on the case where $p(z)$ is a polynomial of degree 2. In this case we show that our map is conformally conjugate to a Blaschke product of degree 4.

This is joint work with A. Garijo.